Content-Type: multipart/mixed; boundary="-------------0504300644432" This is a multi-part message in MIME format. ---------------0504300644432 Content-Type: text/plain; name="05-154.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-154.keywords" Classical Mechanics, Perturbation theory, KAM ---------------0504300644432 Content-Type: application/x-tex; name="mc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mc.tex" % User-ID: math-ph/0504085 % Password: uj2qh \newcount\mgnf %ingrandimento \mgnf=2 \ifnum\mgnf=0 \magnification=1000 \hsize=15truecm\vsize=20.2truecm%\voffset2.truecm\hoffset.5truecm \parindent=0.3cm\baselineskip=0.45cm\fi \ifnum\mgnf=1 \magnification=\magstephalf \voffset=.5truecm % \hoffset=0.truecm % \hsize=15truecm\vsize=20.2truecm \baselineskip=18truept plus0.1pt minus0.1pt \parindent=0.9truecm % \lineskip=0.5truecm\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \fi %\ifnum\mgnf=2% % \magnification=1200% % \hsize=15truecm\vsize=20.2truecm% % \baselineskip=18truept plus0.1pt minus0.1pt \parindent=0.9truecm% % \lineskip=0.5truecm\lineskiplimit=0.1pt \parskip=0.1pt plus1pt% %\fi \ifnum\mgnf=2\magnification=1200\fi \ifnum\mgnf=0 \def\openone{\leavevmode\hbox{\ninerm 1\kern-3.3pt\tenrm1}}% \def\*{\vglue0.2truecm}\fi \ifnum\mgnf=1 \def\openone{\leavevmode\hbox{\ninerm 1\kern-3.63pt\tenrm1}}% \def\*{\vglue0.3truecm}\fi \ifnum\mgnf=2\def\*{\vglue0.7truecm}\fi \openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI DI FONT %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\titolo=cmbx12\font\titolone=cmbx10 scaled\magstep 2\font \titolino=cmbx10% \font\cs=cmcsc10\font\sc=cmcsc10\font\css=cmcsc8% \font\ss=cmss10\font\sss=cmss8% \font\crs=cmbx8% \font\indbf=cmbx10 scaled\magstep2 \font\type=cmtt10% \font\ottorm=cmr8\font\ninerm=cmr9% \font\msxtw=msbm9 scaled\magstep1% \font\msytw=msbm9 scaled\magstep1% \font\msytww=msbm7 scaled\magstep1% \font\msytwww=msbm5 scaled\magstep1% %\font\msytwwww=msbm4 scaled\magstep1% \font\euftw=eufm9 scaled\magstep1% \font\euftww=eufm7 scaled\magstep1% \font\euftwww=eufm5 scaled\magstep1% \def\st{\scriptstyle}% \def\dt{\displaystyle}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% LETTERE GRECHE E LATINE IN NERETTO %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % lettere greche e latine in neretto italico - pag.430 del manuale \font\tenmib=cmmib10 \font\eightmib=cmmib8 \font\sevenmib=cmmib7\font\fivemib=cmmib5 \font\ottoit=cmti8 \font\fiveit=cmti5\font\sixit=cmti6%% %!!!@@@\font\fiveit=cmti7\font\sixit=cmti7%% \font\fivei=cmmi5\font\sixi=cmmi6\font\ottoi=cmmi8 \font\ottorm=cmr8\font\fiverm=cmr5\font\sixrm=cmr6 \font\ottosy=cmsy8\font\sixsy=cmsy6\font\fivesy=cmsy5%% \font\ottobf=cmbx8\font\sixbf=cmbx6\font\fivebf=cmbx5% \font\ottott=cmtt8% \font\ottocss=cmcsc8% \font\ottosl=cmsl8% \def\ottopunti{\def\rm{\fam0\ottorm}\def\it{\fam6\ottoit}% \def\bf{\fam7\ottobf}% \textfont1=\ottoi\scriptfont1=\sixi\scriptscriptfont1=\fivei% \textfont2=\ottosy\scriptfont2=\sixsy\scriptscriptfont2=\fivesy% %\textfont3=\tenex\scriptfont3=\tenex\scriptscriptfont3=\tenex% \textfont4=\ottocss\scriptfont4=\sc\scriptscriptfont4=\sc% %\scriptfont4=\ottocss\scriptscriptfont4=\ottocss% \textfont5=\eightmib\scriptfont5=\sevenmib\scriptscriptfont5=\fivemib% \textfont6=\ottoit\scriptfont6=\sixit\scriptscriptfont6=\fiveit% \textfont7=\ottobf\scriptfont7=\sixbf\scriptscriptfont7=\fivebf% %\textfont\bffam=\eightmib\scriptfont\bffam=\sevenmib% %\scriptscriptfont\bffam=\fivemib% \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt\rm} \let\nota=\ottopunti% \textfont5=\tenmib\scriptfont5=\sevenmib\scriptscriptfont5=\fivemib \mathchardef\Ba = "050B %alfa \mathchardef\Bb = "050C %beta \mathchardef\Bg = "050D %gamma \mathchardef\Bd = "050E %delta \mathchardef\Be = "0522 %varepsilon \mathchardef\Bee = "050F %epsilon \mathchardef\Bz = "0510 %zeta \mathchardef\Bh = "0511 %eta \mathchardef\Bthh = "0512 %teta \mathchardef\Bth = "0523 %varteta \mathchardef\Bi = "0513 %iota \mathchardef\Bk = "0514 %kappa \mathchardef\Bl = "0515 %lambda \mathchardef\Bm = "0516 %mu \mathchardef\Bn = "0517 %nu \mathchardef\Bx = "0518 %xi \mathchardef\Bom = "0530 %omi \mathchardef\Bp = "0519 %pi \mathchardef\Br = "0525 %ro \mathchardef\Bro = "051A %varrho \mathchardef\Bs = "051B %sigma \mathchardef\Bsi = "0526 %varsigma \mathchardef\Bt = "051C %tau \mathchardef\Bu = "051D %upsilon \mathchardef\Bf = "0527 %phi \mathchardef\Bff = "051E %varphi \mathchardef\Bch = "051F %chi \mathchardef\Bps = "0520 %psi \mathchardef\Bo = "0521 %omega \mathchardef\Bome = "0524 %varomega \mathchardef\BG = "0500 %Gamma \mathchardef\BD = "0501 %Delta \mathchardef\BTh = "0502 %Theta \mathchardef\BL = "0503 %Lambda \mathchardef\BX = "0504 %Xi \mathchardef\BP = "0505 %Pi \mathchardef\BS = "0506 %Sigma \mathchardef\BU = "0507 %Upsilon \mathchardef\BF = "0508 %Fi \mathchardef\BPs = "0509 %Psi \mathchardef\BO = "050A %Omega \mathchardef\BDpr = "0540 %Dpr \mathchardef\Bstl = "053F %* \def\BK{\bf K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% RIFERIMENTI SIMBOLICI A FORMULE, PARAGRAFI E FIGURE %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Ogni paragrafo deve iniziare con il comando \section(#1,#2), dove #1 % e' il simbolo associato al paragrafo e #2 e' il titolo. Per le % appendici bisogna pero' usare \appendix(#1,#2). % % Se nel titolo compaiono riferimenti ad altri simboli, questi vanno % racchiusi fra parentesi graffe, per es. {\equ(1.2)}; in caso contrario % si provoca un errore. % % Ogni sottoparagrafo deve iniziare con il comando \sub(#1) o \asub(#1), % nelle appendici. % % I riferimenti a paragrafi e sottoparagrafi si realizzano con il comando % \sec(#1), che produce il numero effettivo preceduto dal simbolo di % paragrafo, o \secc(#1), che produce solo il numero (serve nel caso si % faccia riferimento ad un sottoparagrafo, che e' un Lemma, un Teorema o % altro oggetto suscettibile di una denominazione speciale). % % Le formule sono contrassegnate con \Eq(#1), eccetto che all'interno % del comando \eqalignno, dove si deve usare \eq(#1). Nelle appendici % i comandi corrispondenti sono \Eqa(#1) e \eqa(#1). % I riferimenti alle formule si realizzano con \equ(#1). % % La numerazione delle figure utilizza il comando \eqg(#1), per % contrassegnarle, e \graf(#1) per citarle. % \global\newcount\numsec\global\newcount\numapp \global\newcount\numfor\global\newcount\numfig \global\newcount\numsub \numsec=0\numapp=0\numfig=0 \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\veraappendice{\number\numapp}\def\verasub{\number\numsub} \def\verafigura{\number\numfig} \def\Section(#1,#2){\advance\numsec by 1\numfor=1\numsub=1\numfig=1% \SIA p,#1,{\veroparagrafo} % \write15{\string\Fp (#1){\secc(#1)}}% \write16{ sec. #1 ==> \secc(#1) }% \0\hbox%to \hsize {\titolo\hfill \number\numsec. #2\hfill% \expandafter{\hglue-1truecm\alato(sec. #1)}}} \def\appendix(#1,#2){\advance\numapp by 1\numfor=1\numsub=1\numfig=1% \SIA p,#1,{A\veraappendice} % \write15{\string\Fp (#1){\secc(#1)}}% \write16{ app. #1 ==> \secc(#1) }% \hbox to \hsize{\titolo Appendix A\number\numapp. #2\hfill% \expandafter{\alato(app. #1)}}\*% } \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1#2 e' gia' stato definito !!!!} \fi} \def \Fe(#1)#2{\SIA fe,#1,#2 } \def \Fp(#1)#2{\SIA fp,#1,#2 } \def \Fg(#1)#2{\SIA fg,#1,#2 } \def\etichetta(#1){(\veroparagrafo.\veraformula)% \SIA e,#1,(\veroparagrafo.\veraformula) % \global\advance\numfor by 1% \write15{\string\Fe (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} \def\etichettaa(#1){(A\veraappendice.\veraformula)% \SIA e,#1,(A\veraappendice.\veraformula) % \global\advance\numfor by 1% \write15{\string\Fe (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} %\def\getichetta(#1){Fig. \verafigura% \def\getichetta(#1){\veroparagrafo.\verafigura% \SIA g,#1,{\veroparagrafo.\verafigura} % \global\advance\numfig by 1% \write15{\string\Fg (#1){\graf(#1)}}% \write16{ Fig. #1 ==> \graf(#1) }} \def\etichettap(#1){\veroparagrafo.\verasub% \SIA p,#1,{\veroparagrafo.\verasub} % \global\advance\numsub by 1% \write15{\string\Fp (#1){\secc(#1)}}% \write16{ par #1 ==> \secc(#1) }} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqg(#1){\getichetta(#1)\alato(fig. #1)} \def\sub(#1){\0\palato(p. #1){\bf \etichettap(#1).}} \def\asub(#1){\0\palato(p. #1){\bf \etichettapa(#1).}} \def\apprif(#1){\senondefinito{e#1}% \eqv(#1)\else\csname e#1\endcsname\fi} \def\equv(#1){\senondefinito{fe#1}$\clubsuit$#1% \write16{eq. #1 non e' (ancora) definita}% \else\csname fe#1\endcsname\fi} \def\grafv(#1){\senondefinito{fg#1}$\clubsuit$#1% \write16{fig. #1 non e' (ancora) definito}% \else\csname fg#1\endcsname\fi} \def\secv(#1){\senondefinito{fp#1}$\clubsuit$#1% \write16{par. #1 non e' (ancora) definito}% \else\csname fp#1\endcsname\fi} \def\eqo{{\global\advance\numfor by 1}} \def\equ(#1){\senondefinito{e#1}\equv(#1)\else\csname e#1\endcsname\fi} \def\graf(#1){\senondefinito{g#1}\grafv(#1)\else\csname g#1\endcsname\fi} \def\figura(#1){{\css Figura} \getichetta(#1)} %\def\fig(#1){\0\veroparagrafo.\getichetta(#1)} \def\secc(#1){\senondefinito{p#1}\secv(#1)\else\csname p#1\endcsname\fi} %\def\sec(#1){{\S\secc(#1)}} \def\sec(#1){{\secc(#1)}} \def\refe(#1){{[\secc(#1)]}} \def\BOZZA{%\bz=1 \def\alato(##1){\rlap{\kern-\hsize\kern-.5truecm{$\scriptstyle##1$}}} \def\palato(##1){\rlap{\kern-.5truecm{$\scriptstyle##1$}}} } \def\alato(#1){} \def\galato(#1){} \def\palato(#1){} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% DATA E PIE' DI PAGINA %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}} \def\oramin{\hourmin } \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\ \oramin} \setbox200\hbox{$\scriptscriptstyle \data $} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% INSERIMENTO FIGURE ( se si usa DVIPS ) %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newdimen\xshift \newdimen\xwidth \newdimen\yshift \newdimen\ywidth \def\ins#1#2#3{\vbox to0pt{\kern-#2\hbox{\kern#1 #3}\vss}\nointerlineskip} \def\eqfig#1#2#3#4#5{ \par\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \yshift=#2 \divide\yshift by 2% %\line {\hglue\xshift \vbox to #2{\vfil #3 \special{psfile=#4.ps} }\hfill\raise\yshift\hbox{#5}}} \def\8{\write12} \def\figini#1{ \catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2 \openout12=#1.ps} \def\figfin{ \closeout12 \catcode`\%=14\catcode`\{=1% \catcode`\}=2\catcode`\<=12\catcode`\>=12} \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13\fi \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \immediate\openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% SIMBOLI VARI %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\theta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi\let\c=\chi \let\ch=\chi \let\ps=\psi \let\y=\upsilon \let\o=\omega\let\si=\varsigma \let\G=\Gamma \let\D=\Delta \let\Th=\Theta\let\L=\Lambda \let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\Y=\Upsilon \def\\{\hfill\break} \let\==\equiv \let\txt=\textstyle\let\dis=\displaystyle \let\io=\infty \def\Dpr{\BDpr\,}%\def\Dpr{\V\dpr\,} \def\aps{{\it a posteriori\ }}\def\ap{{\it a priori\ }} \let\0=\noindent\def\pagina{{\vfill\eject}} \def\bra#1{{\langle#1|}}\def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }}\def\eg{\hbox{\it e.g.\ }} %\def\ie{{i.e. }}\def\eg{{e.g. }} \let\dpr=\partial \def\der{{\rm d}} \let\circa=\cong \def\arccot{{\rm arccot}} \def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}\,} \def\circage{\lower2pt\hbox{$\,\buildrel > \over {\scriptstyle \sim}\,$}} \def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,} \def\fra#1#2{{#1\over#2}} \def\PP{{\cal P}}\def\EE{{\cal E}} \def\MM{{\cal M}} \def\VV{{\cal V}} \def\CC{{\cal C}}\def\FF{{\cal F}} \def\HH{{\cal H}} \def\WW{{\cal W}} \def\TT{{\cal T}}\def\NN{{\cal N}} \def\BB{{\cal B}} \def\II{{\cal I}} \def\RR{{\cal R}}\def\LL{{\cal L}} \def\JJ{{\cal J}} \def\OO{{\cal O}} \def\DD{{\cal D}}\def\AA{{\cal A}} \def\GG{{\cal G}} \def\SS{{\cal S}} \def\KK{{\cal K}}\def\UU{{\cal U}} \def\QQ{{\cal Q}} \def\XX{{\cal X}} \def\T#1{{#1_{\kern-3pt\lower7pt\hbox{$\widetilde{}$}}\kern3pt}} \def\VVV#1{{\underline #1}_{\kern-3pt \lower7pt\hbox{$\widetilde{}$}}\kern3pt\,} \def\W#1{#1_{\kern-3pt\lower7.5pt\hbox{$\widetilde{}$}}\kern2pt\,} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,} \def\lis{\overline}\def\tto{\Rightarrow} \def\etc{{\it etc}} \def\acapo{\hfill\break} \def\mod{{\rm mod}\,} \def\per{{\rm per}\,} \def\sign{{\rm sign}\,} \def\indica{\leaders \hbox to 0.5cm{\hss.\hss}\hfill} \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} \let\ciao=\bye \def\qed{\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} \def\hf#1{{\hat \f_{#1}}} \def\barf#1{{\tilde \f_{#1}}} \def\tg#1{{\tilde g_{#1}}} \def\bq{{\bar q}} \def\Val{{\rm Val}} \def\indic{\hbox{\raise-2pt \hbox{\indbf 1}}} \def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} \def\rrr{\hbox{\msytwww R}} \def\CCC{\hbox{\msytw C}} \def\cccc{\hbox{\msytww C}} \def\ccc{\hbox{\msytwww C}} \def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} \def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} \def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} \def\TTT{\hbox{\msytw T}} \def\tttt{\hbox{\msytww T}} \def\ttt{\hbox{\msytwww T}} \def\QQQ{\hbox{\msytw Q}} \def\qqqq{\hbox{\msytww Q}} \def\qqq{\hbox{\msytwww Q}} \def\vvv{\hbox{\euftw v}} \def\vvvv{\hbox{\euftww v}} \def\vvvvv{\hbox{\euftwww v}}\def\www{\hbox{\euftw w}} \def\wwww{\hbox{\euftww w}} \def\wwwww{\hbox{\euftwww w}} \def\vvr{\hbox{\euftw r}} \def\vvvr{\hbox{\euftww r}} \def\Sqrt#1{{\sqrt{#1}}} %\def\Sqrt#1{{{#1}^{\fra{1}{2}}}} \def\defi{\,{\buildrel def\over=}\,} \def\lhs{{\it l.h.s.}\ } \def\rhs{{\it r.h.s.}\ } \def\cfr{{\it c.f.\ }} \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt% \hbox{\vrule width.#2pt height#1pt \kern#1pt% \vrule width.#2pt}% \hrule height.#2pt}}}} \def\dalam{{\,\mathchoice\sqr66\sqr55\sqr44\sqr33\,}} \def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3} \def\didascalia#1#2{\0Fig.#1{ {\nota\it#2}}} \def\ig{\int} \footline={\rlap{\hbox{\copy200}}\tenrm\hss \number\pageno\hss} \def\V#1{{\bf#1}} \let\wt=\widetilde \def\fk{\f^{(\le k)}} \def\atan{{\,\mathop{\rm atan}\,}} \def\tb{{\th}_0}\def\psb{{\ps}_0} \def\fb{{\f}_0} %\usepackage{eqalignno} \def\fiat{} \vglue1.truecm \centerline{\titolone Classical Mechanics} \*\* \centerline {\it Giovanni Gallavotti} \centerline {\it I.N.F.N. Roma 1, Fisica Roma1} \*\* \Section(1, General principles) \* Classical Mechanics is a theory of point particles motions. If $\V X=(\V x_1,\ldots$, $\V x_n)$ are the particles positions in a Cartesian inertial system of coordinates, the equations of motion are determined by their masses $(m_1,\ldots,m_n)$, $m_j>0$, and by the {\it potential energy} of interaction $V(\V x_1,\ldots,\V x_n)$ as $$m_i {\ddot{\V x}}_i=-\dpr_{\V x_i} V(\V x_1,\ldots,\V x_n),\qquad i=1,\ldots,n\Eq(1.1)$$ % here $\V x_i=(x_{i1},\ldots,x_{id})$ are coordinates of the $i$-th particle and $\dpr_{\V x_i}$ is the gradient $(\dpr_{x_{i1}},\ldots,$ $\dpr_{x_{id}})$; $d$ is the space dimension (\ie $d=3$, usually). The potential energy function will be supposed ``smooth'', \ie {\it analytic} except, possibly, when two positions coincide. The latter exception is necessary to include the important cases of gravitational attraction or, when dealing with electrically charged particles, of Coulomb interaction. A basic result is that if $V$ is bounded below the equation \equ(1.1) admits, given initial data $\V X_0=\V X(0),\V {\dot X}_0=\V{\dot X}(0)$, a unique global solution $t\to\V X(t),\, t\in(-\io,\io)$; otherwise a solution can fail to be global if and only if, in a finite time, it reaches infinity or a singularity point (\ie a configuration in which two or more particles occupy the same point: an event called a {\it collision}). In Eq. \equ(1.1) $-\dpr_{\V x_i} V(\V x_1,\ldots,\V x_n)$ is the {\it force} acting on the points. More general forces are often admitted. For instance velocity dependent {\it friction forces}: they are not considered here because of their phenomenological nature as models for microscopic phenomena which should also, in principle, be explained in terms of conservative forces (furthermore, even from a macroscopic viewpoint, they are rather incomplete models as they should be considered together with the important heat generation phenomena that accompany them). Another interesting example of forces not corresponding to a potential are certain velocity dependent forces like the {\it Coriolis force} (which however appears only in non inertial frames of reference) and the closely related {\it Lorentz force} (in electromagnetism): they could be easily accomodated in the upcoming Hamiltonian formulation of mechanics, see Appendix A2. The {\it action principle} states that an equivalent formulation of the equations \equ(1.1) is that a motion $t\to \V X_0(t)$ satisfying \equ(1.1) during a time interval $[t_1,t_2]$ and leading from $\V X^1=\V X_0(t_1)$ to $\V X^2=\V X_0(t_2)$, renders stationary the {\it action} $$\AA(\{\V X\})= \ig_{t_1}^{t_2} \Big(\sum_{i=1}^n \fra12 \,m_i\,{\dot{\V X}_i}(t)^2 -V(\V X(t))\Big)\,dt\Eq(1.2)$$ % within the class $\MM_{t_1,t_2}{(\V X^1,\V X^2)}$ of smooth (\ie analytic) ``motions'' $t\to\V X(t)$ defined for $t\in[t_1,t_2]$ and leading from $\V X^1$ to $\V X^2$. The function $\LL(\V Y,\V X)=\fra12\sum_{i=1}^n m_i \V y^2_i-V(\V X)\defi K(\V Y)-V(\V X)$, $\V Y=(\V y_1,\ldots,\V y_n)$, is called the {\it Lagrangian} function and the action can be written as $\ig_{t_1}^{t_2}\LL(\dot{\V X}(t),\V X(t))\,dt$. The quantity $K(\dot{\V X}(t))$ is called {\it kinetic energy} and motions satisfying \equ(1.1) conserve {\it energy} as time $t$ varies, \ie $$K(\dot{\V X}(t))+V(\V X(t))=E=const\Eq(1.3)$$ % Hence the action principle can be intuitively thought of as saying that motions proceed by keeping constant the energy, sum of the kinetic and potential energies, while trying to share as evenly as possible their (average over time) contribution to the energy. In the special case in which $V$ is translation invariant motions conserve {\it linear momentum} $\V Q\defi\sum_i m_i \dot{\V x}_i$; if $V$ is rotation invariant around the origin $O$ motions conserve {\it angular momentum} $\V M\defi$ $\sum_i m_i\, \V x_i\wedge\V{\dot x}_i$, where $\wedge$ denotes the vector product in $\RRR^d$, \ie it is the tensor $(\V a\wedge \V b)_{ij}=a_ib_j-b_ia_j,\, i,j=1,\ldots,d$: if the dimension $d=3$ the $\V a\wedge\V b$ will be naturally regarded as a vector. More generally to any continuous symmetry group of the Lagrangian correspond conserved quantities: this is formalized in the {\it Noether theorem}. It is convenient to think that the scalar product in $\RRR^{dn}$ is defined in terms of the ordinary scalar product in $\RRR^d$, $\V a\cdot\V b=\sum_{j=1}^d a_j b_j$, by $(\V v,\V w)=\sum_{i=1}^n m_i \V v_i\cdot\V w_i$: so that kinetic energy and line element $ds$ can be written as $K(\V{\dot X})=\fra12 (\V{\dot X},\V{\dot X})$ and, respectively, $ds^2=\sum_{i=1}^n m_i d\V x_i^2$. Therefore the metric generated by the latter scalar product can be called {\it kinetic energy metric}. The interest of the kinetic metric appears from the {\it Maupertuis' principle} ({\it equivalent} to \equ(1.1)): the principle allows us to identify the trajectory traced in $\RRR^d$ by a motion that leads from $\V X^1$ to $\V X^2$ moving with energy $E$. Parameterizing such trajectories as $\t\to \V X(\t)$ by a parameter $\t$ varying in $[0,1]$ so that the line element is $ds^2=(\V{\dpr_\t X},\V{\dpr_\t X})d\t^2$, the principle states that the trajectory of a motion with energy $E$ which leads from $\V X^1$ to $\V X^2$ makes stationary, among the analytic curves $\Bx\in\MM_{0,1}(\V X^1,\V X^2)$, the function $$L(\Bx)=\ig_{\Bx} \sqrt{E-V(\Bx(s))}\,ds\Eq(1.4)$$ % so that the possible trajectories traced by the solutions of \equ(1.1) in $\RRR^{nd}$ and with energy $E$ can be identified with the {\it geodesics} of the metric $d m^2\defi (E-V(\V X))\cdot d s^2$. \* \0{\it References:} [LL68], [Ga83] \* \Section(2,Constraints) \* Often particles are subject to {\it constraints} which force the motion to take place on a surface $M\subset \RRR^{nd}$: \ie $\V X(t)$ is forced to be a point on the manifold $M$. A typical example is provided by {\it rigid systems} in which motions are subject to forces which keep the mutual distances of the particles constant: $|\V x_i-\V x_j|=\r_{ij}$ with $\r_{ij}$ time independent positive quantities. In essentially all cases the forces that imply constraints, called {\it constraints reactions}, are velocity dependent and, therefore, {\it are not} in the class of {\it conservative} forces considered here, cf. \equ(1.1). Hence, from a fundamental viewpoint admitting only conservative forces, constrained systems should be regarded as idealizations of systems subject to conservative forces which {\it approximately} imply the constraints. In general the $\ell$-dimensional manifold $M$ will not admit a global system of coordinates: however it will be possible to describe points in the vicinity of any $\V X^0\in M$ by using $N=nd$ coordinates $\V q=(q_1,\ldots,q_\ell, q_{\ell+1},\ldots,q_{N})$ varying in an open ball $B_{\V X^0}$: $\V X=\V X(q_1,\ldots,q_\ell, q_{\ell+1},$ $\ldots,q_{N})$. The $q$-coordinates can be chosen {\it well adapted} to the surface $M$ and to the kinetic metric, \ie so that the points of $M$ are identified by $q_{\ell+1}=\ldots=q_N=0$ (which is the meaning of ``adapted'') and furthermore infinitesimal displacements $(0,\ldots,0,d\e_{\ell+1},\ldots,d\e_N)$ out of a point $\V X^0\in M$ are orthogonal to $M$ (in the kinetic metric) and have a length independent of the position of $\V X^0$ on $M$ (which is the meaning of `well adapted'' to the kinetic metric). Motions constrained on $M$ arise when the potential $V$ has the form $$V(\V X)= V_a(\V X)+\l\,W(\V X)\Eq(2.1)$$ % where $W$ is a smooth function which reaches its minimum value, say equal to $0$, precisely on the manifold $M$ while $V_a$ is another smooth potential. The factor $\l>0$ is a parameter called the {\it rigidity} of the constraint. A particularly interesting case arises when, furthermore, the level surfaces of $W$ have the geometric property of being ``parallel'' to the surface $M$: in the precise sense that the matrix $\dpr^2_{q_iq_j}W(\V X), \, i,j>\ell$ is positive definite and $\V X$-independent for all $\V X\in M$ in a system of coordinates well adapted to the kinetic metric. A potential $W$ with the latter properties can be called an {\it approximately ideal constraint reaction}. In fact it can be proved that, given an initial datum $\V X^0\in M$ with velocity $\V{\dot X}^0$ tangent to $M$, \ie given an initial datum whose coordinates in a local system of coordinates are $(\V q_0,\V 0)$ and $(\dot{\V q}_0,\V0)$ with $\V q_0=(q_{01},\ldots,q_{0\ell})$ and $\dot{\V q}_0=(\dot q_{01},\ldots,\dot q_{0\ell})$, the motion generated by \equ(1.1) with $V$ given by \equ(2.1) is a motion $t\to \V X_\l(t)$ which \* \0(1) as $\l\to\io$ tends to a motion $t\to \V X_\io(t)$, \0(2) as long as $\V X_\io(t)$ stays in the vicinity of the initial data, say for $0\le t\le t_1$, so that it can be described in the above local adapted coordinates, its coordinates have the form $t\to(\V q(t),\V 0)=(q_{1}(t),\ldots, q_\ell(t),0,\ldots,0)$: \ie it is a motion developing on the constraint surface $M$, \0(3) the curve $t\to \V X_\io(t),\, t\in[0,t_1]$, as an element of the space $\MM_{0,t_1}{(\V X^0,\V X_{\io}(t_1))}$ of analytic curves on $M$ connecting $\V X^0$ to $\V X_{\io}(t_1)$, renders the action $$A(\V X)=\ig_{0}^{t_1} \big(K(\dot{\V X}(t))-V_a(\V X(t))\big)\,dt \Eq(2.2)$$ % stationary. \* The latter property can be formulated ``intrinsically'', \ie referring only to $M$ as a surface, via the restriction of the metric $d s^2$ to line elements $d \V s=(d q_1,\ldots,d q_\ell,0,\ldots,0)$ tangent to $M$ at the point $\V X=(\V q_0,0,\ldots,0)\in M$; we write $d s^2=\sum_{i,j}^{1,\ell} g_{ij}(\V q) dq_i\,dq_j$. The $\ell\times\ell$ symmetric positive definite matrix $g$ is called the {\it metric} on $M$ induced by the kinetic metric. Then the action in \equ(2.2) can be written as $$\AA(\V q)=\ig_0^{t_1}\Big(\fra12 \sum_{i,j}^{1,\ell} g_{ij}(\V q(t)) \dot q_i(t)\dot q_j(t)-\lis V_a(\V q(t))\Big)\,dt\Eq(2.3)$$ % where $\lis V_a(\V q)\defi V_a(\V X(q_1,\ldots,q_\ell,0,\ldots,0))$: the function $$\LL(\V \Bh,\V q)\defi\fra12 \sum_{i,j}^{1,\ell} g_{ij}(\V q)\h_i\h_j-\lis V_a(\V q)\=\fra12 g(\V q)\Bh\cdot\Bh- \lis V_a(\V q)\Eq(2.4)$$ % is called the {\it constrained Lagrangian} of the system. An important property is that the constrained motions conserve the energy defined as $E=\fra12 (g(\V q)\dot{\V q},\dot{\V q})+\lis V_a(\V q)$, see Sec. \sec(3). The constrained motion $\V X_\io(t)$ of energy $E$ satisfies the Maupertuis' principle in the sense that the curve on $M$ on which the motion develops renders $$L(\Bx)=\ig_\Bx \sqrt{E- V_a(\Bx(s))} \,d s\Eq(2.5)$$ % stationary among the (smooth) curves that develop on $M$ connecting two fixed values $\V X_1$ and $\V X_2$. In the particular case in which $\ell=n$ this is again Maupertuis' principle for unconstrained motions under the potential $V(\V X)$. In general $\ell$ is called the {\it number of degrees of freedom} because a complete description of the initial data requires $2\ell$ coordinates $\V q(0),\dot{\V q}(0)$. If $W$ is minimal on $M$ but the condition on $W$ of having level surfaces parallel to $M$ is not satisfied, \ie if $W$ is not an approximate ideal constraint reaction, it remains still true that the limit motion $\V X_\io(t)$ takes place on $M$. However, {\it in general}, it will not satisfy the above variational principles. For this reason motions arising as limits as $\l\to\io$ of motions developing under the potential \equ(2.1) with $W$ having minimum on $M$ and level curves parallel (in the above sense) to $M$ are called {\it ideally constrained motions} or motions subject by ideal constraints to the surface $M$. As an example suppose that $W$ has the form $W(\V X)=\sum_{i, j\in \PP} w_{ij}(|\V x_i-\V x_j|)$ with $w_{ij}(|\Bx|)\ge0$ an analytic function vanishing only when $|\Bx|=\r_{ij}$ for $i,j$ in some set of pairs $\PP$ and for some given distances $\r_{ij}$ (for instance $w_{ij}(\Bx)=(\Bx^2-\r_{ij}^2)^2\,\g$, $\g>0$). Therefore {\it the so constrained motions $\V X_\io(t)$ of the body satisfy the variational principles} mentioned in connection with \equ(2.3) and \equ(2.5): in other words the above natural way of realizing a rather general rigidity constraint is ideal. \* This is the modern viewpoint on the physical meaning of the constraints reactions: looking at motions in an inertial cartesian system it will appear that the system is subject to the {\it applied forces} with potential $V_a(\V X)$ and to {\it constraint forces} which are {\it defined} as the differences $\V R_i=m_i\ddot{\V x}_i+\Dpr_{\V x_i} V_a(\V X)$. The latter reflect the action of the forces with potential $\l W(\V X)$ in the limit of infinite rigidity ($\l\to\io$). In applications {\it sometimes} the action of a constraint can be regarded as ideal: so that the motion will verify the mentioned variational principles and the $\V R$ can be computed as the differences between the $m_i\ddot{\V x}_i$ and the active forces $-\Dpr_{\V x_i} V_a(\V X)$. In dynamics problems it is, {\it however}, a very difficult and important matter, particularly in engineering, to judge whether a system of particles can be considered as subject to ideal constraints: this leads to important decisions in the construction of machines. It simplifies the calculations of the efforts of the materials but a misjudgement can have serious consequences about stability and safety. For statics problems the difficulty is of lower order: usually assuming that the constraints reaction is ideal leads to an overestimate of the requirements for stability of equilibria. Hence employing the action principle to statics problems, where it constitutes the {\it virtual works principle}, generally leads to economic problems rather than to safety issues. Its discovery even predates Newtonian mechanics. \* \0{\it References:} [Ar68], [Ga83]. \* \Section(3,Lagrange and Hamilton form of the equations of motion) \* The stationarity condition for the action $\AA(\V q)$, cf. \equ(2.3),\equ(2.4), is formulated in terms of the Lagrangian $\LL(\Bh,\Bx)$, see \equ(2.4), by $$\fra{d}{dt}\dpr_{\h_i} \LL(\dot {\V q}(t),{\V q}(t)) = \dpr_{q_i} \LL(\dot {\V q}(t),{\V q}(t)), \qquad i=1,\ldots,\ell\Eq(3.1)$$ % which is a second order differential equation called {\it Lagrangian equation of motion}. It can be cast in ``normal form'': for this purpose, adopting the convention of ``summation over repeated indices'', introduce the ``generalized momenta'' $$p_i\defi g(\V q)_{ij} \dot q_j,\qquad i=1,\ldots,\ell\Eq(3.2)$$ % Since $g(\V q)>0$ the motions $t\to\V q(t)$ and the corresponding velocities $t\to\dot{\V q}(t)$ can be described equivalently by $t\to(\V q(t),\V p(t))$: and the equations of motion \equ(3.1) become the first order equations $$\dot{q}_i=\dpr_{p_i} \HH(\V p,\V q),\qquad \dot{p}_i=-\dpr_{q_i} H(\V p,\V q)\Eq(3.3)$$ % where the function $\HH$, called {\it Hamiltonian} of the system, is defined by $$\HH(\V p,\V q)\defi \fra12 (g(\V q)^{-1}\V p,\V p)+ \lis V_a(\V q)\Eq(3.4)$$ % Eq. \equ(3.3), regarded as equations of motion for phase space points $(\V p,\V q)$, are called {\it Hamilton equations}. In general $\V q$ are local coordinates on $M$ and motions are specified by giving $\V q,\dot{\V q}$ or $\V p,\V q$. Looking for a coordinate free representation of motions consider the pairs $\V X,\V Y$ with $\V X \in M$ and $\V Y$ a vector $\V Y\in T_{\V X}$ tangent to $M$ at the point $\V X$. The collection of pairs $(\V Y,\V X)$ is denoted $T(M)=\cup_{\V X\in M}\big(T_{\V X}\times \{\V X\}\big)$ and a motion $t\to (\dot{\V X}(t),\V X(t))\in T(M)$ in local coordinates is represented by $(\dot{\V q}(t),\V q(t))$. The space $T(M)$ can be called the space of initial data for Lagrange's equations of motion: it has $2\ell$ dimensions (more fancily known as the {\it tangent bundle} of $M$). Likewise the space of initial data for the Hamilton equations will be denoted $T^*(M)$ and it consists in pairs $\V X,\V P$ with $\V X\in M$ and $\V P= g(\V X) \V Y$ with $\V Y$ a vector tangent to $M$ at $\V X$. The space $T^*(M)$ is called the {\it phase space} of the system: it has $2\ell$ dimensions (and it is occasionally called the {\it cotangent bundle} of $M$). Immediate consequence of \equ(3.3) is $\fra{d}{dt} \HH(\V p(t),\V q(t))\=0$ and it means that $\HH(\V p(t),\V q(t))$ is constant along the solutions of the \equ(3.3). Remarking that $\HH(\V p,\V q)=\fra12 (g(\V q)\dot{\V q},\dot{\V q})+\lis V_a(\V q)$ is the sum of the kinetic and potential energies it follows that the conservation of $\HH$ along solutions has the meaning of {\it energy conservation} in presence of ideal constraints. Let $S_t$ be the {\it flow} generated on the phase space variables $(\V p,\V q)$ by the solutions of the equations of motion \equ(3.3), \ie let $t\to S_t(\V p,\V q)\=(\V p(t),\V q(t))$ denote a solution of \equ(3.3) with initial data $(\V p,\V q)$. Then a (measurable) set $\D$ in phase space evolves in time $t$ into a new set $S_t\D$ {\it with the same volume}: this is obvious because the Hamilton equations \equ(3.3) have manifestly $0$ divergence (``{\it Liouville's theorem}''). The Hamilton equations also satisfy a variational principle, called {\it Hamilton action principle}: \ie if $\MM_{t_1,t_2}{((\V p_1,\V q_1),(\V p_2,\V q_2);M)}$ denotes the space of the analytic functions $\Bf:\,t\to (\Bp(t),\Bk(t))$ which in the time interval $[t_1,t_2]$ lead from $(\V p_1,\V q_1)$ to $(\V p_2,\V q_2)$, then the condition that $\Bf_0(t)=(\V p(t),\V q(t))$ satisfies \equ(3.3) can be equivalently formulated by requiring that the function $$\AA_\HH(\Bf)\defi\ig_{t_1}^{t_2} \Big(\Bp(t)\cdot\dot\Bk(t)- \HH(\Bp(t),\Bk(t))\Big)\,dt \Eq(3.5)$$ % be stationary for $\Bf=\Bf_0$: in fact the \equ(3.3) are the stationarity conditions for the Hamilton action \equ(3.5) on $\MM_{t_0,t_1} {((\V p_1,\V q_1),(\V p_2,\V q_2);M)}$. And since the derivatives of $\Bp(t)$ do not appear in \equ(3.5) stationarity is even achieved in the larger space $\MM_{t_1,t_2}{(\V q_1,\V q_2;M)}$ of the motions $\Bf:\,t\to (\Bp(t),\Bk(t))$ leading from $\V q_1$ to $\V q_2$ {\it without} any restriction on the initial and final momenta $\V p_1,\V p_2$ (which therefore cannot be prescribed \ap independently of $\V q_1,\V q_2$). If the prescribed data $\V p_1,\V q_1,\V p_2,\V q_2$ are not compatible with the equations of motion, \eg $H(\V p_1,\V q_2)\ne H(\V p_2,\V q_2)$, then the action functional has no stationary trajectory in $\MM_{t_1,t_2}{((\V p_1,\V q_1),(\V p_2\V q_2);M)}$. \* \0{\it References:} [LL68], [Ar68], [Ga83] \* \Section(4, Canonical transformations of phase space coordinates) \* The Hamiltonian form, \equ(3.4), of the equations of motion turns out to be quite useful in several problems. It is therefore important to remark that it is {\it invariant} under a special class of transformations of coordinates, called {\it canonical transformations}. Consider a local change of coordinates on phase space, \ie a smooth smoothly invertible map $\CC(\Bp,\Bk)=(\Bp',\Bk')$ between an open set $U$ in the phase space of a $\ell$ degrees of freedom Hamiltonian system, into an open set $U'$ in a $2\ell$ dimensional space. The change of coordinates is said {\it canonical} if for any solution $t\to(\Bp(t),\Bk(t))$ of equations like \equ(3.3), for any Hamiltonian $\HH(\Bp,\Bk)$ defined on $U$, the $\CC$--image $t\to (\Bp'(t),\Bk'(t))=\CC(\Bp(t),\Bk(t))$ are solutions of the \equ(3.3) with the ``same'' Hamiltonian, \ie with Hamiltonian $\HH'(\Bp',\Bk')\defi \HH(\CC^{-1}(\Bp',\Bk'))$. The condition that a transformation of coordinates is canonical is obtained by using the arbitrariness of the function $\HH$ and is simply expressed as a necessary and sufficient property of the Jacobian $L$ $$\txt L=\pmatrix{A&B\cr C&D\cr},\quad A_{ij}=\dpr_{\p_j} \p'_i, \ B_{ij}=\dpr_{\k_j}\p'_i, \ C_{ij}=\dpr_{\p_j} \k'_i, \ D_{ij}=\dpr_{\k_j}\k'_i, \Eq(4.1)$$ % where $i,j=1,\ldots,\ell$. Let $E=\pmatrix{0&1\cr-1&0}$ denote the $2\ell\times2\ell$ matrix formed by four $\ell\times\ell$ blocks, equal to the $0$ matrix or, as indicated, to the $\pm$ identity matrix; then, if a superscript $T$ denotes matrix transposition, the condition that the map be canonical is that $$L^{-1}=E L^T E^T \qquad{\rm or}\quad L^{-1}=\pmatrix{D^T&-B^T\cr -C^T& A^T\cr}\Eq(4.2)$$ % which immediately implies that $\det L=\pm1$. In fact it is possible to show that \equ(4.2) implies $\det L=1$. The \equ(4.2) is equivalent to the four relations $AD^T-B C^T=1$, $-A B^T+B A^T=0$, $CD^T-D C^T=0$, $-C B^T+D A^T=1$. More explicitly, since the first and the fourth coincide, the relations are $$\big\{\p'_i,\k'_j\big\}=\d_{ij},\quad \big\{\p'_i,\p'_j\big\}=0,\quad \big\{\k'_i,\k'_j\big\}=0\Eq(4.3) $$ % where, for any two functions $F(\Bp,\Bk),G(\Bp,\Bk)$, the {\it Poisson bracket} is $$\big\{F,G\big\}(\Bp,\Bk)\defi \sum_{k=1}^\ell \Big(\dpr_{\p_k} F(\Bp,\Bk)\, \dpr_{\k_k} G(\Bp,\Bk)-\dpr_{\k_k} F(\Bp,\Bk)\, \dpr_{\p_k} G(\Bp,\Bk)\Big)\Eq(4.4)$$ % The latter satisfies {\it Jacobi's identity}: $\big\{\big\{F,G\big\},Q\big\}+ \big\{\big\{G,Q\big\},F\big\}+\big\{\big\{Q,F\big\},G\big\}$ $=0$, for any three functions $F,G,Q$ on phase space. It is quite useful to remark that if $t\to (\V p(t),\V q(t))=S_t(\V p,\V q)$ is a solution to Hamilton equations with Hamiltonian $\HH$ then, given any {\it observable} $F(\V p,\V q)$, it ``evolves'' as $F(t)\defi F(\V p(t),\V q(t))$ satisfying $\dpr_t F(\V p(t),\V q(t))=\,\big\{\HH,F\big\}(\V p(t),\V q(t))$. Requiring the latter identity to hold for all observables $F$ is {\it equivalent} to requiring that the $t\to(\V p(t),\V q(t))$ be a solution of Hamilton's equations for $\HH$. Let $\CC: U\otto U'$ be a smooth, smoothly invertible, transformation between two $2\ell$ dimensional sets: $\CC(\Bp,\Bk)=(\Bp',\Bk')$. Suppose that there is a function $\F(\Bp',\Bk)$ defined on a suitable domain $W$ and such that $$\CC(\Bp,\Bk)=(\Bp',\Bk')\ \tto\ \cases{\Bp=\dpr_{\Bk}\F(\Bp',\Bk)&\cr \Bk'=\dpr_{\Bp'}\F(\Bp',\Bk)&\cr}\Eq(4.5)$$ % then $\CC$ is canonical. This is because \equ(4.5) implies that if $\Bk,\Bp'$ are varied and if $\Bp,\Bk',\Bp',\Bk$ are related by $\CC(\Bp,\Bk)=(\Bp',\Bk')$, then $\Bp\cdot d\Bk + \Bk'\cdot d\Bp'= d\F(\Bp',\Bk)$: which implies $$\Bp\cdot d\Bk-\HH(\Bp,\Bk)dt\= \Bp'\cdot d\Bk'-\HH(\CC^{-1}(\Bp',\Bk'))dt +d \F(\Bp',\Bk)-d(\Bp'\cdot\Bk')\Eq(4.6)$$ % and means that the Hamiltonians $\HH(\V p,\V q))$ and $\HH'(\V p',\V q'))\defi \HH(\CC^{-1}(\V p',\V q'))$ have Hamilton actions $\AA_\HH$ and $\AA_{\HH'}$ differing by a constant, if evaluated on corresponding motions $(\V p(t),\V q(t))$ and $(\V p'(t),\V q'(t))=\CC(\V p(t),\V q(t))$. The constant depends only on the initial and final values $(\V p(t_1),\V q(t_1))$ and $(\V p(t_2),$ $\V q(t_2))$ and, respectively, $(\V p'(t_1),\V q'(t_1))$ and $(\V p'(t_2),\V q'(t_2))$ so that if $(\V p(t)$, $\V q(t))$ makes $\AA_\HH$ extreme also $(\V p'(t),\V q'(t))=\CC(\V p(t),\V q(t))$ makes $\AA_{\HH'}$ extreme. Hence if $t\to (\V p(t),\V q(t))$ solves the Hamilton equations with Hamiltonian $\HH(\V p,\V q)$ then the motion $t\to (\V p'(t),\V q'(t))=\CC(\V p(t),\V q(t))$ solves the Hamilton equations with Hamiltonian $\HH'(\V p',\V q')= \HH(\CC^{-1} (\V p',\V q'))$ {\it no matter which is the function $\HH$}: therefore the transformation is canonical. The function $\F$ is called its {\it generating function}. Eq. \equ(4.5) provides a way to construct canonical maps. Suppose that a function $\F(\Bp',\Bk)$ is given and defined on some domain $W$; then setting $\cases{\Bp=\dpr_{\Bk}\F(\Bp',\Bk)&\cr \Bk'=\dpr_{\Bp'}\F(\Bp',\Bk)&\cr}$ and inverting the first equation in the form $\Bp'=\BX(\Bp,\Bk)$ and substituting the value for $\Bp'$, thus obtained, in the second equation, a map $\CC(\Bp,\Bk)=(\Bp',\Bk')$ is defined on some domain (where the mentioned operations can be performed) and if such domain is open and not empty then $\CC$ is a canonical map. For similar reasons if $\G(\Bk,\Bk')$ is a function defined on some domain then setting $\Bp=\dpr_\Bk \G(\Bk,\Bk')$, $\Bp'=-\dpr_{\Bk'} \G(\Bk,\Bk')$ and solving the first relation to express $\Bk'=\BD(\Bp,\Bk)$ and substituting in the second relation a map $(\Bp',\Bk')=\CC(\Bp,\Bk)$ is defined on some domain (where the mentioned operations can be performed) and if such domain is open and not empty then $\CC$ is a canonical map. Likewise canonical transformations can be constructed starting from \ap given functions $F(\Bp,\Bk')$ or $G(\Bp,\Bp')$. And the most general canonical map can be generated locally (\ie near a given point in phase space) by a single one of the above four ways, possibly composed with a few ``trivial'' canonical maps in which one pair of coordinates $(\p_i,\k_i)$ is transformed into $(-\k_i,\p_i)$. The necessity of including also the trivial maps can be traced to the existence of {\it homogeneous} canonical maps, \ie maps such that $\Bp\cdot d\Bk=\Bp'\cdot d\Bk'$ (\eg the identity map, see below or \equ(9.7) for nontrivial examples) which are action preserving hence canonical, but which evidently cannot be generated by a function $\F(\Bk,\Bk')$ although they can be generated by a function depending on $\Bp',\Bk$. Simple examples of homogeneous canonical maps are maps in which the coordinates $\V q$ are changed into $\V q'=\V R(\V q)$ and, correspondingly, the $\V p$'s are transformed as $\V p'= \big(\dpr_{\V q} \V R(\V q)\big)^{-1\,{T}}\V p$, linearly: indeed this map is generated by the function $F(\V p',\V q)\defi \V p'\cdot\V R(\V q)$. For instance consider the map ``Cartesian--polar'' coordinates $(q_1,q_2)\otto$ $(\r,\th)$ with $(\r,\th)$ the polar coordinates of $\V q$ (namely $\r=\sqrt{q_1^2+q_2^2},\th=\atan q_2/q_1$) and let $\V n\defi {\V q}/\,|\V q|$ $=(n_1,n_2)$ and $\V t=(-n_2,n_1)$. Setting $p_\r\defi\V p\cdot\V n,\,p_\th\defi \r\,\V p\cdot\V t$, the map $(p_1,p_2,q_1,q_2)\otto $ $(p_\r,p_\th,\r,\th)$ is homogeneous canonical (because $\V p\cdot d\V q=\V p\cdot \V n \,d\r+\V p\cdot\V t \,\r\, d\th=p_\r\, d\r+p_\th\, d\th$). As a further example any area preserving map $(p,q)\otto(p',q')$ defined on an open region of the plane $\RRR^2$ is canonical: because in this case the matrices $A,B,C,D$ are just numbers which satisfy $AD-BC=1$ and, therefore, the \equ(4.2) holds. \* \0{\it References:} [LL68], [Ga83] \* \Section(5,Quadratures) \* The simplest mechanical systems are {\it integrable by quadratures}. For instance the Hamiltonian on $\RRR^2$ $$\HH(p,q)=\fra1{2m} p^2+ V(q)\Eq(5.1)$$ % generates a motion $t\to q(t)$ with initial data $q_0,\dot q_0$ such that $\HH(p_0,q_0)=E$, \ie $\fra12m \dot q_0^2+V(q_0)=E$, satisfying $\dot q(t)=\pm\sqrt{\fra2m(E-V(q(t)))}$. If the equation $E=V(q)$ has only two solutions $q_-(E)0$ the motion is periodic with period $$\txt T(E)=2\ig_{q_-(E)}^{q_+(E)} \fra{dx}{\sqrt{\fra2m (E-V(x))}}.\Eq(5.2)$$ % The special solution with initial data $q_0=q_-(E),\dot q_0=0$ will be denoted $Q(t)$ and it is an analytic function (by the general regularity theorem on ordinary differential equations). For $0\le t\le \fra{T}2$ or for $\fra{T}2\le t\le T$ it is given, respectively, by $$\txt t=\ig_{q_-(E)}^{Q(t)} \fra{dx}{\sqrt{\fra2m(E-V(x))}},\qquad{\rm or}\qquad t=\fra{T}2-\ig^{Q(t)}_{q_+(E)} \fra{dx}{\sqrt{\fra2m(E-V(x))}}.\Eq(5.3) $$ % The most general solution with energy $E$ has the form $q(t)=Q(t_0+t)$ where $t_0$ is defined by $q_0=Q(t_0), \dot q_0=\dot Q(t_0)$, \ie it is the time needed to the ``standard solution'' $Q(t)$ to reach the initial data for the new motion. If the derivative of $V$ vanishes in one of the extremes or if one at least of the two solutions $q_\pm(E)$ does not exist the motion is not periodic and it may be unbounded: nevertheless it is still expressible via integrals of the type \equ(5.2). If the potential $V$ is periodic in $q$ and the variable $q$ is considered varying on a circle then essentially {\it all solutions are periodic}: exceptions can occurr if the energy $E$ has a value such that $V(q)=E$ admits a solution where $V$ has zero derivative, Typical examples are the {\it harmonic oscillator}, the {\it pendulum}, the {\it Kepler oscillator}: whose Hamiltonians, if $m,\o,g,h,G,k$ are positive constants, are respectively $$\fra{p^2}{2m}+\fra12 m \o^2 q^2,\qquad \fra{p^2}{2m}+m g(1-\cos \fra{q}h),\qquad \fra{p^2}{2m}-m k \fra1{|q|}+ m \fra{ G^2}{2q^2}\Eq(5.4)$$ % the latter has a potential which is singular at $q=0$ but if $G\ne0$ the energy conservation forbids too close an approach to $q=0$ and the singularity becomes irrelevant. The integral in \equ(5.3) is called a {\it quadrature} and the systems in \equ(5.1) are therefore {\it integrable by quadratures}. Such systems, at least in the cases in which the motion is periodic, are best described in new coordinates in which periodicity is more manifest. Namely when $V(q)=E$ has only two roots $q_\pm(E)$ and $\mp V'(q_\pm(E))>0$ the {\it energy--time} coordinates can be used by replacing $q,\dot q$ or $p,q$ by $E,\t$ where $\t$ is the time needed to the standard solution $t\to Q(t)$ to reach the given data, \ie $Q(\t)=q,\dot Q(\t)=\dot q$. In such coordinates the motion is simply $(E,\t)\to(E,\t+t)$ and, of course, the variable $\t$ has to be regarded as varying on a circle of radius $T/2\p$. The $E,\t$ variables are a kind of polar coordinates as can be checked by drawing the curves of constant $E$, ``energy levels'', in the plane $p,q$ in the cases in \equ(5.4), see Fig. 1. In the harmonic oscillator case all trajectories are periodic. In the pendulum case all motions are periodic except the ones which separate the oscillatory motions (the closed curves in the second drawing) from the rotatory motions (the apparently open curves) which, in fact, are on closed curves as well if the $q$ coordinate, \ie the vertical coordinate in Fig. 1, is regarded as ``periodic'' with period $2\p h$. In the Kepler case only the negative energy trajectories are periodic and a few of them are drawn in Fig. 1. The single dots represent the equilibrium points in phase space. \eqfig{250pt}{105pt} {}{fig1}{(1)} \0Fig. 1: {\nota The energy levels of the harmonic oscillator, the pendulum, and the Kepler motion} \0The region of phase space where motions are periodic is a set of points $(p,q)$ with the topological structure of $\cup_{u\in U}(\{u\}\times C_u)$ where $u$ is a coordinate varying in an open interval $U$, for instance the set of values of the energy, and $C_u$ is a closed curve whose points $(p,q)$ are identified by a coordinate, for instance by the time necessary to an arbitrarily fixed datum with the same energy to evolve into $(p,q)$. In the above cases, \equ(5.4), if the ``radial'' coordinate is chosen to be the energy the set $U$ is the interval $(0,+\io)$ for the harmonic oscillator, $(0,2 mg)$ or $(2mg,+\io)$ for the pendulum, and $(-\fra12 \fra{mk^2 }{G^2},0)$ in the Kepler case. The fixed datum for the reference motion can be taken, in all cases, of the form $(0,q_0)$ with the time coordinate $t_0$ given by \equ(5.3). It is remarkable that the energy-time coordinates are canonical coordinates: for instance in the vicinity of $(p_0,q_0)$ and if $p_0>0$ this can be seen by setting $$S(q,E)=\ig_{q_0}^q \sqrt{2 m(E-V(x))}\,dx\Eq(5.5)$$ % and checking that $p=\dpr_q S(q,E), \,t=\dpr_E S(q,E)$ are identities if $(p,q)$ and $(E,t)$ are coordinates for the same point so that the criterion expressed by \equ(4.6) applies. It is convenient to standardize the coordinates by replacing the time variable by an angle$\a=\fra{2\p}{T(E)}t$; and instead of the energy any invertible function of it can be used. It is natural to look for a coordinate $A=A(E)$ such that the map $(p,q)\otto(A,\a)$ is a canonical map: this is easily done as the function $$\hat S(q,A)=\ig_{q_0}^q \sqrt{2 m(E(A)-V(x))}\,dx\Eq(5.6)$$ % generates (locally) the correspondence between $p=\sqrt{2 m(E(A)-V(q))}$ and $\a= E'(A) \ig_0^q$ $ \fra{dx}{\sqrt{2 m^{-1}(E(A)-V(x))}}$. Therefore, by the criterion \equ(4.6), if $E'(A)=\fra{2\p}{T(E(A))}$, \ie if $A'(E)=\fra{T(E)}{2\p}$, the coordinates $(A,\a)$ will be canonical coordinates. Hence, by \equ(5.2), $A(E)$ can be taken equal to $$A=\fra1{2\p} 2\ig_{q_-(E)}^{q_+(E)} \sqrt{2 m(E-V(q))}\,dq\=\fra1{2\p}\oint pdq\Eq(5.7)$$ % where the last integral is extended to the closed curve of energy $E$, see Fig. 1. The {\it action--angle coordinates} $(A,\a)$ are defined in open regions of phase space covered by periodic motions: in action--angle coordinates such regions have the form $W=J\times \TTT$ of a product of an open interval $J$ and a one dimensional ``torus'' $\TTT=[0,2\p]$ (\ie a unit circle). \* \0{\it References:} [LL68], [Ar68], [Ga83]. \* \Section(6,Quasi periodicity and integrability) \* {\it A Hamiltonian is called {\it integrable} in an open region $W\subset T^*(M)$ of phase space if \0(1) there is an analytic and non singular (\ie with non zero Jacobian) change of coordinates $(\V p,\V q)\otto(\V I,\Bf)$ mapping $W$ into a set of the form $\II\times\TTT^\ell$ with $\II\subset \RRR^\ell$ (open) and {\it furthermore} \0(2) the flow $t\to S_t(\V p,\V q)$ on phase space is transformed into $(\V I,\Bf)\to (\V I,\Bf+\Bo(\V I)\,t)$ where $\Bo(\V I)$ is a smooth function on $\II$.} \* This means that in suitable coordinates, that can be called ``integrating coordinates'', the system appears as a set of $\ell$ points with coordinates $\Bf=(\f_1,\ldots,\f_\ell)$ moving on a unit circle at angular velocities $\Bo(\V I)=(\o_1(\V I),\ldots,\o_{\ell}(\V I))$ depending on the actions of the initial data. A system integrable in a region $W$ which in integrating coordinates $\V I,\Bf$ has the form $\II\times \TTT^\ell$ is said {\it anisochronous} if $\det \dpr_{\V I}\Bo(\V I)\ne0$. It is said {\it isochronous} if $\Bo(\V I)\=\Bo$ is {\it independent on $\V I$}. The motions of integrable systems are called {\it quasi periodic} with frequency {\it spectrum} $\Bo(\V I)$, or with {\it frequencies} $\Bo(\V I)/2\p$, in the coordinates $(\V I,\Bf)$. Clearly an integrable system admits $\ell$ independent constants of motion, the $\V I=(I_1,\ldots,I_\ell)$ and, for each choice of $\V I$, the other coordinates vary on a ``standard'' $\ell$-dimensional torus $\TTT^\ell$: hence it is possible to say that a phase space region of integrability is {\it foliated into $\ell$-dimensional invariant tori} $\TT(\V I)$ parameterized by the values of the constants of motion $\V I\in \II$. {\it If an integrable system is anisochronous then it is canonically integrable}: \ie it is possible to define on $W$ a canonical change of coordinates $(\V p,\V q)=\CC (\V A,\Ba)$ mapping $W$ onto $J\times\TTT^{\ell}$ and such that $\HH(\CC(\V A,\Ba))=h(\V A)$ for a suitable $h$: so that, if $\Bo(\V A)\defi \dpr_{\V A} h(\V A)$, the equations of motion become $$\dot{\V A}=\V0,\qquad\dot\Ba=\Bo(\V A)\Eq(6.1)$$ % Given a system $(\V I,\Bf)$ of coordinates integrating an anisochronous system the construction of action--angle coordinates can be performed, {\it in principle}, via a classical procedure (under a few extra assumptions). Let $\g_1,\ldots,\g_\ell$ be $\ell$ {\it topologically independent} circles on $\TTT^\ell$, for definiteness let $\g_i(\V I)=\{\Bf\,|\,\f_1=\f_2=\ldots=\f_{i-1}=\f_{i+1}=\ldots=0,\ \f_i\in[0,2\p]\}$, and set $$A_i(\V I)=\fra1{2\p}\oint_{\g_i(\V I)} \V p\cdot d\V q\Eq(6.2)$$ % If the map $\V I\otto \V A(\V I)$ is analytically invertible as $\V I=\V I(\V A)$ the function $$S(\V A,\Bf)= (\l)\ig_{\V 0}^{\,\Bf} \V p\cdot d\V q \Eq(6.3)$$ % is well defined if the integral is over any path $\l$ joining the points $(p(\V I(\V A),\V0)$, $q(\V I(\V A),\V0))$ and $(p(\V I(\V A),$ $\Bf)),$ $q(\V I(\V A),\Bf)$ lying on the torus parameterized by $\V I(\V A)$. The key remark in the proof that \equ(6.3) really defines a function of the only variables $\V A,\Bf$ is that {\it anisochrony implies the vanishing of the Poisson brackets} (cf. \equ(4.4)): $\{I_i,I_j\}=0$ (hence also $\{ A_i, A_j\}\=\sum_{h,k}\dpr_{I_k}\dpr_{I_h}\{I_k,I_h\}=0$). And the property $\{I_i,I_j\}=0$ can be checked to be precisely the integrability condition for the differential form $\V p\cdot d\V q$ restricted to the surface obtained by varying $\V q$ while $\V p$ is constrained so that $(\V p,\V q)$ stays on the surface $\V I$=constant, \ie on the invariant torus of the points with fixed $\V I$. The latter property is necessary and sufficient in order that the function $S(\V A,\Bf)$ be well defined (\ie be independent on the integration path $\l$) up to an additive quantity of the form $\sum_i 2\p n_iA_i$ with $\V n=(n_1,\ldots,n_\ell)$ integers.. Then the action--angle variables are defined by the canonical change of coordinates with $S(\V A,\Bf)$ as generating function, \ie by setting $$\a_i=\dpr_{A_i} S(\V A,\Bf), \qquad I_i=\dpr_{\f_i} S(\V A,\Bf).\Eq(6.4)$$ % and, since the computation of $S(\V A,\Bf)$ is ``reduced to integrations'' which can be regarded as a natural extension of the quadratures discussed in the one dimensional cases, also such systems are called {\it integrable by quadratures}. The just described construction is a version of the more general {\it Arnold-Liouville theorem}. In practice, however, the actual evaluation of the integrals in \equ(6.2),\equ(6.3) can be difficult: its analysis in various cases (even as ``elementary'' as the pendulum) has in fact led to key progress in various domains, \eg in the theory of special functions and in group theory. In general any surface on phase space on which the restriction of the differential form $\V p\cdot d\V q$ is locally integrable is called a {\it Lagrangian manifold}: hence the invariant tori of an anisochronous integrable system are Lagrangian manifolds. If an integrable system is anisochronous it cannot admit more than $\ell$ independent constants of motion; furthermore it does not admit invariant tori of dimension $> \ell$. Hence $\ell$-dimensional invariant tori are called {\it maximal}. Of course invariant tori of dimension $<\ell$ can also exist: this happens when the variables $\V I$ are such that the frequencies $\Bo(\V I)$ admit nontrivial rational relations; \ie there is an integer components vector $\Bn\in\ZZZ^\ell, \,\Bn=(\n_1,\ldots,\n_\ell)\ne0$ such that $$\Bo(\V I)\cdot\Bn=\sum_i \o_i(\V I)\,\n_i=\V0\Eq(6.5)$$ % in this case the invariant torus $\TT(\V I)$ is called {\it resonant}. If the system is anisochronous then $\det\dpr_{\V I}\Bo(\V I)\ne0$ and, therefore, the resonant tori are associated with values of the constants of motion $\V I$ which {\it form a set of $0$-measure} in the space $\II$ but {\it which is not empty and dense}. Examples of isochronous systems are the systems of harmonic oscillators, \ie systems with Hamiltonian $\sum_{i=1}^\ell \fra1{2m_i} p_i^2+ \fra12 \sum_{i,j}^{1,\ell} c_{ij}q_i q_j$ where the matrix $v$ is a positive definite matrix. This is an isochronous system with frequencies $\Bo=(\o_1,\ldots,\o_\ell)$ whose squares are the eigenvalues of the matrix $m_i^{-\fra12}c_{ij}m_j^{-\fra12}$. It is integrable in the region $W$ of the data $\V x=(\V p,\V q)\in \RRR^{2\ell}$ such that, setting $A_\b= \fra 1{2 \, \o_\b}\, \big((\sum_{i=1}^\ell \fra{v_{\b,i} p_i}{\sqrt{m_i}})^2+ \o_\b^2 (\sum_{i=1}^\ell \fra{v_{\b,i} q_i}{\sqrt{{m_i}^{-1}}})^2\big)$ for all eigenvectors $\V v_\b$, $\b=1,\ldots,\ell$, of the above matrix, the vectors $\V A$ have all components $>0$ . Even though this system is isochronous it, nevertheless, admits a system of canonical action--angle coordinates in which the Hamiltonian takes the simplest form $$h(\V A)=\sum_{\b=1}^\ell \o_\b A_\b\=\,\Bo\cdot\V A\Eq(6.6)$$ % with $\a_\b=-\atan \fra{\sum_{i=1}^\ell \fra{v_{\b,i} p_i}{\sqrt{m_i}}}{ \sum_{i=1}^\ell \sqrt{m_i}\o_\b{v_{\b,i} q_i}} $ as conjugate angles. An example of anisochronous system is the {\it free rotators} or {\it free wheels}: \ie $\ell$ noninteracting points on a circle of radius $R$ or $\ell$ noninteracting coaxial wheels of radius $R$. If $J_i=m_i R^2$ or, respectively, $J_i=\fra12 m_i R^2$ are the inertia moments and if the positions are determined by $\ell$ angles $\Ba=(\a_1,\ldots,\a_\ell)$ the angular velocities are constants related to the angular momenta $\V A=(A_1,\ldots,A_\ell)$ by $\o_i=A_i/J_i$. The Hamiltonian and the specrum are $$h(\V A)=\sum_{i=1}^\ell\fra1{2J_i} A_i^2,\qquad \Bo(\V A)=\Big(\fra1{J_i} A_i\Big)_{i=1,\ldots,\ell}\Eq(6.7)$$ \0{\it References:} [LL68], [Ar68], [Ga83], [Fa98]. \* \Section(7,Multidimensional quadratures: central motion) \* Important mechanical systems with more than one degree of freedom are integrable by canonical quadratures in vast regions of phase space. This is checked by showing that there is a foliation into invariant tori $\TT(\V I)$ of dimension equal to the number $\ell$ of degrees of freedom parameterized by $\ell$ constants of motion $\V I$ {\it in involution}, \ie such that $\{I_i,I_j\}=0$. One then performs, if possible, the construction of the action--angle variables by the quadratures discussed in the previous section. The above procedure is well exemplified by the theory of the planar motion of a unit mass attracted by a coplanar center of force: the Lagrangian is, in polar coordinates $(\r,\th)$, $\LL=\fra{m}{2} (\dot\r^2+\r^2\dot\th^2)-V(\r)$. The planarity of the motion is not a strong resttriction as central motion takes always place on a plane Hence the equations of motion are $\fra{d}{dt} m\r^2\dot\th=0$, \ie $m\r^2\dot\th=G$ is a constant of motion (it is the angular momentum), and $\ddot\r=-\dpr_\r V(\r)+\dpr_\r \fra{m}2\r^2\dot \th^2=-\dpr_\r V(\r)+ \fra{G^2}{m\,\r^3}\defi-\dpr_\r V_G(\r)$. So that energy conservation yields a second constant of motion $E$ $$\fra{m}2\dot\r^2 +\fra12 \fra{G^2}{m\r^2}+V(\r)=E= \fra1{2m} p_\r^2+\fra1{2m}\fra{p_\th^2}{\r^2}+V(\r)\Eq(7.1)$$ % The \rhs is the Hamiltonian for the system, derived from $\LL$, if $p_\r,p_\th$ denote conjugate momenta of $\r,\th$: $p_\r=m\dot\r$ and $p_\th=m\r^2\dot \th$ (note that $p_\th=G$). Suppose $\r^2 V(\r)\tende{\r\to0}0$: then the singularity at the origin cannot be reach\-ed by any motion starting with $\r>0$ if $G>0$. Assume also that the function $V_G(\r)\defi \fra12 \fra{G^2}{m\r^2}+V(\r)$ has only one minimum $E_0(G)$, no maximum and no horizontal inflection and tends to a limit $E_\io(G)\le \io$ when $\r\to\io$. Then the system is integrable in the domain $W=\{(\V p,\V