Content-Type: multipart/mixed; boundary="-------------0802250401769" This is a multi-part message in MIME format. ---------------0802250401769 Content-Type: text/plain; name="08-35.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-35.keywords" lambda-symmetries; reduction procedures; dynamical systems ---------------0802250401769 Content-Type: application/x-tex; name="Rho.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Rho.tex" \documentclass[10pt]{article} \def\a{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} %% NON ridefinire come \d !!!! \def\eps{\varepsilon} \def\la{\lambda} \def\s{\sigma} \def\ze{\zeta} \def\om{\omega} \def\th{\theta} \def\vth{\vartheta} \def\vphi{\varphi} \def\De{\Delta} \def\La{\Lambda} \def\phi{\varphi} \def\cd{\cdot} %\def\xb{{\bf x}} %\def\yb{{\bf y}} %\newtheorem{theorem}{Theorem} \def\pd{\partial} \def\d{{\rm d}} %% derivative \def\na{\nabla} \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\~#1{\widetilde #1} \def\.#1{\dot #1} \def\^#1{\widehat #1} \def\beq{\begin{equation}} \def\eeq{\end{equation}} %\def\Am{{\mathcal A}} %\def\qd{{\dot q}} \def \ov{\over} %\def \bar{\overline} \def \lb{\label} \def\={\, =\, } %\def \id{\! :=} \def\dst{\displaystyle} %%%%%%%% abbrev \def \sy {symmetry} \def \sys {symmetries} \def \so {solution} \def \eq{equation} \def\ls{$\la$-\sy} \def\lss{$\la$-\sys} \def \qq{\qquad} \def \q{\quad} %\def \pn{\paragraph\noindent} \def \sk{\medskip} \def \ni{\noindent} \def\vf {vector field} \def\R{{\bf R}} \date{} \begin{document} %\parindent0pt \title{{\bf Reduction of systems of first-order differential equations via $\Lambda$-symmetries}} \author{ Giampaolo Cicogna\thanks{Email: cicogna@df.unipi.it} \\~\\ Dipartimento di Fisica ``E.Fermi'' dell'Universit\`a di Pisa\\ and Istituto Nazionale di Fisica Nucleare, Sez. di Pisa \\~\\ Largo B. Pontecorvo 3, Ed. B-C, I-56127, Pisa, Italy } \maketitle \ni{\bf Abstract} The notion of $\la$-symmetries, originally introduced by C. Muriel and J.L. Romero, is extended to the case of systems of first-order ODE's (and of dynamical systems in particular). It is shown that the existence of a symmetry of this type produces a reduction of the differential equations, restricting the presence of the variables involved in the problem. The results are compared with the case of standard (i.e. exact) Lie-point symmetries and are also illustrated by some examples. \bigskip \ni {\it PACS}: 02.20.Sv; 02.30.Hq \bigskip \ni {\it Keywords}: $\lambda$-symmetries; reduction procedures; dynamical systems \section{Introduction} It is a well known property that if an ordinary differential \eq\ (ODE) admits a Lie point-\sy , then the order of the \eq\ can be lowered by one (see e.g. \cite{Ol}). The notion of $\la$-\sy\ has been introduced in 2001 by Muriel and Romero \cite{MR1,MR2} with the main purpose of obtaining this reduction even in the absence of standard Lie \sys . The idea consists in introducing a suitable modification of the prolongation rules of the \vf\ in such a way that the lowering procedure still works, even if $\la$-\sys\ are not \sys\ in the proper sense, as they do not map in general \so s into \so s. $\la$-Symmetries are related to \sys\ of integral exponential type \cite{Ol,MR1,MR2}, to hidden and to some classes of potential \sys\ (see \cite{MR2,ASG,GMM} and references therein). The meaning of $\la$-prolongation has been clarified (together with a possible generalization of the procedure) by means of classical theory of characteristics of \vf s \cite{PS}. $\la$-Symmetries