q)\,|\, E_0(G)0$) \vskip-1mm$$S(A_1,A_2,\r,\th)=G\th+\ig_{\r_{E,-}}^\r \sqrt{2m(E-V_G(x))}dx\Eq(7.5)$$ % In terms of the above $\o_0,\ch_0$ the Jacobian matrix $\fra{\dpr(E,G)}{\dpr(A_1,A_2)}$ is computed from \equ(7.4),\equ(7.5) to be $\pmatrix{\o_0&\ch_0\cr0&1\cr}$. It follows $\dpr_E S=t, \,\dpr_G S=\th-\lis\th(t)-\ch_0t$ so that, see \equ(6.4), \kern-1mm $$\a_1\defi\dpr_{A_1} S=\o_0 t, \qquad\a_2=\defi\dpr_{A_2} S=\th-\lis\th(t) \Eq(7.6)$$ % and $(A_1,\a_1),(A_2,\a_2)$ are the action--angle pairs. \* \0{\it References:} [LL68], [Ga83]. \* \Section(8, Newtonian potential and Kepler's laws) \* The anisochrony property, \ie $\det\fra{\dpr(\o_0,\ch_0)}{\dpr(A_1,A_2)}\ne0$ or, equivalently, $\det\fra{\dpr(\o_0,\ch_0)}{\dpr(E,G)}$ $\ne0$, {\it is not satisfied} in the important cases of the harmonic potential and of the Newtonian potential. Anisochrony being only a sufficient condition for canonical integrability it is still possible (and true) that, nevertheless, in both cases the canonical transformation generated by \equ(7.5) integrates the system. This is expected since the two potentials are limiting cases of anisochronous ones (\eg $|\V q|^{2+\e}$ and $|q|^{-1-\e}$ with $\e\to0$). The Newtonian potential $\HH(\V p,\V q)=\fra1{2m}\V p^2-\fra{k\,m}{|\V q|}$ is integrable in the region $G\ne0, \, E_0(G)=-\fra{k^2m^3}{2G^2}0$ and $\k,\o,g\ge 0$. They describe the motion of $n$ interacting particles on a line. The integration method for the above systems is again to find first the constants of motion and later to look for quadratures, when appropriate. The constants of motion can be found with the method of the {\it Lax pairs}. One shows that there is a pair of self adjoint $n\times n$ matrices $M(\V p,\V q), N(\V p,\V q)$ such that the equations of motion become $$\txt\fra{d}{dt} M(\V p,\V q)\, = \,i\, \big[M(\V p,\V q), N(\V p,\V q)\big],\qquad i=\sqrt{-1}\Eq(10.3)$$ % which imply that $M(t)=U(t) M(0)U(t)^{-1}$, with $U(t)$ a unitary matrix. When the equations can be written in the above form it is clear that the $n$ eigenvalues of the matrix $M(0)=M(\V p_0,\V q_0)$ are constants of motion. When appropriate, \eg in the Calogero lattice case with $\o>0$, it is possible to proceed to find canonical action--angle coordinates: a task that is quite difficult due to the arbitrariness of $n$, but which is possible. The Lax pairs for the Calogero lattice (with $\o=0,g=m=1$) are $$M_{hh}= p_h,\ \ N_{hh}=0, \ {\rm and}\ M_{hk}=\fra{i}{(q_h-q_k)},\ N_{hk}= \fra{1}{(q_h-q_k)^2} \ h\ne k\Eq(10.4)$$ % while for the Toda lattice (with $m=g=\fra12\k=1$) the non zero matrix elements of $M,N$ are $$\eqalign{ M_{hh}&= p_h,\qquad M_{h,h+1}=M_{h+1,h}=e^{-(q_h-q_{h+1})},\qquad\cr N_{h,h+1}&= -N_{h+1,h}=\,i\,e^{-(q_h-q_{h+1})}\cr} \Eq(10.5)$$ % which are checked by first trying the case $n=2$. Another integrable system ({\cs Sutherland}) is $$\HH_S(\V p,\V q)=\fra1{2m}\sum_{i=k}^n p_k^2+\sum_{h1 \Eq(12.2)$$ % with $c_\n$ the Fourier transform of the cosine ($c_{\pm1}=\fra12$, $c_\n=0$ if $\n\ne\pm1$), and (of course) $h^{(1)}_\n=-i\n c_\n$. Eq. \equ(12.2) is obtained by expanding the \rhs of \equ(12.1) in powers of $h$ and then taking the Fourier transform of both sides retaining only terms of order $k$ in $\e$. Iterating the above relation imagine to draw all trees $\th$ with $k$ ``branches'', or ``lines'', {\it distinguished by a label taking $k$ values}, and $k$ nodes and attach to each node $v$ a {\it harmonic} label $\n_v=\pm1$ as in Fig. 5. The trees will assumed to start with a {\it root line} $vr$ linking a point $r$ and the ``first node'' $v$, see Fig. 5, and then bifurcate arbitrarily (such trees are sometimes called ``rooted trees''). \* \eqfig{270pt}{70pt}{ \ins{54pt}{53pt}{$\st\Bn$} \ins{87pt}{36pt}{$\st\Bn_0$} \ins{137pt}{72pt}{$\st\Bn_1$} \ins{186pt}{86pt}{$\st\Bn_4$} \ins{136pt}{55pt}{$\st\Bn_2$} \ins{137pt}{25pt}{$\st\Bn_{3}$} \ins{187pt}{2pt}{$\st\Bn_{10}$} \ins{187pt}{15pt}{$\st\Bn_{9}$} \ins{187pt}{30pt}{$\st\Bn_8$} \ins{187pt}{43pt}{$\st\Bn_7$} \ins{187pt}{57pt}{$\st\Bn_6$} \ins{187pt}{72pt}{$\st\Bn_5$} }{fig5}{(5)} \0Fig. 5: {\nota An example of a tree graph and of its labels. It contains only one simple node ($\st1$). Harmonics are indicated next to their nodes. Labels distinguishing lines are not marked.} Imagine the tree oriented from the endpoints towards the root $r$ (not to be considered a node) and given a node $v$ call $v'$ the node immediately following it. If $v$ is the first node before the root $r$ let $v'=r$ and $\n_{v'}=1$. For each such decorated tree define its {\it numerical value} $$\Val(\th)= \fra{-i}{k!} \prod_{lines\, l= \,v'v} (\n_{v'}\n_v)\prod_{nodes} c_{\n_v}\Eq(12.3)$$ % and define a {\it current} $\n(l)$ on a line $l=v'v$ to be the sum of the harmonics of the nodes preceding $v'$: $\n(l)=\sum_{w\le v}\n_v$. Call $\n(\th)$ the current flowing in the root branch and call order of $\th$ the number of nodes (or branches). Then $$h^{(k)}_\n=\sum_{\th,\,\n(\th)=\n\atop order(\th)=k} \Val (\th)\Eq(12.4)$$ % provided trees are considered identical if they can be overlapped (labels included) after suitably scaling the lengths of their branches and pivoting them around the nodes out of which they emerge (the root is always imagined fixed at the origin). If the trees are stripped of the harmonic labels their number is finite and it can be estimated $\le k! 4^k$ (because the labels which distinguish the lines can be attached to an unlabeled tree in many ways). The harmonic labels (\ie $\n_v=\pm1$) can be laid down in $2^k$ ways, and the value of each tree can be bounded by $\fra1{k!} 2^{-k}$ (because $c_{\pm1}=\fra12$). Hence $\sum_{\n}| h^{(k)}_\n|\le 4^k$, which gives a (rough) estimate of the radius of convergence of the expansion of $h$ in powers of $\e$: namely $.25$ (easily improvable to $0.3678$ if $4^kk!$ is replaced by $k^{k-1}$ using Cayley's formula for the enumeration of rooted trees). A simple expression for $h^{(k)}(\ps)$ ({\cs Lagrange}) is $h^{(k)}(\ps)=\fra1{k!} \dpr_\ps^{k-1}\sin^k\ps$ (also readable from the tree representation): the actual radius of convergence, first determined by {\cs Laplace}, of the series for $h$ can be also determined from the latter expression for $h$ ({\cs Rouch\'e}) or directly from the tree representation: it is $\sim .6627$. One can find better estimates or at least more efficient methods for evaluating the sums in \equ(12.4): in fact in performing the sum in \equ(12.4) important {\it cancellations} occurr. For instance the harmonic labels can be subject to the further strong constraint that {\it no line carries zero current} because the sum of the values of the trees of fixed order and with at least one line carrying $0$ current vanishes. The above expansion can also be simplified by ``partial resummations''. For the purpose of an example, call {\it simple} the nodes with one entering and one exiting line (see Fig. 5). Then all tree graphs which on any line between two non simple nodes contain any number of simple nodes can be eliminated. This is done by replacing, in evaluating the (remaining) trees value, the factors $\n_{v'}\n_v$ in \equ(12.3) by $\n_{v'}\n_v/(1-\e\cos\ps )$: then the value of a tree $\th$ becomes a function of $\ps$ and $\e$ to be denoted $\Val(\th)_\ps$ {\it and} \equ(12.4) is replaced by $$h(\ps)=\sum_{k=1}^\io {\sum_{\th,\,\n(\th)=\n\atop order(\th)=k}}^{\kern-.3cm *} \kern3mm\e^k\, e^{i\,\n\,\ps}\, \Val(\th)_\ps\Eq(12.5)$$ % where the $*$ means that the trees are {\it subject to the further restriction of not containing any simple node}. It should be noted that the above graphical representation of the solution of the Kepler equation is strongly reminiscent of the representations of quantities in terms of graphs that occurr often in quantum field theory. Here the trees correspond to ``Feynman graphs'', the factors associated with the nodes are the ``couplings'', the factors associated with the lines are the ``propagators'' and the resummations are analogous to the ``self-energy resummations'', while the mentioned cancellations can be related to the class of identities called ``Ward identities''. Not only the analogy can be shown to be not superficial, but it turns out to be even very helpful in key mechanical problems: see Appendix \sec(A). The existence of a vast number of identities relating the tree values is shown already by the ``simple'' form of Lagrange's series and by the even more remarkable resummation ({\cs Levi-Civita}) leading to $$h(\ps)=\sum_{k=1}^\io \fra{(\e\sin\ps)^k}{k!} \Big(\fra1{1-\e\cos\ps}\dpr_\ps\Big)^k\,\ps\Eq(12.6)$$ % It is even possible further collection of the series terms to express it as series with much better convergence properties, for instance its terms can be reorganized and collected ({\it resummed}) so that $h$ is expressed as a power series in the parameter $$\h=\fra{\e\,e^{\sqrt{1-\e^2}}}{1+\sqrt{1-\e^2}}\Eq(12.7)$$ % with radius of convergence $1$, which corresponds to $\e=1$ (via a simple argument by {\cs Levi-Civita}). The analyticity domain for the Lagrange series is $|\h|<1$. This also determines the Laplace radius value: it is the point closest to the origin of the complex curve $|\h(\e)|=1$: it is imaginary so that it is the root of the equation $\e e^{\sqrt{1+\e^2}}/(1+\sqrt{1+\e^2})=1$. The analysis provides an example, in a simple case of great interest in applications, of the kind of computations actually necessary to represent the perturbing function in terms of action--angle variables. The property that the function $c(\l)$ in \equ(12.1) is the cosine has been used only to limit the range of the label $\n$ to be $\pm1$; hence the same method, with similar results, can be applied to study the inversion of the relation between the average anomaly $\l$ and the true anomaly $\th$ and to get, for instance, quickly the properties of $f,g$ in \equ(8.2). \* \0{\it References:} [LC]. \* \Section(13, Lindstedt and Birkhoff series. Divergences.) \* Nonexistence of constants of motion, rather than being the end of the attempts to study by perturbation methods motions close to integrable ones, marks the beginning of renewed efforts to understand their nature. Let $(\V A,\Ba)\in U\times \TTT^\ell$ be action--angle variables defined in the integrability region for an analytic Hamiltonian and let $h(\V A)$ be its value in the action--angle coordinates. Suppose that $h(\V A)$ is {\it anisochronous} and let $f(\V A,\Ba)$ be an analytic perturbing function. Consider, for $\e$ small, the Hamiltonian $\HH_\e(\V A,\Ba)=\HH_0(\V A)+\e f(\V A,\Ba)$. Let $\Bo_0=\Bo(\V A_0)\=\Dpr_{\V A} \HH_0(\V A)$ be the frequency spectrum, see Section \sec(6), of one of the invariant tori of the unperturbed system corresponding to an action $\V A_0$. Short of integrability the question to ask at this point is whether the perturbed system admits an analytic invariant torus on which motion is quasi periodic with\\ (1) the same spectrum $\Bo_0$ and \\ (2) depends analytically on $\e$ at least for $\e$ small and \\ (3) reduces to the ``unperturbed torus'' $\{\V A_0\}\times\TTT^\ell$ as $\e\to0$. More concretely the question is \* {\it Are there functions $\V H_\e(\Bps),\V h_\e(\Bps)$ analytic in $\Bps\in\TTT^\ell$ and in $\e$ near $0$, {\it vanishing as $\e\to0$} and such that the torus with parametric equations $$\V A=\V A_0+{\V H}_\e(\Bps),\qquad \Ba=\Bps+\V h_\e(\Bps)\qquad \Bps\in\TTT^\ell \Eq(13.1)$$ % is invariant and, if $\Bo_0\defi\Bo(\V A_0)$, the motion on it is simply $\Bps\to\Bps+\Bo_0 t$, \ie it is quasi periodic with spectum $\Bo_0$?} \* In this context {\cs Poincar\'e}'s theorem of Sect.\sec(11) had followed another key result, earlier developed in particular cases and completed by him, which provides a {\it partial} answer to the question. Suppose that $\Bo_0=\Bo(\V A_0)\in\RRR^\ell$ satisfies a {\it Diophantine property}, namely suppose that there exist constants $C,\t>0$ such that $$ |\Bo_0\cdot\Bn|\ge \fra1{C |\Bn|^\t},\qquad\hbox{for all} \ \V0\ne \Bn\in\ZZZ^\ell\Eq(13.2)$$ % which, for each $\t>\ell-1$ {\it fixed}, is a property enjoyed by all $\Bo\in\RRR^\ell$ {\it but for a set of zero measure}. Then the motions on the unperturbed torus run over trajectories that {\it fill the torus densely} because of the ``irrationality'' of $\Bo_0$ implied by \equ(13.2). Writing Hamilton's equations,, $\dot\Ba=\dpr_{\V A}\HH_0(\V A)+\e\Dpr_{\V A} f(\V A,\Ba)$, $\dot{\V A}=-\e\Dpr_{\Ba}f(\V A,\Ba)$ with $\V A,\Ba$ given by \equ(13.1) with $\Bps$ replaced by $\Bps+\Bo t$, and using the density of the unperturbed trajectories implied by \equ(13.2), the condition that \equ(13.1) are equations for an invariant torus on which the motion is $\Bps\to\Bps+\Bo_0 t$ are $$\eqalign{ &\Bo_0+(\Bo_0\cdot\Dpr_{\Bps}) \V h_\e(\Bps)= \Dpr_{\V A} \HH_0(\V A_0+\V H_\e(\Bps))+\e\Dpr_{\V A} f(\V A_0+\V H_\e(\Bps), \Bps+\V h_\e(\Bps))\cr & (\Bo_0\cdot\Dpr_{\Bps}) \V H_\e(\Bps)= -\e \Dpr_{\Ba} f(\V A_0+\V H_\e(\Bps), \Bps+\V h_\e(\Bps))\cr}\Eq(13.3)$$ % The theorem referred above ({\cs Poincar\'e}) is that \* \0{\it If the unperturbed system is anisochronous and $\Bo_0=\Bo(\V A_0)$ satisfies \equ(13.2) for some $C,\t>0$ there exist two well defined power series $\V h_\e(\Bps)=\sum_{k=1}^\io$ $ \e^k\,\V h^{(k)}(\Bps)$ and $\V H_\e(\Bps)=\sum_{k=1}^\io \e^k\,\V H^{(k)}(\Bps)$ which solve \equ(13.3) to all orders in $\e$. The series for $\V H_\e$ is uniquely determined, and such is also the series for $\V h_\e$ up to the addition of an arbitrary constant at each order, so that it is unique if $\V h_\e$ is required, as henceforth done with no loss of generality, to have zero average over $\Bps$.} \* The algorithm for the construction is illustrated in a simple case in Section \sec(14), \equ(14.4),\equ(14.5). Convergence of the above series, called {\it Lindstedt series}, even for $\e$ small has been a problem for a rather long time. {\cs Poincar\'e} proved existence of the formal solution; but his other result, discussed in Sect.\sec(11), casts doubts on convergence although {\it it does not exclude it} as was immediately stressed by several authors (including {\cs Poincar\'e} himself). The result of Sect. 11 shows impossibility of solving \equ(13.3) {\it for all} $\Bo_0$'s near a given spectrum, analytically and uniformly, but it does not exclude the possibility of solving it {\it for a single} $\Bo_0$. % The theorem admits several extensions or analogues: an interesting one is to the case of isochronous unperturbed systems: \* \0{\it Given the Hamiltonian $\HH_\e(\V A,\Ba)=\Bo_0\cdot\V A+\e f(\V A,\Ba)$, with $\Bo_0$ satisfying \equ(13.2) and $f$ analytic, there exist power series $\CC_\e(\V A',\Ba'),\,u_\e(\V A')$ such that $\HH_\e(\CC_\e(\V A',\Ba'))=\Bo_0\cdot\V A'+u_\e(\V A')$ holds as an equality between formal power series (\ie order by order in $\e$) and at the same time the $\CC_\e$ regarded as a map satisfies order by order the condition (\ie {\rm \equ(4.3)}) that it is a canonical map.} \* This means that there is a generating function $\V A'\cdot\Ba+\F_\e(\V A',\Ba)$ also defined by a formal power series $\F_\e(\V A',\Ba)=\sum_{k=1}^\io \e^k\, \F^{(k)}(\V A',\Ba)$, \ie such that if $\CC_\e(\V A',\Ba')=(\V A,\Ba)$ then it is true, order by order in powers of $\e$, that $\V A=\V A'+\Dpr_{\Ba}\F_\e(\V A',\Ba)$ and $\Ba'=\Ba+\Dpr_{\V A'}\F_\e(\V A',\Ba)$. The series for $\F_\e,u_\e$ are called {\it Birkhoff series}. In this isochronous case {\it if Birkhoff series were convergent} for small $\e$ and $(\V A',\Ba)$ in a region of the form $U\times \TTT^\ell$, with $U\subset\RRR^\ell$ open and bounded, it would follow that, for small $\e$, $\HH_\e$ would be integrable in a large region of phase space (\ie where the generating function can be used to build a canonical map: this would essentially be $U\times \TTT^\ell$ deprived of a small layer of points near the boundary of $U$). However convergence for $\e$ small is false (in general) as shown by the simple two dimensional example $$\HH_\e(\V A,\Ba)=\Bo_0\cdot\V A+\e\,(A_2+f(\Ba)),\qquad (\V A,\Ba)\in\RRR^2\times \TTT^2\Eq(13.4)$$ % with $f(\Ba)$ an arbitrary analytic function with {\it all} Fourier coefficients $f_\Bn$ positive for $\Bn\ne\V0$ and $f_{\V0}=0$. In the latter case the solution is $$u_\e(\V A')=\e A_2,\quad \F_\e(\V A',\Ba)=\sum_{k=1}^\io \e^k \sum_{\V0\ne \Bn\in\zzz^2} {f_\Bn\,e^{i\Ba\cdot\Bn}}\fra{(i\,\n_2)^k} {(i(\o_{01}\n_1+\o_{02}\n_2))^{k+1}}\Eq(13.5) $$ % The series does not converge: in fact its convergence would imply integrability and, consequently, bounded trajectories in phase space: however the equations of motion for \equ(13.4) can be easily solved {\it explicitly} and in any open region near a given initial data there are other data which have unbounded trajectories if $\o_{01}/(\o_{02}+\e)$ is rational. Nevertheless even in this elementary case a formal sum of the series yields $$u(\V A')=\e A'_2,\quad \F_\e(\V A',\Ba)=\e \sum_{\V 0\ne\Bn\in\zzz^2} \fra{f_\Bn\,e^{i\Ba\cdot\Bn}}{i(\o_{01}\n_1+(\o_{20}+\e)\n_2)}\Eq(13.6)$$ % and the series in \equ(13.6) (no longer a power series in $\e$) is really convergent if $\Bo=(\o_{01},\o_{02}+\e)$ is a Diophantine vector (by \equ(13.2), because analyticity implies exponential decay of $|f_\Bn|$). Remarkably for such values of $\e$ the Hamiltonian $\HH_\e$ {\it is integrable and it is integrated} by the canonical map generated by \equ(13.6), in spite of the fact that \equ(13.6) is obtained, from \equ(13.5), via the {\it non rigorous sum rule} $$\sum_{k=0}^\io z^k=\fra1{1-z} \qquad{\rm for}\ z\ne1\Eq(13.7)$$ % (applied to cases with $|z|\ge1$, which are certainly realized for a dense set of $\e$'s even if $\Bo$ is Diophantine because the $z$'s have values $z=\fra{\n_2}{\Bo_0\cdot\Bn}$). In other words the integration of the equations is elementary and once performed it becomes apparent that, if $\Bo$ is diophantine, the solutions can be rigorously found from \equ(13.6). Note that, for instance, this means that relations like $\sum_{k=0}^\io 2^k=-1$ are really used to obtain \equ(13.6) from \equ(13.5). Another extension of Lindstedt series arises in a perturbation of an an\-iso\-chro\-nous system when asking the question of what happens to the unperturbed invariant tori $\TT_{\Bo_0}$ on which the spectrum {\it is resonant}, \ie $\Bo_0\cdot\Bn=0$ for some $\Bn\ne\V0, \, \Bn\in\ZZZ^\ell$. The result is that even in such case there is a formal power series solutions showing that {\it at least a few} of the (infinitely many) invariant tori into which $\TT_{\Bo_0}$ is in turn foliated in the unperturbed case can be formally continued at $\e\ne0$, see Section \sec(15). \* \0{\it References:} [Po]. \* \Section(14, Quasi periodicity and KAM stability) \* To discuss more advanced results it is convenient to restrict attention to a special (non trivial) paradigmatic case $$\HH_\e(\V A,\Ba)=\fra12 \V A^2+\e\,f(\Ba)\Eq(14.1)$$ % In this simple case (called {\it Thirring model}: representing $\ell$ particles on a circle interacting via a potential $\e f(\Ba)$) the equations for the maximal tori \equ(13.3) reduce to equations for the only functions $\V h_\e$: $$(\Bo\cdot\Dpr_{\Bps})^2\V h_\e(\Bps)=-\e\,\Dpr_\Ba f(\Bps+\V h_\e(\Bps)),\qquad \Bps\in\TTT^\ell\Eq(14.2)$$ % as the second of \equ(13.3) simply becomes the definition of $\V H_\e$ because the \rhs does not involve $\V H_\e$. The real problem is therefore whether the formal series considered in Section \sec(13) converge at least for small $\e$: and the example \equ(13.4) on the Birkhoff series shows that sometimes {\it sum rules} might be needed in order to give a meaning to the series. In fact whenever a problem (of physical interest) admits a formal power series solution which is not convergent, or which it is not known whether it is convergent, then one should look for sum rules for it. The modern theory of perturbations starts with the proof of the convergence for $\e$ small enough of the Lindstedt series ({\cs Kolmogorov}). The general ``KAM'' result is \* \0{\it Consider the Hamiltonian $\HH_\e(\V A,\Ba)=h(\V A)+\e f(\V A,\Ba)$, defined in $U=V\times \TTT^\ell$ with $V\subset\RRR^\ell$ open and bounded and with $f(\V A,\Ba),h(\V A)$ analytic in the closure $\lis V\times \TTT^\ell$ where $h(\V A)$ is also anisochronous; let $\Bo_0\defi$ $ \Bo(\V A_0)=\dpr_{\V A}h(\V A_0)$ and assume that $\Bo_0$ satisfies \equ(13.2). Then \\ (1) there is $\e_{C,\t}>0$ such that the Lindstedt series converges for $|\e|<\e_{C,\t}$, \\ (2) its sum yields two function $\V H_\e(\Bps),\V h_\e(\Bps)$ on $\TTT^\ell$ which parameterize an invariant torus $\TT_{C,\t}(\V A_0,\e)$, \\ (3) on $\TT_{C,\t}(\V A_0,\e)$ the motion is $\Bps \to\Bps+\Bo_0t$, see \equ(13.1). \\ (4) the set of data in $U$ which belong to invariant tori $\TT_{C,\t}(\V A_0,\e)$ with $\Bo(\V A_0)$ satisfying \equ(13.2) with prefixed $C,\t$ has complement with volume $<\, const\, C^{-a}$ for a suitable $a>0$ and with area also $<\, const\, C^{-a}$ on each nontrivial surface of constant energy $\HH_\e=E$. } \* In other words for $\e$ small the spectra of most unperturbed quasi periodic motions can still be found as spectra of perturbed quasi periodic motions developing on tori which are close to the corresponding unperturbed ones (\ie with the same spectrum). This is a {\it stability result}: for instance in systems with two degrees of freedom the invariant tori of dimension $2$ which lie on a given energy surface, which has $3$ dimensions, will separate the points on the energy surface into the set which is ``inside'' the torus and the set which is ``outside'': hence an initial datum starting (say) inside cannot reach the outside. Likewise a point starting between two tori has to stay in between forever: and if the two tori are close this means that motion will stay very localized in action space, with a trajectory accessing only points close to the tori and coming close to all such points, within a distance of the order of the distance between the confining tori. The case of three or more degrees of freedom is quite different, see Sect.\sec(17),\sec(19). In the simple case of the rotators system \equ(14.1) the equations for the parametric representation of the tori are the \equ(14.2). The latter bear some analogy with the easier problem in \equ(12.1): but the \equ(14.2) are $\ell$ equations instead of one and they are differential equations rather than ordinary equations. Furthermore the function $f(\Ba)$ which plays here the role of $c(\l)$ in \equ(12.1) has Fourier coefficients $f_{\Bn}$ with no restrictions on $\Bn$, while the Fourier coefficiens $c_\n$ for $c$ in \equ(12.1) do not vanish only for $\n=\pm1$. The above differences are, to some extent, ``minor'' and the power series solution to \equ(14.2) can be constructed {\it by the same algorithm} used in the case of \equ(12.1): namely one forms trees as in Fig. 5 with the harmonic labels $\n_v\in\ZZZ$ replaced by $\Bn_v\in\ZZZ^\ell$ (still to be thought of as possible harmonic indices in the Fourier expansion of the perturbing function $f$). All other labels affixed to the trees in Sec. 11 will be the same. In particular the {\it current flowing on a branch} $l=v'v$ will be defined as the sum of the harmonics of the nodes $w\le v$ preceding $v$: $$\Bn(l)\defi \sum_{w\le v}\Bn_w\Eq(14.3)$$ % and we call $\Bn(\th)$ the current flowing in the root branch. This time the value $\Val(\th)$ of a tree has to be defined differently because the equation \equ(14.2) to be solved contains the differential operator $(\Bo_0\cdot\Dpr_\Bps)^2$ which in Fourier transform becomes multiplication of the Fourier component with harmonic $\Bn$ by $(i\Bo\cdot\Bn)^2$. The variation due to the presence of the operator $(\Bo_0\cdot\Dpr_\Bps)^2$ and the necessity of its inversion in the evaluation of $\V u\cdot\V h^{(k)}_{\Bn}$, \ie of the component of $\V h_\Bn^{(k)}$ along an arbitrary unit vector $\V u$, is nevertheless quite simple: the value of a tree graph $\th$ of order $k$ (\ie with $k$ nodes and $k$ branches) has to be defined by (cf. \equ(12.3)) $$\Val(\th)\defi \fra{-i\,(-1)^k}{k!} \Big(\prod_{lines\, l= \,v'v} \fra{\Bn_{v'}\cdot \Bn_v}{\big(\Bo_0\cdot\Bn(l)\big)^2}\Big) \Big(\prod_{nodes\,v} f_{\Bn_v}\Big)\Eq(14.4)$$ % where the $\Bn_{v'}$ appearing in the factor relative to the root line $rv$ from the first node $v$ to the root $r$, see Fig. 5, is interpreted as an unit vector $\V u$ (it was interpreted as $1$ in the ``one dimensional'' case \equ(12.1)). The \equ(14.4) makes sense only for trees in which no line carries $\V0$ current. Then the component along $\V u$ (the harmonic label attached to the root of a tree) of $\V h^{(k)}$ is given, see also \equ(12.4), by $$\V u\cdot \V h^{(k)}_\Bn= \sum^*_{\th,\,\Bn(\th)=\Bn\atop order(\th)=k} \Val (\th)\Eq(14.5)$$ % where the $*$ means that the sum is only over trees in which a non zero current $\Bn(l)$ flows on the lines $l\in \th$. The quantity $\V u\cdot\V h^{(k)}_{\V0}$ will be $\V0$, see Section \sec(13). In the case of \equ(12.1) zero current lines could appear: {\it but} the contributions from tree graphs containing at least one zero current line would cancel. In the present case the statement that the above algorithm actually gives $\V h^{(k)}_\Bn$ {\it by simply ignoring} trees with lines with $\V 0$ current is non trivial. It has been {\cs Poincar\'e}'s contribution to the theory of Lindstedt series to show that {\it even in the general case}, cf.\equ(13.3), the equations for the invariant tori can be solved by a formal power series. The \equ(14.5) is proved by induction on $k$ after checking it for the first few orders. The algorithm just described leading to \equ(14.4) can be extended to the case of the general Hamiltonian considered in the KAM theorem. The convergence proof is more delicate than the (elementary) one for the equation \equ(12.1). In fact the values of trees of order $k$ can give large contributions to $\V h^{(k)}_\Bn$: because the ``new'' factors $\big(\Bo_0\cdot\Bn(l)\big)^{2}$, although not zero, can be quite small and their small size can overwhelm the smallness of the factors $f_{\Bn}$ and $\e$: in fact even if $f$ is a trigonometric polynomial (so that $f_{\Bn}$ vanishes identically for $|\Bn|$ large enough) the currents flowing in the branches can be very large, of the order of the number $k$ of nodes in the tree, see \equ(14.3). This is called the {\it small divisors} problem. The key to its solution goes back to a related work ({\cs Siegel}) which shows that \* \0{\it Consider the contribution to the sum in \equ(14.3) from graphs $\th$ in which no pairs of lines which lie on the same path to the root carry the same current and, furthermore, the node harmonics are bounded by $|\Bn|\le N$ for some $N$. Then the number of lines $\ell$ in $\th$ with divisor $\Bo_0\cdot\Bn_\ell$ satisfying $2^{-n}0 \Eq(15.1)$$ % where $(\V A,\Ba)\in U\times \TTT^r$, $U\in\RRR^r$, $(\V p,\V q)\in V\subset \RRR^{2 s_1}$, $(\Bp,\Bk)\in V'\subset \RRR^{2 s_2}$ with $V,V'$ neighborhoods of the origin and $\ell=r+s_1+s_2$, $s_i\ge0,s_1+s_2>0$ and $\pm\sqrt{\l_j},\pm\sqrt{\m_j}$ are called {\it Lyapunov coefficients} of the resonance. The perturbations considered are supposed to have the form $\e f(\V A,\Ba,\V p,\V q,\Bp,\Bk)$. The denomination of \ap stable or unstable refers to the properties of the ``\ap given unperturbed Hamiltonian''. The name of \ap unstable is certainly appropriate if $s_1>0$: here also $s_1=0$ is allowed for notational convenience implying that the Lyapunov coefficients in \ap unstable cases are all of order $1$ (whether real, $\l_j$ or imaginary $i\sqrt{\m_j}$). In other words the \ap stable case, $s_1=s_2=0$ in \equ(15.1), is the only excluded case. Of course the stability properties of the motions when a perturbation acts will depend on the perturbation {\it in both cases}. The {\it a priori stable} systems have usually a great variety of resonances (\eg in the anisochro\-nous case resonances of any dimension are dense). The {\it a priori unstable} systems have (among possible other resonances) some {\it very special} $r$-dimensional resonances occurring when the {\it unstable coordinates} $(\V p,\V q)$ and $(\Bp,\Bk)$ are zero and the frequencies of the $r$ action--angle coordinates are rationally independent. In the first case, \ap stable, the general question is whether the resonant motions, which form invariant tori of dimension $r$ arranged into families that fill $\ell$ dimensional invariant tori, continue to exists, in presence of small enough perturbations $\e f(\V A,\Ba)$, on slightly deformed invariant tori. Similar questions can be asked in the \ap unstable cases. To examine more closely the matter consider the formulation of the simplest problems. \* \0{\it A priori stable resonances}: More precisely let $\{\V A_0\}\times\TTT^\ell$ be the unperturbed invariant torus $\TT_{\V A_0}$ with spectrum $\Bo_0=\Bo(\V A_0)=\V\dpr_{\V A} \HH_0(\V A_0)$ with only $r$ rationally independent components. For simplicity suppose that $\Bo_0=(\o_1,\ldots,\o_r,0,\ldots,0)\defi(\Bo,\V0)$ with $\Bo\in\RRR^r$. The more general case in which $\Bo$ has only $r$ rationally independent components can be reduced to special case above by a canonical linear change of coordinates at the price of changing the $\HH_0$ to a new one, still quadratic in the actions but containing mixed products $A_iB_j$: the proofs of the results that are discussed here would not be really affected by such more general form of $\HH$. It is convenient to distinguish between the ``fast'' angles $\a_1,\ldots,\a_r$ and the ``resonant'' angles $\a_{r+1},\ldots,\a_\ell$ (also called ``slow'' or ``secular'') and call $\Ba=(\Ba',\Bb)$ with $\Ba'\in\TTT^r$ and $\Bb\in \TTT^s$. Likewise we distinguish the fast actions $\V A'=(A_1,\ldots,A_r)$ and the resonant ones $A_{r+1},\ldots,A_{\ell}$ and set $\V A=(\V A',\V B)$ with $\V A'\in \RRR^r$ and $\V B\in\RRR^s$. Therefore the torus $\TT_{\V A_0}$, $\V A_0=(\V A'_0,\V B_0)$, is in turn a {\it continuum of invariant tori} $\TT_{\V A_0,\Bb}$ with trivial parametric equations: $\Bb$ fixed, $\Ba'= \Bps$, $\Bps \in \TTT^r$, and $\V A'=\V A'_0,\,\V B=\V B_0$. On each of them the motion is: $\V A',\V B,\Bb$ constant and $\Ba'\to \Ba'+ \Bo\,t$, with rationally independent $\Bo\in \RRR^r$. Then the natural question is whether there exist functions $\V h_\e,\V k_\e,\V H_\e,\V K_\e$ smooth in $\e$ near $\e=0$ and in $\Bps\in\TTT^r$, vanishing for $\e=0$, and such that the torus $\TT_{\V A_0,\Bb_0,\e}$ with parametric equations $$\eqalign{ \V A'=&\V A'_0+\V H_\e(\Bps),\qquad \Ba'=\Bps+\V h_\e(\Bps),\cr \V B=&\V B_0+\V K_\e(\Bps),\qquad \Bb=\Bb_0+\V k_\e(\Bps),\cr}\qquad\Bps\in\TTT^r\Eq(15.2)$$ % is invariant for the motions with Hamiltonian $\HH_\e(\V A,\Ba)= \fra12{\V A'}^2+\fra12\V B^2+\e f(\Ba',\Bb)$ and the motions on it are $\Bps\to \Bps+ \Bo\,t$. The above property, when satisfied, is summarized by saying that the unperturbed resonant motions $\V A=(\V A'_0,\V B_0)$, $\Ba=(\Ba'_0+\Bo' t,\Bb_0)$ can be {\it continued} in presence of perturbation $\e f$, for small $\e$, to quasi periodic motions with the same spectrum and on a slightly deformed torus $\TT_{\V A'_0,\Bb_0,\e}$. \* \0{\it A priori unstable resonances}: here the question is whether the {\it special} invariant tori continue to exist in presence of small enough perturbations, of course slightly deformed. This means asking whether, given $\V A_0$ such that $\Bo(\V A_0)=\V\dpr_{\V A}\HH_0(\V A_0)$ has rationally independent components, there are functions $(\V H_\e(\Bps),$ $\V h_\e(\Bps))$, $(\V P_\e(\Bps),\V Q_\e(\Bps))$ and $(\BP_\e(\Bps), \BK_\e (\Bps))$ smooth in $\e$ near $\e=0$, vanishing for $\e=0$, analytic in $\Bps\in\TTT^r$ and such that the $r$-dimensional surface $$\eqalign{ \V A=&\V A_0+\V H_\e(\Bps),\qquad \Ba=\Bps+\V h_\e(\Bps)\cr \V p=&\V P_\e(\Bps),\kern1.7cm \V q=\V Q_\e(\Bps)\cr \Bp=&\BP_\e(\Bps),\kern1.8cm \Bk=\BK_\e(\Bps)\cr} \qquad \Bps\in \TTT^r\Eq(15.3)$$ % is an invariant torus $\TT_{\V A_0,\e}$ on which the motion is $\Bps\to\Bps+\Bo(\V A_0)\,t$. Again the above property is summarized by saying that the unperturbed special resonant motions can be {\it continued} in presence of perturbation $\e f$ for small $\e$ to quasi periodic motions with the same spectrum and on a slightly deformed torus $\TT_{\V A_0,\e}$. Some answers to the above questions are presented in the following section. \* \0{\it References:} [GBG04]. \* \Section(16, Resonances and Lindstedt series) \* We discuss the equations \equ(15.2) in the paradigmatic case in which the Hamiltonian $\HH_0(\V A)$ is $\fra12\V A^2$ (cf. \equ(14.1)). It will be $\Bo(\V A')\=\V A'$ so that $\V A_0=\Bo,\V B_0=\V0$ and the perturbation $f(\Ba)$ can be considered as a function of $\Ba=(\Ba',\Bb)$: let $\lis f(\Bb)$ be defined as its average over $\Ba'$. The determination of the invariant torus of dimension $r$ which can be continued in the sense discussed in Sect. \sec(15) is easily understood in this case. A resonant invariant torus which, among the tori $\TT_{\V A_0,\Bb}$, has parameric equations that can be continued as a {\it formal powers series} in $\e$ is the torus $\TT_{\V A_0,\Bb_0}$ with $\Bb_0$ a stationarity point for $\lis f(\Bb)$, \ie an {\it equilibrium point for the average perturbation}: $\dpr_\Bb\lis f(\Bb_0)=0$. In fact the following theorem holds \kern3mm \0{\it If $\Bo\in \RRR^r$ satisfies a Diophantine property and if $\Bb_0$ is a nondegenerate stationarity point for the ``fast angle average'' $\lis f(\Bb)$ (\ie such that $\det \dpr^2_{\Bb\,\Bb}\lis f(\Bb_0)\ne0$), then the equations for the functions $\V h_\e,\V k_\e$: $$\eqalign{ (\Bo\cdot\dpr_\Bps)^2 {\V h}_\e(\Bps)=&-\e\dpr_{\Ba'} f(\Bps+\V h_\e(\Bps),\Bb_0+\V k_\e(\Bps))\cr (\Bo\cdot\dpr_\Bps)^2 {\V k}_\e(\Bps)=&-\e\dpr_{\Bb} f(\Bps+\V h_\e(\Bps)+\V k_\e(\Bps))\cr}\Eq(16.1)$$ % can be formally solved in powers of $\e$.} \kern3mm Given the simplicity of the Hamiltonian that we are considering, \ie \equ(14.1), it is not necessary to discuss the functions $\V H_\e,\V K_\e$ because the equations that they should obey reduce to their definitions as in the case of Sec. \sec(14), and for the same reason. In other words also the resonant tori admit a Lindstedt series representation. {\it It is however very unlikely that the series are, in general, convergent}. Physically this {\it new} aspect is due to the fact that the linearization of the motion near the torus $\TT_{\V A_0,\Bb_0}$ introduces {\it oscillatory motions} around $\TT_{\V A'_0,\Bb_0}$ with frequencies proportional to the square roots of the {\it positive} eigenvalues of the matrix $\e\dpr^2_{\Bb\,\Bb}\lis f(\Bb_0)$: therefore it is naively expected that it has to be necessary that a Diophantine property be required on the vector $(\Bo,\sqrt{\e \m_1},\ldots)$ where $\e\m_j$ are the positive eigenvalues. Hence some values of $\e$, those for which $(\Bo,\sqrt{\e \m_1},\ldots)$ is not a Diophantine vector or is too close to a non Diophantine vector, should be excluded or at least should be expected to generate difficulties. Note that the problem arises no matter what is supposed on the non degenerate matrix $\dpr^2_{\Bb\,\Bb}\lis f(\Bb_0)$ since $\e$ can have either sign; and no matter how $|\e|$ is supposed small. But we can expect that if the matrix $\dpr^2_{\Bb\Bb}\lis f(\Bb_0)$ is (say) positive definite (\ie $\Bb_0$ is a minimum point for $\lis f(\Bb)$) then the problem should be easier for $\e<0$ and viceversa if $\Bb_0$ is a maximum it should be easier for $\e>0$ (\ie in the cases in which the eigenvalues of $\e \dpr^2_{\Bb\Bb}\lis f(\Bb_0)$ are negative and their roots do not have the interpretation of frequencies). Technically the sums of the formal series can be given (so far) a meaning only via summation rules involving divergent series: typically one has to identify in the formal expressions (denumerably many) geometric series which although divergent can be given a meaning by applying the rule \equ(13.7). Since the rule can only be applied if $z\ne1$ this {\it leads to conditions on the parameter $\e$}, in order to exclude that the various $z$ that have to be considered are too close to $1$. Hence this stability result turns out to be {\it rather different} from the KAM result for the maximal tori. Namely the series can be given a meaning via summation rules provided $f$ and $\Bb_0$ satisfy certain additional conditions and provided certain values of $\e$ are excluded. An example of a theorem is the following: \* \0{\it Given the Hamiltonian \equ(14.1) and a resonant torus $\TT_{\V A'_0,\Bb_0}$ with $\Bo=\V A'_0\in\RRR^r$ satisfying a Diophantine property let $\Bb_0$ be a non degenerate maximum point for the average potential $\lis f(\Bb)\defi(2\p)^{-r}\ig_{\tttt^r} f(\Ba',\Bb)d^{\,r}\Ba'$. Consider the Lindstedt series solution for the equations \equ(16.1) of the perturbed resonant torus with spectrum $(\Bo,\V0)$. It is possible to express the single $n$-th order term of the series as a sum of many terms and then rearrange the series thus obtained so that the resummed series converges for $\e$ in a domain $\EE$ which contains a segment $[0,\e_0]$ and also contains a subset of $[-\e_0,0]$ which, although with {\it open dense complement}, is so large that it has $0$ as a Lebesgue density point. Furthermore the resummed series for $\V h_\e,\V k_\e$ define an invariant $r$ dimensional analytic torus with spectrum $\Bo$.