have been extended to partial differential \eq s \cite{GM,CGM} (and called in that context $\mu$-\sys ), and also interpreted in terms of a deformed Lie derivative in a more geometrical approach \cite{PMo}. A nontrivial relationship with nonlocal \sys\ has been recently pointed out \cite{DCF}; an interpretation in terms of appropriately defined changes of reference frames has been also proposed \cite{Gfr}. For the implications of $\la$- and $\mu$-\sys\ in Noether-type conservation rules see \cite{MRO,CG}. In the case of first-order ODE's, Lie \sys\ cannot lower the order of the \eq s, but they provide a sort of ``reduction'' of the complexity of the system, or -- more precisely -- a reduction of the number of the involved variables (see \cite{Ol}, Theorem 2.66). In this paper, we will restrict precisely to the case of {\it systems} of {\it first-order} ODE's (with usual regularity and nondegeneracy assumptions: see e.g. \cite{Ol,BA}), and of dynamical systems (DS) in particular, where the application of \lss\ requires some attention and where they exhibit some relevant peculiarities. We shall prove that some forms of reduction are allowed also in these cases. An application of \lss\ to systems of ODE's has been already considered in a particular case \cite{MR3}; in the present paper we want to examine more general situations. \section{Systems of ODE's} Let us recall first of all that in the case of a {\it single} dependent variable $u=u(t)$ (we shall always denote by $t\in\R$ the independent variable, according to its natural interpretation as the time variable in the case of DS), the first-order $\la$-prolongation $X_\la^{(1)}$ of a \vf\ $X$ \beq\lb{X0} X\=\tau(t,u){\pd\ov{\pd t}}+\phi(t,u){\pd\ov{\pd u}}\eeq is defined as \beq\lb{defXl} X_\la^{(1)}\=X^{(1)}+\la\, Q{\pd\ov{\pd\.u}}\eeq where $X^{(1)}$ is the {\it standard} prolongation \cite{Ol,BA}, $\.u=\d u/\d t$, $\la\=\la(t,u,\.u)$ is an arbitrary $C^\infty$ function, and $Q\=\phi-\tau \,\.u$. Considering systems of \eq s, and then $q>1$ dependent variables $u_a=u_a(t)$, the natural extension of definition (\ref{defXl}) is \beq\lb{defXL} X_\La^{(1)}\=X^{(1)}+(\La Q)_a{\pd\ov{\pd\.u_a}}\eeq where the sum over $a=1,\ldots,q$ is understood, with \beq\lb{X} X\=\tau{\pd\ov{\pd t}}+\phi_a{\pd\ov{\pd u_a}}\, , \q\q\q Q_a\=\phi_a-\tau\,\.u_a\eeq and where now $\La$ is a $q\times q$ matrix of $C^\infty$ functions depending on $t,u_a,\.u_a$. The case $\La=\la\,I$ is the one considered, in the context of DS and also for systems of ODE's of any order, by Muriel and Romero \cite{MR3}. Given a system of $q$ first-order ODE's (we shall assume for simplicity that the number of the \eq s is the same as the number of dependent variables $u_a(t)$) \beq\lb{Fa} F_a(t,u_b,\.u_b)\=0 \q\q\q a,b=1,\ldots,q\eeq we shall say that this system is $\La$-{\it symmetric} under a \vf\ $X$ if there is a matrix $\La$ such that \beq\lb{Lasy} X^{(1)}_\La\,F_a\big|_{F_a=0}\=0 \ .\eeq It is clear from (\ref{defXL}) that $\La$ is not uniquely defined: indeed, for any matrix $R$ such that $RQ=Q$ then also $\La'=\La R$ satisfies the above condition. This arbitrariness in the definition of $\La$, far from being disturbing, may be useful in practice, as it allows the choice of the more convenient matrix $\La$ in view of the given problem. We shall say that the system (\ref{Fa}) is $\La$-{\it invariant} under $X$ if there is a matrix $\La$ such that \beq X^{(1)}_\La\,F_a\=0 \ .