} \* More generally if $\Bb_0$ is only a nondegenerate stationarity point for $\lis f(\Bb)$ the domain of definition of the resummed series is a set $\EE\subset [-\e_0,\e_0]$ which on {\it both sides} of the origin has an open dense complement although it has $0$ as a Lebesgue density point. The above theorem can be naturally extended to the general case in which the Hamiltonian is the most general perturbation of an anisochronous integrable system $\HH_\e(\V A,\Ba)=h(\V A)+\e f (\V A,\Ba)$ if $\dpr^2_{\V A \V A}h$ is a non singular matrix and the resonance arises from a spectrum $\Bo(\V A_0)$ which has $r$ independent components (while the remaining are not necessarily $\V0$). We see that the convergence is a delicate problem for the Lindstedt series for nearly integrable resonant motions. They might even be divergent (mathematically a proof of divergence is an open problem but it is a very resonable conjecture in view of the above physical interpretation), {\it nevertheless} the above theorem shows that sum rules can be given that ``sometimes'', \ie for $\e$ in a large set near $\e=0$, yield a true solution to the problem. This is reminiscent of the phenomenon met in discussing perturbations of isochro\-nous systems in \equ(13.4), but it is a much more complex situation. And it leaves many open problems: {\it foremost of them is the question of uniqueness}. The sum rules of divergent series always contain some arbitrary choices: which lead to doubt about the uniqueness of the functions parameterizing the invariant tori constructed in this way. It might even be that the convergence set $\EE$ may depend upon the arbitrary choices, and that considering several of them no $\e$ with $|\e|<\e_0$ is left out. The case of \ap unstable systems has also been widely studied: in this case too resonances with Diophantine $r$-dimensional spectrum $\Bo$ are considered. However in the case $s_2=0$ (called \ap unstable {\it hyperbolic resonance}) the Lindstedt series can be shown to be convergent while in the case $s_1=0$ (called \ap unstable {\it elliptic resonance}) or in the {\it mixed} cases $s_1,s_2>0$ extra conditions are needed. They involve $\Bo$ and $\Bm=(\m_1,\ldots,\m_{s_2})$, cf. \equ(15.1), and properties of the perturbations as well. It is also possible to study a slightly different problem: namely to look for conditions on $\Bo,\Bm,f$ which imply that for small $\e$ invariant tori with spectrum $\e$-dependent but {\it close}, in a suitable sense, to $\Bo$ exist. The literature is vast but it seems fair to say that, given the above comments, {\it particularly those concerning uniqueness and analyticity}, the situation is still quite unsatisfactory. \* \0{\it References:} [GBG04]. %\ifnum\tipo=2\pagina\fi \* \Section(17, Diffusion in phase space) \* The KAM theorem implies that a perturbation of an analytic anisochro\-nous integrable system, \ie with an analytic Hamiltonian $\HH_\e(\V A,\Ba)=\HH_0(\V A)$ $+\e f(\V A,\Ba)$ and non degenerate Hessian matrix $\dpr^2_{\V A\V A}h(\V A)$, generates large families of maximal invariant tori. Such tori lie on the energy surfaces but do not have codimension $1$ on them, \ie do not split the $(2\ell-1)$--dimensional energy surfaces into disconnected regions {\it except}, of course, in the case of $2$--degrees of freedom systems, see Sect.\sec(14).. Therefore there might exist trajectories with initial data close in action space to $\V A^i$ which reach phase space points close in action space to $\V A^f\ne \V A^i$ for $\e\ne0$, {\it no matter how small}. Such {\it diffusion} phenomenon would occurr in spite of the fact that the corresponding trajectory has to move in a space in which very close to each $\{\V A\}\times \TTT^\ell$ there is an invariant surface on which points move keeping $\V A$ constant within $O(\e)$, which for $\e$ small can be $\ll|\V A^f-\V A^i|$. In \ap unstable systems (cf. Sect. 15) with $s_1=1,s_2=0$ it is not difficult to see that the corresponding phenomenon can actually happen: the paradigmatic example ({\cs Arnold}) is the \ap unstable system % $$\HH_\e=\fra{A_1^2}2+A_2+\fra{p^2}2+ g\, (\cos q-1)+\e\, (\cos \a_1+\sin\a_2)(\cos q-1) \Eq(17.1)$$ % This is a system describing a motion of a ``pendulum'' ($(p,q)$ coordinates) interacting with a ``rotating wheel'' ($(A_1,\a_1)$ coordinates) and a ``clock'' ($(A_2,\a_2)$ coordinates) \ap unstable near the points $p=0,q=0,2\p$, ($s_1=1,s_2=0$, $\l_1=\sqrt{g}$, cf. \equ(15.1)). And it can be proved that on the energy surface of energy $E$ and for each $\e\ne0$ small enough (no matter how small) {\it there are initial data with action coordinates close to $\V A^i=(A^i_1,A^i_2)$ with $\fra12 {A^{i\,2}_1}+ {A^i_2}$ close to $E$ eventually evolving to a datum $\V A'=(A'_1,A'_2)$} with $A'_1$ at distance from $A^f_1$ smaller than an {\it arbitrarily} prefixed distance (of course with energy $E$). {\it Furthermore} during the whole process the pendulum energy stays close to $0$ within $o(\e)$ (\ie the pendulum swings following closely the unperturbed separatrices). In other words \equ(17.1) describes a machine (the pendulum) which, working approximately in a cycle, extracts energy from a reservoir (the clock) to transfer it to a mechanical device (the wheel). The statement that diffusion is possible means that the machine can work as soon as $\e\ne0$, if the initial actions and the initial phases (\ie $\a_1,\a_2,p,q$) are suitably tuned (as functions of $\e$). The peculiarity of the system \equ(17.1) is that the unperturbed pendulum fixed points $P_\pm$ (\ie the equilibria $p=0,q=0,2\p$) {\it remain unstable equilibria even when $\e\ne0$}: and this is an important simplifying feature. \eqfig{240pt}{100pt}{}{fig6}{(6)} \kern-4truemm \0Fig. 6: {\nota The first drawing represents the $\e=0$ geometry: the ``partial energy'' lines are parabolae, $\fra12 A_1^2+A_2=const$. The vertical lines are the resonances $A_1=rational$ (\ie $\n_1 A_1+\n_2=0$). The disks are neighborhoods of the points $\V A^i$ and $\V A^f$ (the dots at their centers). The second drawing ($\e\ne0$) is an artist rendering of a trajectory in $\V A$ space, driven by the pendulum swings to accelerate the wheel from $A^i_1$ to $A^f_1$ at the expenses of the clock energy, sneaking through invariant tori (not represented and approximately) located ``away'' from the intersections between resonances and partial energy lines (a dense set, however). The pendulum coordinates are not shown: its energy stays close to $0$ within a power of $\e$. Hence the pendulum swings staying close to the separatrix. The oscillations symbolize the wiggly behavior of the partial energy $\fra12 A_1^2+A_2$ in the process of sneaking between invariant tori which, because of their invariance, would be impossible without the pendulum. The energy $\fra12A^2_1$ of the wheel increases slightly at each pendulum swing: accurate estimates yield an increase of the wheel speed $A_1$ of the order of $\e/\hbox{\nota log}\, \e^{-1}$ at each swing of the pendulum implying a transition time of the order of $g^{-\fra12}\e^{-1}\hbox{\nota log}\,\e^{-1}$.\vfil} It permits bypassing the obstacle, arising in the analysis of more general cases, represented by the resonance surfaces consisting in the $\V A$'s with $A_1\n_1+\n_2=0$: the latter correspond to harmonics $(\n_1,\n_2)$ present in the perturbing function, \ie the harmonics which would lead to division by zero in an attempt to construct (as necessary in the proof by Arnold's method) the parametric equations of the perturbed invariant tori with action close such $\V A$'s. In the case of \equ(17.1) the problem arises only on the resonance marked in Fig.6 by a heavy line \ie $A_1=0$ corresponding to $\cos \a_1$ in \equ(17.1). If $\e=0$ the points $P_-$ with $p=0,q=0$ and the point $P_+$ with $p=0,q=2\p$ are both unstable equilibria (and they are of course the same point, if $q$ is an angular variable). The unstable manifold (it is a curve) of $P_+$ coincides with the stable manifold of $P_-$ and viceversa. So that the unperturbed system admits non trivial motions leading from $P_+$ to $P_-$ and from $P_-$ to $P_+$, both in a biinfinite time interval $(-\io,\io)$: the $p,q$ variables describe a pendulum and $P_\pm$ are its unstable equilibria which are connected by the separatrices (which constitute the $0$-energy surfaces for the pendulum). The latter property remains true for more general \ap unstable Hamiltonians $$\HH_\e=\HH_0(\V A)+\HH_u(p,q)+\e\, f(\V A,\Ba,p,q), \qquad {\rm in}\ (U\times\TTT^\ell)\times(\RRR^2)\Eq(17.2)$$ % where $\HH_u$ is a one dimensional Hamiltonian which has two unstable equilibrium points $P_+$ and $P_-$ linearly repulsive in one direction and linearly attractive in another which are connected by two {\it heteroclinic} trajectories which, as time tends to $\pm\io$, approach $P_-$ and $P_+$ and viceversa. Actually the points need not be different but, if coinciding, the trajectories linking them must be nontrivial: in the case \equ(17.1) the variable $q$ can be considered an angle and then $P_+$ and $P_-$ would coincide (but are connected by nontrivial trajectories, \ie by trajectories that visit points different from $P_\pm$). Such trajectories are called {\it heteroclinic} if $P_+\ne P_-$ and {\it homoclinc} if $P_+=P_-$. In the general case besides the homoclinicity (or heteroclinicity) condition certain weak genericity conditions, automatically satisfied in the example \equ(17.1), have to be imposed in order to show that given $\V A^i$ and $\V A^f$ with the same unperturbed energy $E$ one can find, for all $\e$ small enough but not equal to $0$, initial data ($\e$-dependent) with actions arbitrarily close to $\V A^i$ which evolve to data with actions arbitrarily close to $\V A^f$. This is a phenomenon called {\it Arnold diffusion}. Simple sufficient conditions for a transition from near $\V A^i$ to near $\V A^f$ are expressed by the following result \vskip2mm \0{\it Given the Hamiltonian \equ(17.2) with $\HH_u$ admitting two hyperbolic fixed points $P_\pm$ with heteroclinic connections, $t\to (p_a(t),q_a(t)),\,a=1,2$, suppose that \\ \0(1) On the unperturbed energy surface of energy $E=\HH(\V A^i)+\HH_u(P_\pm)$ there is a regular curve $\g\,:\,s\to \V A(s)$ joining $\V A^i$ to $\V A^f$ such that the unperturbed tori $\{\V A(s)\}\times \TTT^\ell$ can be continued at $\e\ne0$ into invariant tori $\TT_{\V A(s),\e}$ for a set of values of $s$ which fills the curve $\g$ leaving only gaps of size of order $o(\e)$. \0(2) The $\ell\times\ell$ matrix $D_{ij}$ of the second derivatives of the integral of $f$ over the heteroclinic motions is not degenerate, \ie \vskip-1.5mm $$|\det D|=\big|\det \,\Big(\ig_{-\io}^\io dt \, \dpr_{\a_i\a_j} f(\V A,\Ba +\Bo(\V A)\,t, p_a(t),q_a(t))\Big)\big|>c>0\Eq(17.3)$$ \vskip-1mm \0for all $\V A$'s on the curve $\g$ and all $\Ba\in\TTT^2$. \\ Given arbitrarily $\r>0$, for $\e\ne0$ small enough there are initial data with action and energy closer than $\r$ to $\V A^i$ and $E$, respectively, which after a long enough time acquire an action closer than $\r$ to $\V A^f$ (keeping the initial energy).} \vskip2mm The above two conditions can be shown to hold generically for many pairs $\V A^i\ne \V A^f$ (and many choices of the curves $\g$ connecting them) if the number of degree of freedom is $\ge3$. Thus the result, obtained by a simple extension of the argument originally outlined by Arnold to discuss the paradigmatic example \equ(17.1), proves the existence of diffusion in {\it a priori unstable} system. The integral in \equ(17.3) is called {\it Melnikov integral}. The real difficulty is to estimate the time needed for the transition: it is a time that obviously has to diverge as $\e\to0$. Assuming $g$ fixed (\ie $\e$--independent) a naive approach easily leads to estimates which can even be worse than $O(e^{a\e^{-b}})$ with some $a,b>0$. It has finally been shown that in such cases the minimum time can be, for rather general perturbations $\e f(\Ba,q)$, estimated above by $O(\e^{-1}\log\e^{-1})$, which is the best that can be hoped for under generic assumptions. \* \0{\it References:} [Ar68], [CV00]. \* \Section(18, Long time stability of quasi periodic motions) \* A more difficult problem is whether the same phenomenon of migration in action space occurs in \ap stable systems. The root of the difficulty is a remarkable stability property of quasi periodic motions. Consider Hamiltonians $\HH_\e(\V A,\Ba)= h(\V A)+\e f(\V A,\Ba)$ with $\HH_0(\V A)=h(\V A)$ {\it strictly convex, analytic and anisochronous} on the closure $\lis U$ of an open {\it bounded} region $U\subset\RRR^\ell$, and a perturbation $\e f(\V A,\Ba)$ analytic in $\lis U\times\TTT^\ell$. Then \ap bounds are available on how long it can possibly take to migrate from an action close to $\V A_1$ to one close to $\V A_2$: and the bound is of ``exponential type'' as $\e\to0$ (\ie it admits a lower bound which behaves as the exponential of an inverse power of $\e$). The simplest theorem is ({\cs Nekhorossev}): \vskip2mm \0{\it There are constants $00$ constants. The three scales are $\o_1^{-1},\sqrt{g^{-1}},\o_2^{-1}$. In this case there are many (although by no means all) pairs $\V A_1,\V A_2$ which can be connected within a time that can be estimated to be of order $O(\e^{-1}\log \e^{-1})$. This is a rapid diffusion case in a \ap unstable system in which condition \equ(17.3) is {\it not} satisfied: because the $\e$--dependence of $\Bo(\V A)$ implies that the lower bound $c$ in \equ(17.3) must depend on $\e$ (and be exponentiallly small with an inverse power of $\e$ as $\e\to0$). The unperturbed system in \equ(18.1) is non resonant in the part $\HH_0$ for $\e>0$ outside a set of zero measure (\ie where the vector $\Bo_\e$ satisfies a suitable Diophantine property) and, furthermore, it is \ap unstable: cases met in applications can be \ap stable and resonant (and often not anisochronous) in the part $\HH_0$. And in such system not only the speed of diffusion is not understood but proposals to prove its existence, if present (as expected), have so far not given really satisfactory results. \* \0{\it References:} [Ne77]. \* \Section(19, The three bodies problem) \* Mechanics and the three bodies problem can be almost identified in the sense that the motion of three gravitating masses has been since the beginning a key astronomical problem and at the same time the source of inspiration for many techniques: foremost among them the theory of perturbations. As an introduction consider a special case. Let three masses $m_S=m_0,m_J=m_1,m_M=m_2$ interact via gravity, \ie with interaction potential $-k m_i m_j |\V x_i-\V x_j|^{-1}$: the simplest problem arises when the third body has a neglegible mass compared to the two others and the latter are supposed to be on a circular orbit; furthermore the mass $m_J$ is $\e m_S$ with $\e$ small and the mass $m_M$ moves in the plane of the circular orbit. This will be called the {\it circular restricted three body problem}. In a reference system with center $S$ and rotating at the angular speed of $J$ around $S$ inertial forces (centrifugal and Coriolis') act. Supposing that the body $J$ is located on the axis with unit vector $\V i$ at distance $R$ from the origin $S$, the acceleration of the point $M$ is $\ddot\Br={\V F}+\o^2_0(\Br-\fra{\e R}{1+\e}\V i)-2\Bo_0\wedge\dot\Br$ if $\V F$ is the force of attraction and $\Bo_0\wedge\dot\Br\=\o_0\dot\Br^\perp$ where $\Bo_0$ is a vector with $|\Bo_0|=\o_0$ and perpendicular to the orbital plane and $\Br^\perp\defi (-\r_2,\r_1)$ if $\Br=(\r_1,\r_2)$. Here, taking into account that the origin $S$ rotates around the fixed center of mass, $\o^2_0(\Br-\fra{\e R}{1+\e}\V i)$ is the centrifugal force while $-2\Bo_0\wedge\dot\Br$ is the Coriolis force. The equations of motion can therefore be derived from a Lagrangian $$\LL=\fra12\dot{\Br}^2-W + \o_0\Br^\perp\cdot\dot{\Br}+\fra12\o_0^2{\Br}^2- \o_0^2\fra{\e \,R}{1+\e}\Br\cdot\V i\Eq(19.1)$$ % where $\o_0^2R^3=k m_S (1+\e)\defi g_0$ and $W=-\fra{k m_S}{|\Br|} -\fra{k\,m_S\,\e}{|\Br-R\V i|}$ if $k$ is the gravitational constant, $R$ the distance between $S$ and $J$ and finally the last three terms come from the Coriolis force (the first) and from the centripetal force (the other two, taking into account that the origin $S$ rotates around the fixed center of mass). Setting $g=g_0/(1+\e)\= k m_S$, the Hamiltonian of the system is $$\txt\HH=\fra12(\V p-\o_0 \Br^\perp)^2 -\fra{g}{|\Br|}-\fra12\o_0^2\Br^2-\e\fra{g}{R} \big({|\fra\Br{R}-\V i|^{-1}} -\fra{\Br}R\cdot\V i\big)\Eq(19.2) $$ % The first part can be expressed immediately in the action--angle coordinates for the two body problem, cf. Sect. \sec(8). Calling such coordinates $(L_0,\l_0,G_0,\g_0)$ and $\th_0$ the polar angle of $M$ with respect to the ellipse major axis and $\l_0$ the mean anomaly of $M$ on its ellipse, the Hamiltonian becomes, taking into account that for $\e=0$ the ellipse axis rotates at speed $-\o_0$, $$\txt \HH=-\fra{g^2}{2L_0^2}-\o_0G_0 -\e\fra{g}{R} \big({|\fra\Br{R}-\V i|^{-1}} -\fra{\Br}R\cdot\V i\big)\Eq(19.3)$$ % which is convenient if we study the {\it interior problem}, \ie $|\Br|< R$. This can be expressed in the action--angle coordinates via \equ(8.1), \equ(8.2): $$\eqalign{\txt \th_0&\txt=\l_0+f_{\l_0},\kern2cm \th_0+\g_0=\l_0+\g_0+ f_{\l_0},\cr \txt e&\txt=\big(1-\fra{G^2_0}{L^2_0}\big)^{\fra12},\kern1cm\fra{|\Br|}{R}= \fra{ G_0^2}{gR}\fra1{1+e\cos(\l_0+f_{\l_0})},\cr}\Eq(19.4)$$ % where, see \equ(8.2), $f_\l=(f(e \sin\l, e\cos\l)$ and $f(x,y)=2x(1+\fra54y+\ldots)$ with the $\ldots$ denoting higher orders in $x,y$ even in $x$. The Hamiltonian takes the form, if $\o^2=g R^{-3}$, $$\HH_\e=-\fra{g^2}{2 L_0^2}-\o G_0+ \e\fra{g}{R} F(G_0,L_0,\l_0,\l_0+\g_0)\Eq(19.5)$$ % where the only important feature (for our purposes) is that $F(L,G,\a,\b)$ is an {\it analytic function} of $L,G,\a,\b$ near a datum with $|G|0$) and $|\Br|< R$. {\it However} the domain of analyticity in $G$ is rather small as it is constrained by $|G|0$, with $\o_1/\o_2$ Diophantine, then the perturbed system admits a motion which is quasi periodic with spectrum proportional to $\Bo$ and takes place on an orbit which wraps around a torus remaining {\it forever close} to the unperturbed torus (which can be visualized as described by a point moving, according to the area law on an ellipse rotating at rate $-\o_0$) with actions $(L_0,G_0)$, provided $\e$ is small enough. Hence \* \0{\it The KAM theorem answers, at least conceptually, the classical question: can a solution of the three body problem remain close to an unperturbed one forever? \ie is it possible that a solar system is stable forever?} \* Assuming $e, |\Br|/R\ll 1$ and retaining only the lowest orders in $e$ and $|\Br|/R\ll 1$ the Hamiltonian \equ(19.5) simplifies into \vskip-3mm $$\eqalign{ \txt\HH=&\txt-\fra{g^2}{2L_0^2}- \o G_0 +\d_\e(G_0)-\fra{\e g}{2R} \fra{G_0^4}{g^2R^2} \big(\txt3 \cos2(\l_0+\g_0)-\cr &\txt- e\, \cos\l_0-\fra92 e \cos(\l_0+2\g_0)+\fra32 e \cos(3\l_0+2\g_0) %%%???