\eeq It is not too restrictive to assume that the system of ODE's we are going to consider can be put into a $\La$-invariant form. Indeed, extending to $\La$-\sys\ a well known result \cite{Ol,PS}, it can be shown that if a system is $\La$-symmetric, then there exists a $q\times q$ matrix $A=A(t,u,\.u)$ such that $$X^{(1)}_\La\,F_a\=A_{ab}F_b \ .$$ It is now enough to prove, applying standard arguments (cf. e.g. \cite{CLlib,CDW,CL}), the existence of some some $q\times q$ invertible matrix $S$, possibly depending on $t,u,\.u$, such that $$X^{(1)}_\La\,S+S\,A\=0 $$ and the (locally) equivalent system $G_a\equiv S_{ab}F_b\=0$ turns out to be $\La$-invariant. We will consider in the following, unless otherwise stated, only $\La$-invariant systems. The matrix $\La$ plays the role of an additional ``unknown'' in the determining \eq s which are deduced from the $\La$-invariance condition \beq\tau{\pd F_a\ov{\pd t}}+\phi_b{\pd F_a\ov{\pd u_b}}+\(\phi_b^{(1)}+(\La Q)_b\){\pd F_a\ov{\pd \.u_b}}\=0\eeq where $\phi_b^{(1)}$ is the coefficient of the standard first-order prolongation, and which clearly strongly depend on the explicit form of the functions $F_a$. For instance, in the case where $F_a=\.u_a-f_a(t,u)$, i.e. the case of dynamical systems (see Section 3), these \eq s are $$\{\phi\, ,\, f\}_a\={\pd\ov{\pd t}}(\phi_a-\tau\,f_a)-{\pd\tau\ov{\pd u_b}}f_af_b+\La_{ab}Q_b \ , $$ where $\dst{\{\phi\, ,\, f\}_a=\phi_b{\pd f_a\ov{\pd u_b}}-f_b{\pd \phi_a\ov{\pd u_b}}}$, which clearly become \beq\lb{LPTI} \{\phi\, ,\, f\}_a\=(\La\,\phi)_a \eeq in the case of autonomous systems and time-independent \vf s $X$ with $\tau\equiv 0$ (see \cite{cglib}). \sk Let us now introduce ``\sy -adapted'' coordinates $w_a$ (sometimes also called canonical coordinates) characterized by the property of being invariant under the action of the \vf\ $X$: \beq\lb{defwa} X\,w_a\=\tau{\pd w_a\ov{\pd t}} + \phi_b{\pd w_a\ov{\pd u_b}}\=0 \ ;\eeq they are obtained through the associated characteristic \eq s %\ {\tt (Lagrange?)} $${\d t\ov{\tau}}\={\d u_a\ov{\phi_a}} \ .$$ In this way we introduce exactly $q$ new variables $w_a$; one at least of these, say $w_q$, will depend explicitly on $t$, and we will choose this as the new independent variable and call it $\eta$. In particular, if $\tau\equiv 0$, we can choose $\eta=t$. As $(q+1)$-th variable, which will be called $z$, we will take the coordinate ``along the action of $X$'', i.e. such that $X\,z=1$. Summarizing, the new set of variables is \beq\lb{wz} \eta,\,w_\a(\eta),\, z(\eta),\ \q\q\q \a=1,\ldots,q-1 \eeq (among these, $w_\a$ and $w_q\equiv\eta$ are invariant under $X$) and clearly do {\it not} depend on $\La$. We have now to write the given \vf\ $X$ and its first $\La$-prolongation (\ref{defXL}) in terms of these coordinates. We get first $X\=X^{(1)}\=\pd /\pd z$, and we then find that eq. (\ref{defXL}) takes the form \beq\lb{XtLa} X_\La^{(1)}\= {\pd\ov{\pd z}}+M_\a{\pd\ov{\pd w'_\a}}+M_q{\pd\ov{\pd z'}} \eeq where $w_\a'=\d w_\a/\d\eta,\, z'=\d z/\d\eta$ and (here and in the following the sum will be always understood over the repeated indices $\a=1,\ldots,q-1$ and $a=1,\ldots,q$; $D_t$ is the total derivative) \beq\lb{ma} M_\a\=(D_t\eta)^{-2}\Big(D_t\eta{\pd w_\a\ov{\pd u_a}}-D_tw_\a{\pd\eta\ov{\pd u_a}}\Big)(\La Q)_a \eeq \beq\lb{mq} M_q\=(D_t\eta)^{-2}\Big(D_t\eta{\pd z\ov{\pd u_a}}-D_tz{\pd\eta\ov{\pd u_a}}\Big)\ (\La Q)_a \ .