-3e\sin\l_0\sin 2(\l_0+\g_0) \big)\cr}\Eq(19.7)$$ % where $\d_\e(G_0)=-((1+\e)^{\fra12}-1)\,\o \,G_0-\fra{\e\,g}{2R} \fra{G_0^4}{g^2 R^2}$ and $e=(1-G_0^2/L_0^2)^{\fra12}$. It is an interesting exercise to estimate, assuming as model the \equ(19.7) and following the proof of the KAM theorem, how small has $\e$ to be if a planet with the data of Mercury can be stable forever on a (slowly precessing) orbit with actions close to the present day values under the influence of a mass $\e$ times the solar mass orbiting on a circle, at a distance from the Sun equal to that of Jupiter. It is possible to follow either the above reduction to the ordinary KAM theorem or to apply directly to \equ(19.7) the Lindstedt series expansion, proceeding along the lines of Sect. \sec(14). The first approach is easy but the second is more efficient: unless the estimates are done in a particularly careful manner the value found for $\e m_S$ does not have astronomical interest. \* \0{\it References:} [Ar68]. \* \Section(20, Rationalization and regularization of singularities) \* Often integrable systems have interesting data which lie on the boundary of the integrability domain. For instance the central motion when $L=G$ (circular orbits) or the rigid body in a rotation around one of the principal axes or the two body problem when $G=0$ (collisional data). In such cases perturbation theory cannot be applied as discussed above. Typically the perturbation depends on quantities like $\sqrt{L-G}$ and is {\it not analytic} at $L=G$. Nevertheless it is sometimes possible to enlarge phase space and introduce new coordinates in the vicinity of the data which in the initial phase space are singular. A notable example is the failure of the analysis of the circular restricted three body problem: it apparently fails when the orbit that we want to perturb is circular. It is convenient to introduce the canonical coordinates $L,\l$ and $G,\g$ $$L=L_0,\ G=L_0-G_0,\ \l=\l_0+\g_0,\ \g=-\g_0\Eq(20.1)$$ % so that $e=\sqrt{2G L^{-1}}\sqrt{1-G(2L)^{-1}}$ and $\l_0=\l+\g$ and $\th_0=\l_0+f_{\l_0}$ where $f_{\l_0}$ is defined in \equ(8.2) (see also \equ(19.4)). Hence \kern-3mm $$\eqalign{\txt \th_0&\txt=\l+\g+f_{\l+\g},\qquad \th_0+\g_0=\l+ f_{\l+\g},\cr \txt e=&\txt\sqrt{2G} \sqrt{\fra{1}{L}\big(1-\fra{G}{2L}\big)},\qquad \fra{|\Br|}{R}= \fra{L^2 (1-e^2)}{gR}\fra1{1+e\cos(\l+\g+f_{\l+\g})},\cr}\Eq(20.2) $$ % and the Hamiltonian \equ(19.7) takes the form $$\HH_\e=-\fra{g^2}{2 L^2}-\o L+\o G+\e\fra{g}{R} F(L-G,L,\l+\g,\l)\Eq(20.3)$$ % In the coordinates $L,G$ of \equ(20.1) the unperturbed circular case corresponds to $G=0$ and the \equ(19.3) once expressed in the action--angle variables $G,L,\g,\l$ is analytic in a domain whose size is controlled by $\sqrt{G}$. Nevertheless very often problems of perturbation theory can be ``{\it regularized}''. This is done by ``enlarging the integrability'' domain by adding to it points (one or more) around the singularity (a boundary point of the domain of the coordinates) and introduce new coordinates to describe simultaneously the data close to the singularity and the newly added points: in many interesting cases the equations of motion are no longer singular, \ie become analytic, in the new coordinates and therefore apt to describe the motions that reach the singularity in a finite time. {\it One can say that the singul;arity was only apparent}. Perhaps this is best illustrated precisely in the above circular restricted three body problem. There the singularity is where $G=0$, \ie at a circular unperturbed orbit. If we describe the points with $G$ small in a new system of coordinates obtained from the one in \equ(20.1) by letting alone $L, \l$ and setting \vskip-2mm $$p=\sqrt{2G}\cos\g,\qquad q=\sqrt{2G}\sin\g\Eq(20.4)$$ % then $p,q$ vary in a neighborhood of the origin with the origin itself {\it excluded}. Adding the origin of the $p,q$ plane then in a full neighborhood of the origin the Hamiltonian \equ(19.3) is {\it analytic} in $L,\l,p,q$. This is because it is analytic, cf. \equ(19.3),\equ(19.4), as a function of $L,\l$ and $e\cos\th_0$ and of $\cos(\l_0+\th_0)$. Since $\th_0=\l+\g+f_{\l+\g}$ and $\th_0+\l_0=\l+ f_{\l+\g}$ by \equ(19.4), the Hamiltonian \equ(19.3) is analytic in $L,\l, e\cos(\l+\g+f_{\l+\g}),\, \cos(\l+f_{\l+\g})$ for $e$ small (\ie for $G$ small) and, by \equ(8.2), $f_{\l+\g}$ is analytic in $e\sin(\l+\g)$ and $e\cos(\l+\g)$. Hence the trigonometric identities \vskip-3mm $$ \txt e\sin (\l+\g)\,= \,\fra {p \sin\l+q\cos\l}{\sqrt L}\,\sqrt{1-\fra{G}{2L}}, \qquad e\cos(\l+\g)\,= \,\fra {p \cos\l-q\sin\l}{\sqrt L}\,\sqrt{1-\fra{G}{2L}} \Eq(20.5)$$ % together with $G=\fra12(p^2+q^2)$ imply that \equ(20.3) is analytic near $p=q=0$ and $L>0,\l\in[0,2\p]$. The Hamiltonian becomes analytic and the new coordinates are suitable to describe motions crossing the origin: \eg setting $C\defi\fra12(1-\fra{p^2+q^2}{4L})\,L^{-\fra12}$ \equ(19.7) becomes \vskip-3truemm $$\eqalign{ \txt\HH=&\txt-\fra{g^2}{2L^2}-\o L+\o \fra12(p^2+q^2)+\d_\e(\fra12(p^2+q^2))-\fra{\e g}{2R} \fra{(L-\fra12(p^2+q^2))^4}{g^2 R^2}\cdot\cr &\txt\cdot \Big(3\cos2\l -\big( (-11\cos\l +3\cos3\l)\,p-\,(7\sin\l+3\sin3\l)\,q\big)\, C\Big) \cr}\Eq(20.6)$$ % The KAM theorem does not apply in the form discussed above to ``cartesian coordinates'' \ie when, as in \equ(20.6), the unperturbed system is not assigned in action--angle variables: however there are versions of the theorem (actually corollaries of it) which do apply and therefore it becomes possible to obtain some results even for the perturbations of circular motions by the techniques that have been illustrated here. \vskip1mm Likewise the Hamiltonian of the rigid body with a fixed point $O$ and subject to analytic external forces becomes singular, if expressed in the action--angle coordinates of Deprit, when the body motion nears a rotation around a principal axis or more generally nears a configuration in which any two of the axes $\V i_3$, $\V z$, $\V z_0$ coincide (\ie any two among the principal axis, the angular momentum axis and the inertial $z$-axis coincide, see Section \sec(9)). Nevertheless by imitating the procedure just described in the simpler cases of the circular three body problem, it is possible to enlarge phase space so that in the new coordinates the Hamiltonian is analytic near the singular configurations. \vskip1mm A regularization also arises when considering collisional orbits in the unrestricted planar three body problem. In this respect a very remarkable result is the regularization of collisional orbits in the planar three body problem. After proving that if the total angular momentum does not vanish simoultaneous collisions of the three masses cannot happen within any finite time interval the question is reduced to the regularization of two bodies collisions, under the assumption that the total angular momentum does not vanish. The {\it local} change of coordinates which changes the relative position coordinates $(x,y)$ of two colliding bodies as $(x,y)\to(\x,\h)$ with $x+i y=(\x+i\h)^2$ is not one to one, hence it has to be regarded as an enlargement of the positions space, if points with different $(\x,\h)$ are considered different. However the equations of motion written in the variables $\x,\h$ have no singularity at $\x,\h=0$, ({\cs Levi-Civita}). \vskip1mm Another celebrated regularization is the regularization of the {\it Schwar\-tz\-schild metric}, \ie of the general relativity version of the two body problem: it is however somewhat out of the scope of this review ({\cs Synge, Kruskal}). \* \0{\it References:} [LC]. \* \appendix(A, KAM resummation scheme) \* The idea to control the ``remaining contributions'' is to reduce the problem to the case in which there are no pairs of lines that follow each other in the tree order and which have the {\it same current}. Mark by a {\it scale label} ``$0$'' the lines of a tree whose divisors are $>1$: these are lines which give no problems in the estimates. Then mark by a scale label ``$\ge1$'' the lines with current $\Bn(l)$ such that $|\Bo_0\cdot\Bn(l)|\le 2^{-n+1}$ for $n=1$ (\ie the remaining lines). The lines labeled $0$ are said to be {\it on scale $0$}, while those labeled $\ge1$ are said to be {\it on scale $\ge1$}. A {\it cluster of scale $0$} will be a {\it maximal} collection of lines of scale $0$ forming a connected subgraph of a tree $\th$. Consider only trees $\th_0\in \Th_0$ of the family $\Th_0$ of trees {\it containing no clusters of lines with scale label $0$ which have only one line entering the cluster and one exiting it with equal current}. It is useful to introduce the notion of a line $\ell_1$ situated ``{\it between}'' two lines $\ell,\ell'$ with $\ell'>\ell$: this will mean that $\ell_1$ precedes $\ell'$ but not $\ell$. All trees $\th$ in which there are some pairs $l'> l$ of consecutive lines of scale label $\ge1$ which have equal current and such that all lines between them bear scale label $0$ are obtained by ``inserting'' on the lines of trees in $\Th_0$ with label $\ge1$ any number of clusters of lines and nodes, with lines of scale $0$ and with the property that the sum of the harmonics of the nodes inserted {\it vanishes}. Consider a line $l_0\in \th_0\in\Th_0$ linking nodes $v_1\fra12$ of the propagator, so that it certainly converges for $\e$ small enough (by the estimates in Section \sec(12) where the propagators were identically $1$) and the sum is of order $\e$ (actually $\e^2$), hence $<1$. {\it However} such an argument cannot be repeated when dealing with lines with smaller propagators (which still have to be discussed). Therefore a method not relying on so trivial a remark on the size of the propagators has eventually to be used when considering lines of scale higher than $1$, as it will soon become necessary. The advantage of the collection of terms achieved with \equ(A1.2) is that we can represent $\V h$ as a sum of values of trees which are {\it simpler} because they contain no pair of lines of scale $\ge1$ with in between lines of scale $0$ with total sum of the node harmonics vanishing. The price is that the divisors are now more involved and we even have a problem due to the fact that we have not proved that the series in \equ(A1.2) converges. In fact it is a geometric series whose value is the \rhs of \equ(A1.2) obtained by the sum rule \equ(13.7) {\it unless we can prove that the ratio of the geometric series is $<1$}. This is trivial in this case by the previous remark: but it is better to remark that there is anothr reason for convergence, whose use is not really necesary here but it will become essential later. The property that the ratio of the geometric series is $<1$ can be regarded as due to the consequence of the {\it cancellation} mentioned in Section \sec(14) which can be shown to imply that the ratio is $<1$ because $M^{(0)}(\Bn)=\e^2 \,(\Bo_0\cdot\Bn)^2 m^{(0)}(\Bn)$ with $|m^{(0)}(\Bn)|< D_0$ for some $D_0>0$ and for all $|\e|<\e_0$ for some $\e_0$: so that for small $\e$ the divisor in \equ(A1.2) is essentially still what it was before starting the resummation. At this point an induction can be started. Consider trees evaluated with the new rule and place a scale lavel ``$\ge2$'' on the lines with $|\Bo_0\cdot\Bn(l)|\le 2^{-n+1}$ for $n=2$: leave the label ``$0$'' on the lines already marked so and label by ``$1$'' the other lines. The lines of scale ``$1$'' will satisfy $2^{-n}< |\Bo_0\cdot\Bn(l)|\le 2^{-n+1}$ for $n=1$. And the graphs will now possibly contain lines {\it of scale $0,1$ or $\ge2$ while lines with label ``$\ge1$'' no longer can appear}, by construction. A {\it cluster of scale $1$} will be a maximal collection of lines of scales $0,1$ forming a connected subgraph of a tree $\th$ and containing at least one line of scale $1$. The construction carried considering clusters of scale $0$ can be repeated by considering trees $\th_1\in\Th_1$ with $\Th_1$ the collection of trees with lines marked $0,1$ or $\ge2$ and in which no pairs of lines with equal momentum appear to follow each other if between them there are only lines marked $0$ or $1$. Insertion of connected clusters $\g$ of such lines on a line $l_0$ of $\th_1$ leads to define a matrix $M^{(1)}$ formed by summing tree values of clusters $\g$ with lines of scales $0$ or $1$ evaluated with the line factors defined in \equ(A1.1) and with the restriction that {\it in $\g$ there are no pairs of lines $\ell<\ell'$ with the same current and which follow each other while any line between them has lower scale (\ie $0$}), here between means preceding $l'$ but not preceding $l$, as above. Therefore a scale independent method has to devised to check convergence for $M^{(1)}$ and for the matrices to be introduced later to deal with even smaller propagators. This is achieved by the following extension of Siegel's theorem mentioned in Section \sec(14): \vskip2mm \0{\it Let $\Bo_0$ satisfy \equ(13.2) and set $\Bo= C\Bo_0$. Consider the contribution to the sum in \equ(14.3) from graphs $\th$ in which \\ (1) no pairs $\ell'>\ell$ of lines which lie on the same path to the root carry the same current $\Bn$ if all lines $\ell_1$ between them have current $\Bn(\ell_1)$ such that $|\Bo\cdot\Bn(\ell_1)|> 2 |\Bo\cdot\Bn|$. \\ (2) the node harmonics are bounded by $|\Bn|\le N$ for some $N$. \\ Then the number of lines $\ell$ in $\th$ with divisor $\Bo\cdot\Bn_\ell$ satisfying $2^{-n}<|\Bo\cdot\Bn_\ell|\le 2^{-n+1}$ does not exceed $4\, N \, k\, 2^{-n/\t}$, $n=1,2,\ldots$.} \vskip2mm%\* This implies, by the same estimates in \equ(14.6), that the series defining $M^{(1)}$ converges. Again it must be checked that there are cancellations implying that $M^{(1)}(\Bn)=\e^2 \,(\Bo_0\cdot\Bn)^2 m^{(1)}(\Bn)$ with $|m^{(1)}(\Bn)|< D_0$ for the {\it same} $D_0>0$ and the same $\e_0$. At this point one deals with trees containing only lines carrying labels $0,1,\ge2$ and the line factors for the lines $\ell=v'v$ of scale $0$ are $\Bn_{v'}\cdot\Bn_{v}/(\Bo_0\cdot\Bn(\ell))^2$, those of the lines $\ell=v'v$ of scale $1$ have line factors $\Bn_{v'}\cdot(\Bo_0\cdot\Bn(\ell)^2-M^{(0)}(\Bn(\ell)))^{-1} \Bn_v$ and those of the lines $\ell=v'v$ of scale $\ge2$ have line factors $\Bn_{v'}\cdot(\Bo_0\cdot\Bn(\ell)^2-M^{(1)}(\Bn(\ell)))^{-1} \Bn_v$. {\it Furthermore no pair of lines of scale ``$1$'' or of scale ``$\ge 2$'' with the same momentum and with only lines of lower scale (\ie of scale ``$0$'' in the first case or of scale ``$0$'',''$1$'' in the second) between then can follow each other.} And so on until, after infinitely many steps, the problem is reduced to the evaluation of tree values in which each line carries a scale label $n$ and there are no pairs of lines which follow each other and which have only lines of lower scale in between. Then the Siegel argument applies once more and the series so resummed is an absolutely convergent series of functions analytic in $\e$: hence the original series is convergent. Although at each step there is a lower bound on the denominators it would not be possible to avoid using Siegel's theorem. In fact the lower bound would becomes worse and worse as the scale increases. In order to check the estimates of the constants $D_0,\e_0$ which control the scale independence of the convergence of the various series it is necessary to take advantage of the theorem, and of the absence at each step of the necessity of considering trees with pairs of consecutive lines with equal momentum and intermediate lines of higher scale. One could also perform the analysis by bounding $h^{(k)}$ order by order with no resummations (\ie without changing the line factors) and exhibiting the necessary cancellations. Or the paths that {\cs Kolmogorov}, {\cs Arnold} and {\cs Moser} used to prove the first three (somewhat different) versions of the theorem, by successive approximations of th equations for the tori, can be followed. The invariant tori are {\it Lagrangian manifolds} just as the unperturbed ones (cf. comments after \equ(6.4)) and, in the case of the Hamiltonian \equ(14.1) the generating function $\V A\cdot\Bps+\F(\V A,\Bps)$ can be expressed in terms of their parametric equations \pagina %\vskip-2mm $$\eqalign{ &\F(\V A,\Bps)=G(\Bps)+\V a\cdot\Bps+ \V h(\Bps)\cdot(\V A-\Bo-\D \V h(\Bps))\cr & \Dpr_\Bps G(\Bps)\defi-\D \V h(\Bps)+\T h(\Bps)\Dpr_\Bps \D \T h(\Bps)-\V a\cr & \V a\defi \ig (-\D \V h(\Bps)+\T h(\Bps)\Dpr_\Bps \D \T h(\Bps))\fra{d\Bps}{(2\p)^\ell}=\ig \T h(\Bps)\Dpr_\Bps \D \T h(\Bps)\fra{d\Bps}{(2\p)^\ell} \cr }\Eqa(A1.3) $$ % where $\D=(\Bo\cdot \Dpr_\Bps)$ and the invariant torus corresponds to $\V A'=\Bo$ in the map $\Ba=\Bps+\Dpr_{\V A}\F(\V A,\Bps)$ and $\V A'=\V A+\Dpr_\Bps\F(\V A,\Bps)$. In fact by \equ(A1.3) the latter becomes $\V A'=\V A-\D\V h$ and, from the second of \equ(13.3) written for $f$ depending only on he angles $\Ba$, it is $\V A=\Bo+\D\V h$ when $\V A,\Ba$ are on the ivariant torus. Note that if $\V a$ exists it is necessarily determined by the third relation but the check that the second equation in \equ(A1.3) is soluble (\ie that the \rhs is an exact gradient up to a constant) is nontrivial. The canonical map generated by $\V A\cdot\Bps +\F(\V A,\Bps)$ is {\it also} defined for $\V A'$ close to $\Bo$ and foliates the neighborhood of the invariant torus with other tori: of course for $\V A'\ne \Bo$ the tori defined in this way are, in general, not invariant. \* \0{\it References:} [GBG94]. \* \appendix(B, Coriolis and Lorentz forces. Larmor precession) {\it Larmor precession} is the part of the motion of an electrically charged particle caused by the action of a magnetic field $\V H$ (in an inertial frame of reference). It is due to the {\it Lorentz force} which, on a unit mass with unit charge, produces an acceleration $\ddot\Br=\V v\wedge \V H$ if the speed of light is $c=1$. Therefore if $\V H=H \V k$ is directed along the $\V k$ axis the acceleration it produces is the same that the Coriolis force would impress on a unit mass located in a reference frame which rotates with angular velocity $\o_0\V k$ around the $\V k$ axis if $\V H=2\o_0 \V k$. The above remarks imply that a homogeneous sphere homogeneoulsy electrically charged with a unit charge and freely pivoting about its center in a constant magnetic field $H$ directed along the $\V k$ axis undergoes the same motion it would follow if not subject to the magnetic field but seen in a non inertial reference frame rotating at constant angular velocity $\o_0$ around the $\V k$ axis if $H$ and $\o_0$ are related by $H=2\o_0$: in this frame the Coriolis force is interpreted as a magnetic field. This holds, however, only if the centrifugal force has zero moment with rerspect to the center: true in the spherical symmetry case only. In spherically non symmetric cases the centrifugal forces have in general non zero moment so the equivalence between Coriolis forces and magnetic fields is only approximate. The {\it Larmor theorem} makes this more precise. It gives a quantitative estimate of the difference between the motion of a general system of particles of mass $m$ in a magnetic field and the motion of the same particles in a rotating frame of reference but in absence of a magnetic field. The approximation is estimated in terms of the size of the {\it Larmor frequency} $eH/2mc\,$: which should be small compared to the other characteristic frequencies of the motion of the system: the physical meaning is that the centrifugal force should be small compared to the other forces. The vector potential $\V A$ for a constant magnetic field in the $\V k$-direction $\V H=2\o_0 \V k$ is $\V A= 2\o_0\V k\wedge\Br\= 2\o_0\Br^\perp$. Therefore, from the treatment of the Coriolis force in Section \sec(19), see \equ(19.2), the motion of a charge $e$ with mass $m$ in a magnetic field $\V H$ with vector potential $\V A$ and subject to other forces with potential $W$ can be described, in an inertial frame and in generic units in which the speed of light is $c$, by a Hamiltonian $$\HH =\fra1{2m}(\V p-\fra{e}c\V A)^2+ W(\Br)\Eqa(A2.1)$$ % where $\V p=m\dot\Br+\fra{e}c \V A$ and $\Br$ are canonically conjugated. %\ifnum\tipo=0\pagina\fi \* \def\*{\vskip.6mm} \0{\bf References} \* \nota \0{\bf [Ar68] Arnold, V.I.:} {\it Mathematical methods of classical mechanics}, Sprin\-ger-Verlag, 1989. \* \0{\bf [CD82] Calogero, F., Degasperis, A.:} {\it Spectral transform and solitons}, North Holland, 1982. \* \0{\bf [CV00] Chierchia, L., Valdinoci, E.:} {\it A note on the construction of Hamiltonian trajectories along heteroclinic chains}, Forum Mathematicum, {\bf12}, 247-255, 2000. \* \0{\bf [Fa98] Fass\`o, F.