\eeq In particular, if $\tau\equiv 0$, and then with $\eta=t$, we have more simply $${\pd w'_\a\ov{\pd\.u_a}}\={\pd w_\a\ov{\pd u_a}} \q\q , \q\q {\pd z'\ov{\pd \.u_a}}\={\pd z\ov{\pd u_a}}$$ and \beq\lb{mt0}M_\a\={\pd w_\a\ov{\pd u_a}}(\La\,Q)_a\q\q , \q\q M_q\={\pd z\ov{\pd u_a}}(\La\,Q)_a \ . \eeq The above expressions (\ref{ma},\ref{mq}) can be obtained either by direct calculation expressing by the chain rule the operators $\pd/\pd\.u_a$ in terms of $\pd/\pd w'_\a,\,\pd/\pd z'$, or -- more elegantly -- starting from the algebraic relation \beq\lb{XD} [X^{(1)}_\La,\,D_t]\=-D_t(\tau)D_t+(\La Q)_a{\pd\ov{\pd u_a}}\eeq which can be easily proved and generalizes to $\La$-\sys\ other similar known identities \cite{MR1,MR2}. From this, one directly gets indeed $$X_\La^{(1)}(w'_\a)\=X_\La^{(1)}\Big({D_tw_\a\ov{D_t\eta}}\Big)\= {X_\La^{(1)}(D_tw_\a)(D_t\eta)-(D_tw_\a)X_\La^{(1)}(D_t\eta)\ov{(D_t\eta)^2}}\=$$ \beq\lb{reob}\hskip-.5cm=(D_t\eta)^{-2}\Big(D_t\eta{\pd w_\a\ov{\pd u_a}}- D_tw_\a{\pd\eta\ov{\pd u_a}}\Big)(\La Q)_a\=M_\a\eeq thanks to $X^{(1)}_\La w_\a=X w_\a=0$, $X^{(1)}_\La\eta=X \eta=0$; similarly for $X_\La^{(1)}(z')$. \sk It can be interesting to point out that eq. (\ref{reob}) puts in clear evidence the difference with respect to exact \sys: indeed, starting from the $q$ ($X$-invariant) variables $w_\a,\, \eta$ one obtains $q-1$ first-order differential quantities $w'_\a$ which are invariant under $X^{(1)}$, but in general {\it not} under $X_\La^{(1)}$. \sk In turn, the given system of differential \eq s will take the form (we will use the $\,\~\cdot\, $ to denote the expressions in the new variables) \beq\lb{Ft}\~F_a(\eta,w_\a,w'_\a,z,z')\=0\eeq and the condition of its $\La$-invariance under $X$ now becomes \beq\lb{Xtinv}{\pd\~F_a\ov{\pd z}}+M_\a{\pd \~F_a\ov{\pd w'_\a}}+M_q{\pd\~F_a\ov{\pd z'}}\=0 \ .\eeq This allows us to state the following first form of reduction: \sk \ni {\bf Theorem 1}. {\it If the system (\ref{Fa}) is $\La$-invariant under a \vf\ $X$, then, once written in the \sy-adapted coordinates $\eta,w_\a,w'_\a,z,z'$, it turns out to depend on only $2q$ quantities (instead of $2q+1$): i.e. on the $q$ variables $w_\a,\,\eta$ and on other $q$ first-order differential $\La$-invariant quantities $\ze_a=\ze_a(\eta,z,w_\a,w'_\a,z')$ which are obtained from the characteristic \eq s \beq\lb{fodi} \d z\={\d w'_\a\ov{M_\a}}\={\d z'\ov{M_q}}\eeq coming from condition (\ref{Xtinv}).} \sk Examples 1 and 2 will illustrate this result. \section{The case of Dynamical Systems} Let us now consider the particularly important case of the dynamical systems, i.e. the systems of first-order ODE's which are written ``in explicit form'': \beq\lb{DS} \.u_a\=f_a(t,u) \ .\eeq Clearly, once \sy-adapted coordinates are introduced, the system becomes ``automatically'' a function of the $2q$ quantities $w_\a,\eta,\zeta_a$, as granted by Theorem 1. But it can be preferable or more convenient (e.g. in view of the physical interpretation in terms of ``evolution'' problem, or also if the explicit expression of the $\zeta_a$ is not known\footnote{If one is interested to know ``a priori'' the expressions of the $q$ differential $\La$-invariant quantities $\ze_a=\ze_a(\eta,z,w_\a,w'_\a,z')$, one has to express $M_\a,M_q$ in terms of $w_\a,\eta$ in order to solve (\ref{fodi}). }) to adopt a different point of view, i.e. to preserve the form of the system as an explicit DS, i.e. to rewrite it as follows \beq\lb{DStd} w'_\a\=\~f_\a(\eta,w,z) \eeq \beq\lb{DStz} \,\,\,z'\,\,\=\~f_q(\eta,w,z)\ . \eeq and to look for the dependence on $z$ of the r.h.s. This point of view will be elucidated by Examples 3 and 4. Recalling the expression (\ref{XtLa}) of the first $\La$-prolongation of $X$, we then easily deduce in this case: \sk \ni{\bf Theorem 2.