}: {\it Quasi-periodicity of motions and complete integrability of Hamiltonian systems}, Ergodic Theory and Dynamical Systems, {\bf 18}, 1349-1362, 1998. \* \0{\bf [Ga83] Gallavotti, G.:} {\it The elements of mechanics}, Sringer Verlag, New York, 1983. \* \0{\bf [GBG04] Gallavotti, G., Bonetto, F., Gentile, G.:} {\it Aspects of the ergodic, qualitative and statistical properties of motion}, Springer--Verlag, Berlin, 2004. \* \0{\bf [Ko55] N. Kolmogorov:} {\it On the preservation of conditionally periodic motions}, Doklady Akade\-mia Nauk SSSR, {\bf 96}, 527-- 530, 1954. \* \0{\bf [LL68] Landau, L.D., Lifshitz, E.M.}: {\it Mechanics}, Pergamon Press, 1976. \* \0{\bf [LC] Levi-Civita, T.} {\it Opere Matematiche}, Accademia Nazionale dei Lincei, Zani\-chelli, Bologna, 1956. \* \0{\bf [Mo62] J. Moser:} {\it On invariant curves of an area preserving mapping of the annulus}, Nachricten Akadenie Wissenschaften G\"ottingen, {\bf 11}, 1--20, 1962. \* \0{\bf [Ne77] Nekhorossev, V.} {\it An exponential estimate of the time of stability of nearly integrable Hamiltonian systems}, Russian Mathematical Surveys, 32 (6), 1 {\it65}, 1977. \* \0{\bf [Po] Poincar\'e, H.:} {\it M\'ethodes nouvelles de la m\'ecanique cel\`este}, Vol. I, Gauthier-Villars, reprinted by Gabay, Paris, 1987. \end{document} ---------------0504300644432 Content-Type: application/postscript; name="fig1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig1.ps" /punto { % x y punto gsave 2.4 0 360 newpath arc fill stroke grestore} def .7 .7 scale /L {150} def /h {xp xm sub 50 div 0.05 add} def 50 70 translate /armonico { 0 xm moveto xm h xp {/q exch def /p {E q q mul 2 div sub 2 mul abs sqrt} def p q lineto } for stroke clear 0 xm moveto xm h xp {/q exch def /p {E q q mul 2 div sub 2 mul abs sqrt neg} def p q lineto } for stroke clear } def /armonico1 { 0 xp moveto xp h neg xm {/q exch def /p {E q q mul 2 div sub 2 mul abs sqrt} def p q lineto } for stroke clear 0 xp moveto xp h neg xm {/q exch def /p {E q q mul 2 div sub 2 mul abs sqrt neg} def p q lineto } for stroke clear } def L L L 6 mul {/E exch def /xm {2 E mul sqrt neg } def /xp {2 E mul sqrt } def armonico armonico1} for 0 0 punto 110 0 translate /L {15} def /pendolo {%/h {xp xm sub 50 div 0.0001 add} def /z h /z stack pop pop pop 0 xm L mul moveto xm h xp {/x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul} def p q lineto} for stroke clear 0 xm L mul moveto xm h xp {/x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul neg} def p q lineto} for stroke clear } def /pendolo1 { 0 xp L mul moveto xp h neg xm {/x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul} def p q lineto} for stroke clear 0 xp L mul moveto xp h neg xm {/x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul neg} def p q lineto} for stroke clear } def /pendolo2 { /xm {3.1415 neg} def /xp {xm neg} def /q {L xm mul} def /p {180 3.1415 div xm mul cos E add 2 mul abs sqrt L mul} def p q moveto /ddd {xp xm sub 50 div} def xm ddd xp { /x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul} def p q lineto} for stroke clear} def /pendolo3 { /xm {3.1415 neg} def /xp {xm neg} def /q {L xm mul} def /p {180 3.1415 div xm mul cos E add 2 mul abs sqrt L mul} def p neg q moveto /ddd {xp xm sub 50 div} def xm ddd xp { /x exch def /q {L x mul} def /p {180 3.1415 div x mul cos E add 2 mul abs sqrt L mul} def p neg q lineto} for stroke clear} def -1 2 7 div 1 {/E exch def /xm {3.1415 180 div E neg arccos mul neg} def /xp {xm neg} def pendolo pendolo1 } for 1 1 2 div 2 {/E exch def pendolo2 pendolo3 } for 0 0 punto 0 3.1415 L mul punto 0 3.1415 L mul neg punto 70 -70 translate /keplero { /d {xp xm sub 50 div} def 0 xm L mul moveto xm d xp {/q exch def /p {1 q abs div 1 q q mul div sub E add 2 mul abs sqrt} def p H mul q L mul lineto} for stroke clear 0 xm L mul moveto xm d xp {/q exch def /p {1 q abs div 1 q q mul div sub E add 2 mul abs sqrt} def p H mul neg q L mul lineto} for stroke clear } def /keplero1 { /d {xp xm sub 80 div} def 0 xp L mul moveto xp d neg xm {/q exch def /p {1 q abs div 1 q q mul div sub E add 2 mul abs sqrt} def p H mul q L mul lineto} for stroke clear 0 xp L mul moveto xp d neg xm {/q exch def /p {1 q abs div 1 q q mul div sub E add 2 mul abs sqrt} def p H mul neg q L mul lineto} for stroke clear } def %/E {-0.07} def 45 -50 translate /L {40} def /H {100} def %1 2 scale /E1 {-.20} def /E2 {-.15} def /dd {E2 E1 sub 4 div} def E1 dd E2 {/E exch def /xp {-1 1 4 E mul add abs sqrt sub 2 div E div} def /xm {-1 1 4 E mul add abs sqrt add 2 div E div} def %xm xp /z stack clear keplero keplero1} for 0 2 L mul punto ---------------0504300644432 Content-Type: application/postscript; name="fig2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig2.ps" /punto { % x y punto gsave 1.5 0 360 newpath arc fill stroke grestore} def /linea { %x1 y1 x2 y2 gsave moveto lineto stroke grestore} def /tlinea {gsave 4 2 roll moveto lineto [4 4] 2 setdash stroke [] 0 setdash grestore} def /origine1assexper2pilacon|P_2-P_1| {% P1 P2 ori.... 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /ellisse0 {% Ro Rv ellisse0: Ro=raggio orizzontale Rv= r. vert. exch dup 0 moveto 0 1 360 { 3 1 roll dup 3 2 roll dup 5 1 roll exch 5 2 roll 3 1 roll dup 4 1 roll sin mul 3 1 roll exch cos mul exch lineto} for stroke pop pop} def /ellisse {% Ro Rv x y X Y gsave origine1assexper2pilacon|P_2-P_1| pop ellisse0 grestore} def /espressione{% ########1_########2 P /y exch def /x exch def /Symbols findfont 8 scalefont setfont x y moveto exch show x 5 add y 3 sub moveto /Symbols findfont 7 scalefont setfont show /Symbols findfont 7 scalefont setfont} def % esempio (a) (b) 100 100 espressione %/Times-Roman findfont 12 scalefont setfont gsave /o {0 0} def /a {40} def /e {0.75} def /S { a e mul 0} def /xx {40} def a a translate /b {a 1 e e mul sub sqrt mul} def [4 4] 2 setdash o a 0 360 arc stroke o b 0 360 arc stroke [] 0 setdash a b 0 0 10 0 ellisse /P {a xx cos mul b xx sin mul} def /P1 {a xx cos mul a xx sin mul} def /P2 {b xx cos mul b xx sin mul} def /Q- {a neg 0} def /Q+ {a 0} def o P1 tlinea P P1 linea P P2 linea Q- Q+ linea o 0.3 a mul 0 xx arc stroke (x) () 0.4 a mul xx 2 div cos mul 0.4 a mul xx 2 div sin mul espressione o punto P punto P1 punto P2 punto S punto /Times-Roman findfont 8 scalefont setfont o exch 4 sub exch 8 sub moveto (O) show S exch 0 sub exch 8 sub moveto (S) show P exch 4 sub exch 8 sub moveto (P) show Q+ exch 5 sub exch 30 sub moveto (e=0.75) show newpath 2 a mul 20 add 0 translate /r {a a b add 2 div sub} def /ang {40} def /xy {a b add 2 div ang cos mul a b add 2 div ang sin mul} def xy punto 0 0 punto 0 0 xy tlinea 0 0 a b add 2 div 0 linea xy r ang 40 sub ang 220 add arc stroke 0 0 a b add 2 div ang 30 add 360 ang add 30 sub arc stroke /E1 {b ang cos mul b ang sin mul} def /E2 {a ang cos mul a ang sin mul} def /EP {a ang cos mul b ang sin mul} def E1 EP linea E2 EP linea E1 punto E2 punto EP punto o 0.3 a mul 0 ang arc stroke /S {e a mul 0} def S punto /Times-Roman findfont 8 scalefont setfont 0 b moveto (D) show E2 exch 5 add exch moveto (E) show o exch 4 sub exch 8 sub moveto (O) show S exch 5 sub exch 8 sub moveto (S) show EP exch 4 sub exch 8 sub moveto (P) show Q+ exch 15 sub exch 30 sub moveto (e=0.75) show /Times-Roman findfont 7 scalefont setfont xy exch 1 sub exch 2 add moveto (c) show (x) () 0.4 a mul ang 2 div cos mul 0.4 a mul ang 2 div sin mul espressione newpath 2 a mul 20 add 0 translate /e {0.3} def a b 0 0 10 0 ellisse o punto S punto P punto S P linea Q- Q+ linea newpath S 0.2 a mul 0 58 arc stroke [4 4] 2 setdash %o a 0 360 arc stroke o b 0 360 arc stroke [] 0 setdash /Times-Roman findfont 8 scalefont setfont o exch 4 sub exch 8 sub moveto (O) show S exch 0 sub exch 8 sub moveto (S) show P exch 4 sub exch 12 sub moveto (P) show Q+ 30 sub moveto (e=0.3) show (q) () 0.5 a mul 58 2 div cos mul 5 add 0.5 a mul 58 2 div sin mul 5 sub espressione grestore ---------------0504300644432 Content-Type: application/postscript; name="fig3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig3.ps" %PS-Giovanni /linea {4 2 roll moveto lineto stroke} def /tlinea {1 setlinewidth [4 4] 0 setdash linea stroke [] 0 setdash} def /punto {newpath 3 0 360 arc closepath fill stroke } def /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /TR {/FO exch def /Times-Roman findfont FO scalefont setfont} def /SI {/FO exch def /Symbols findfont FO scalefont setfont} def /TRB {/FO exch def /Times-Roman-Bold findfont FO scalefont setfont} def /SIB {/FO exch def /Symbols-Bold findfont FO scalefont setfont} def /espressioner{% ####1_####2 P /y exch def /x exch def 14 TR x y moveto exch show x 6 add y 3 sub moveto 9 TR show} def /espressiones{% ####1_####2 P /y exch def /x exch def 12 SI x y moveto exch show x 7 add y 5 sub moveto 9 SI show} def % esempio (a) (b) 100 100 espressiones %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi /L {200} def /P1 {0 L 5 div} def /P2 {L .1 mul 0 } def /P3 {L .25 mul 0} def /P4 {L .8 mul L .5 mul} def /P5 {L 0.75 mul L 0.65 mul} def /P6 {L .35 mul L .50 mul} def /P7 {L 0.15 mul L .90 mul} def /P8 {L .35 mul L} def gsave %100 100 translate 0.5 0.5 scale P6 P1 linea P6 P1 dirpunta P6 P2 tlinea P6 P2 dirpunta P6 P3 linea P6 P3 dirpunta P6 P4 linea P6 P4 dirpunta P6 P5 linea P6 P5 dirpunta P6 P7 linea P6 P7 dirpunta P6 P8 linea P6 P8 dirpunta (i) (1) P3 exch 10 add exch 5 add espressioner (i) (2) P5 10 add espressioner (i) (3) P7 10 add espressioner (x) () P1 10 add espressioner (n) () P2 exch 5 sub exch 10 add espressioner (y) () P4 10 add espressioner (O) () P6 exch 5 add exch 10 add espressioner (z) () P8 exch 10 add exch espressioner %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi (j) (0) 35 60 espressiones (y) (0) 39 25 espressiones (q) (0) 54 143 espressiones /R {25} def newpath P6 R 90 118 arc stroke P6 R 220 243 arc stroke P6 R 2 mul 243 258 arc stroke grestore ---------------0504300644432 Content-Type: application/postscript; name="fig4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig4.ps" %PS! Giovanni /Times-Roman findfont 12 scalefont setfont /linea {4 2 roll moveto lineto stroke} def /tlinea {1 setlinewidth [4 4] 0 setdash linea stroke [] 0 setdash} def /tarc {1 setlinewidth [4 4] 0 setdash arc stroke [] 0 setdash} def /punto {newpath 3 0 360 arc closepath fill stroke } def /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /espressione{% ####1_####2 P /y exch def /x exch def /Symbols findfont 7 scalefont setfont x y moveto exch show x 5 add y 3 sub moveto /Symbols findfont 5 scalefont setfont show /Symbols findfont 7 scalefont setfont} def % esempio (a) (b) 100 100 espressione %/Times-Roman findfont 12 scalefont setfont %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi /L {200} def /P1 {-10 L .2 mul } def /P2 {L .25 mul 0 } def /P3 {L .45 mul 0} def /P4 {L .8 mul 0.8 mul L 0.05 mul 0.8 mul} def /P5 {L 0.90 mul 0.8 mul L 0.3 mul 0.8 mul exch 20 add exch -10 add} def /P6 {L .35 mul L .50 mul} def %/P7 {L 0.85 mul 0.9 mul L .85 mul 0.9 mul } def /P7 {L 0.85 mul 0.75 mul L .85 mul 1.0 mul exch 1.0 mul exch exch 20 add exch 30 sub} def /P8 {L .35 mul L} def /P9 {L .10 mul L 0.9 mul} def /P10 {0 L 1.1 mul 0.65 mul 1.1 mul } def /P66 {L .8 mul L .50 mul} def /P77 {136 170} def gsave 0.5 0.5 scale 20 10 translate P6 P1 linea P6 P1 dirpunta P6 P2 linea P6 P2 dirpunta %m P6 P3 linea P6 P3 dirpunta %n P6 P4 tlinea P6 P4 dirpunta P6 P5 linea P6 P5 dirpunta P6 P7 linea P6 P7 dirpunta P6 P8 linea P6 P8 dirpunta P6 P9 linea P6 P9 dirpunta P6 P10 linea P6 P10 dirpunta P6 P66 linea P6 P66 dirpunta /R {35} def P6 R 90 140 arc stroke P6 R 2 mul 90 122 arc stroke P6 R 1.5 mul 122 140 arc stroke P6 R 218 260 arc stroke %\g P6 R 1.5 mul 260 281 arc stroke %%%%%%%%%%%%%%%%%P6 R 303 325 tarc stroke P6 R 281 325 arc stroke %\ps P6 R 2.2 mul 303 325 tarc stroke P6 R 2 mul 217 260 arc stroke %\mathaccent "7016 \ps P6 R 2 mul 260 303 tarc stroke P6 P77 linea P6 P77 dirpunta /Times-Roman-Bold findfont 14 scalefont setfont P1 14 add moveto (x) show P66 10 add moveto (y) show P8 exch 10 add exch moveto (z) show P2 exch -35 add exch -1 add moveto (x=m) show P3 exch 3 add exch 10 add moveto (n) show P4 exch 8 add exch 5 add moveto (n) show P5 10 add moveto (1) show P6 exch 1 add exch 13 add moveto (O) show P7 10 add moveto (y) show P9 10 add moveto (3) show P10 exch -35 add exch 10 add moveto (M || z) show P77 10 add moveto (2) show /Times-Roman-Bold findfont 10 scalefont setfont P4 exch 16 add exch 0 add moveto (0) show P1 14 add exch 7 add exch -5 add moveto (0) show P66 exch 5 add exch 5 add moveto (0) show P8 exch 17 add exch -5 add moveto (0) show grestore (g) () 31 34 espressione (j) (0) 28 18 espressione (j) () 44 23 espressione (y) () 51 33 espressione (y) (0) 73 24 espressione (q) (0) 30 95 espressione (z) () 35 74 espressione (q) () 20 78 espressione %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi ---------------0504300644432 Content-Type: application/postscript; name="fig5.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig5.ps" /puntox {% x P1 P2 puntox : punto su segmento P1 P2 a frazione x /y2 exch def /x2 exch def /y1 exch def /x1 exch def /x exch def x1 x2 x1 sub x mul add y1 y2 y1 sub x mul add} def /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /puntatore {%P1 P2 puntatore (da P2 a P1) 4 copy 4 copy 4 copy 4 copy dirpunta 4 2 roll moveto lineto stroke} def /punto { % x y punto gsave 1.5 0 360 newpath arc fill stroke grestore} def /linea { %x1 y1 x2 y2 gsave 4 2 roll moveto lineto stroke grestore} def /Freccia{% xx XY1 XY22 : da P2 a P1 a distanza x /y2 exch def /x2 exch def /y1 exch def /x1 exch def /xx exch def /XY2 {x2 y2} def /XY1 {x1 y1} def XY2 xx XY1 XY2 puntox puntatore XY1 XY2 linea} def %/P1 {0 0} def %/P2 {100 100} def %/x {.5} def % %Freccia /linea { %x1 y1 x2 y2 gsave moveto lineto stroke grestore} def /tlinea {gsave 4 2 roll moveto lineto [4 4] 2 setdash stroke [] 0 setdash grestore} def /espressione{% ################1_################2 P /y exch def /x exch def /Symbols findfont 8 scalefont setfont x y moveto exch show x 5 add y 3 sub moveto /Symbols findfont 6 scalefont setfont show /Symbols findfont 8 scalefont setfont} def % esempio (a) (b) 100 100 espressione %/Times-Roman findfont 12 scalefont setfont %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi /TR {/FO exch def /Times-Roman findfont FO scalefont setfont} def /SI {/FO exch def /Symbols findfont FO scalefont setfont} def /TRB {/FO exch def /Times-Roman-Bold findfont FO scalefont setfont} def /L {90} def /H {80} def /P0 {20 H 2 div} def /P1 {L H 2 div} def /P2 {L 2 mul H} def /P3 {L 2 mul 0} def /P4 {3 5 div P1 P2 puntox} def /P6 {1 6 div P2 P3 puntox} def /P7 {2 6 div P2 P3 puntox} def /P8 {3 6 div P2 P3 puntox} def /P9 {4 6 div P2 P3 puntox} def /P10 {5 6 div P2 P3 puntox} def /P5 {0.5 P1 P7 puntox} def /P11 {0.5 P1 P3 puntox} def gsave %100 100 translate 0.5 P0 P1 Freccia 0.5 P4 P2 Freccia 0.5 P1 P4 Freccia 0.7 P4 P6 Freccia 0.8 P1 P5 Freccia 0.7 P5 P7 Freccia 0.7 P5 P8 Freccia 0.7 P5 P9 Freccia 0.7 P5 P10 Freccia 0.5 P1 P11 Freccia 0.5 P11 P3 Freccia P11 punto P1 punto P2 punto P3 punto P4 punto P5 punto P6 punto P7 punto P8 punto P9 punto P10 punto grestore ---------------0504300644432 Content-Type: application/postscript; name="fig6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig6.ps" %PS Giovanni /linea { %x1 y1 x2 y2 gsave moveto lineto stroke grestore} def /punto { % x y punto gsave 1 0 360 newpath arc fill stroke grestore} def %100 100 translate 30 30 translate /R {30} def /r {4} def /f {1 30 div} def R 2 div neg 10 R %2 div {/c exch def R 2 div neg dup dup mul 2 div f mul c add moveto R 2 div neg 1 R 1.5 mul {/x exch def x x x mul -2 div f mul c add R add lineto} for stroke} for R 2 div neg 10 R 1.5 mul {/x exch def x R 2 div neg x R 2. mul linea} for R 2 div neg 0 R 1.5 mul 0 linea R 1.3 mul R 2 div neg R 1.3 mul R 2 mul linea 2 setlinewidth 0 R 2 div neg 0 R 2 mul linea 1 setlinewidth /P0 { /x {R 4.2 div} def /c {R 5 div} def x x x mul -2 div f mul c add } def /P1 { /x {R 1. div} def /c {R 5 div} def x x x mul -2 div f mul c add } def .7 setgray P0 R add r 0 360 arc fill stroke P1 R add r 0 360 arc fill stroke 0. setgray P0 R add punto P1 R add punto /TR {/FO exch def /Times-Roman findfont FO scalefont setfont} def /SI {/FO exch def /Symbols findfont FO scalefont setfont} def /TRB {/FO exch def /Times-Roman-Bold findfont FO scalefont setfont} def /SIB {/FO exch def /Symbols-Bold findfont FO scalefont setfont} def /espressioner{% ################1_################2 P /y exch def /x exch def 7 TR x 5 add y 5.4 add moveto show 10 TR x y moveto show } def % esempio (a) (b) 100 100 espressiones %j=\varphi c=\ch q=\th w=\o y=\psi f=\phi (A)(f) R neg 5 R add espressioner (A)(i) 1.8 R mul 18 espressioner 0 R translate 120 0 translate /c {R 5 div} def /g {1} def R 2 div neg dup dup mul -2 div f mul c add moveto R 2 div neg % da 1 % a passi di R 1.5 mul % a {/x exch def x x x mul -2 div f mul x 50 mul sin g mul add c add lineto} for stroke 1 setlinewidth /P0 { /x {R 4.2 div} def /c {R 5 div} def x x x mul -2 div f mul c add } def /P1 { /x {R 1. div} def /c {R 5 div} def x x x mul -2 div f mul c add } def .7 setgray P0 r 0 360 arc fill stroke P1 r 0 360 arc fill stroke 0. setgray P0 punto P1 punto (A)(f) P0 15 neg add exch R 2 div sub exch espressioner (A)(i) P1 5 sub exch 7 add exch espressioner ---------------0504300644432--