} {\it If a DS is $\La$-invariant under $X$, the dependence on $z$ of the r.h.s. of eq.s (\ref{DStd},\ref{DStz}) is given by $${\pd \~f_\a\ov{\pd z}}\=M_\a\q\q ;\q\q {\pd \~f_q\ov{\pd z}}\=M_q \ .$$ Then, if for some $\overline{\a}$ one has $M_{\overline{\a}}=0$, the corresponding $\~f_{\overline{\a}}$ does not depend on $z$. If $M_\a=0$ for all $\a=1,\ldots,q-1$, then only $\~f_q$ depends on $z$ and the system splits into a system for the $q-1$ variables $w_\a=w_\a(\eta)$ and the last \eq\ (\ref{DStz}) which is an ODE for the variables $z$ and $\eta$.} \sk It is useful to compare the situation covered by Theorems 1 and 2 with the case of exact \sy : the difference is that in the case of exact \sy\ all terms of eq.\,(\ref{Ft}) are independent of $z$; the same is true for all the terms at the r.h.s. of (\ref{DStd},\ref{DStz}): then, in this case, the last \eq\ for $z$ and $\eta$ turns out to be a quadrature, as is well known \cite{Ol}. Clearly, if $\La=0$ i.e. if $X$ is an exact \sy , then $M_\a=M_q=0$. Conversely, it can be shown that if $M_\a=M_q=0$ then the \sy\ $X$ is exact. This is particularly clear in the case $\tau\equiv 0$ (and then $\eta=t$): indeed, in this case the conditions $M_\a=M_q=0$ can be written (see (\ref{mt0})) ${\bf J}_{ab}(\La\,Q)_b=0$ where ${\bf J}$ is the (invertible !) Jacobian matrix of the transformation from $u_a$ to $w_\a,z$. Then $\La\,Q=0$, which is the same as $\La=0$ (recall that $\La$ is not uniquely defined). Notice in particular that the term $(\La\,Q)_a$ appearing in the expressions (\ref{ma},\ref{mq}), when written in the new coordinates, becomes $$(\~\La\,\~Q)_a\=\~\La_{aq}$$ indeed $\~Q\equiv(0,0,\ldots,1)$. This shows that only the last column of $\~\La$ is relevant. \sk Finally, let us recall the following result: \sk \ni{\bf Theorem 3.} (Muriel-Romero \cite{MR3}) {\it If $\La=\la\,I$, then $M_\a=0$ for all $\a=1,\ldots,q-1$, and the conclusion of the last part of Theorem 2 holds.} \sk Indeed, from $X\,w_\a=X\,\eta=0$, $X\,z=1$ and the definition of $Q$, one easily deduces $$Q_a{\pd w_\a\ov{\pd u_a}}\=-\tau\,D_tw_\a \q\q ; \q\q Q_a{\pd \eta\ov{\pd u_a}}\=-\tau\,D_t\eta$$ and $$Q_a{\pd z\ov{\pd u_a}}\=-\tau\,D_t z+1$$ hence, in the case $(\La\,Q)_a=\la Q_a$ considered in \cite{MR3}, one gets $$M_\a\=0 \q\q ;\q\q M_q\=(D_t\eta)^{-2}\la \ .$$ Notice also that (\ref{XD}) becomes in this case $$ [X^{(1)}_\la,\,D_t]\=-D_t(\tau)D_t+ \la Q_a{\pd\ov{\pd u_a}}\=-D_t(\tau)D_t+\la X-\la\tau D_t \ . $$ \section{Examples} \sk\ni {\bf Example 1}. This is a very simple example, which can provide a clear illustration of Theorem 1. Consider any system $F_a(t,u_1,u_2,\.u_1,\.u_2)=0\ (a=1,2)$ of two first-order ODE's for the variables $u_1=u_1(t)\,,u_2=u_2(t)$ and consider the \vf\ $$X\={\pd\ov{\pd u_2}}\ .$$ It is easily seen that if one chooses $$ \La\=\pmatrix{0 & 1\cr 0& 1}$$ then, with our notation, $w_1=u_1,\, \eta=t$ and $z=u_2$; eq. (\ref{ma},\ref{mq}) give $M_1=M_2=1$ and therefore from (\ref{fodi}) $\zeta_1=\.w_1-z=\.u_1-u_2,\, \zeta_2=\.z-z=\.u_2-u_2$. Then, $\La$-invariance under $X$ gives that the $F_a$ depend only on the quantities $t,\,\.u_1-u_2,\,\.u_2-u_2$, in agreement with Theorem~1. Extension to more than 2 variables $u_a$ is immediate. \sk\ni {\bf Example 2}. Consider a system of ODE's for the two variables $u_1=u_1(t),\, u_2=u_2(t)$ of the form $$h(s_1,s_2)\,(\.u_1-u_1u_2)+a(t)(u_1^2+u_2^2)u_1+b^2(t)(u_1^2+u_2^2)u_2\=0$$ $$\hskip -.3cm h(s_1,s_2)\,(\.u_2+u_1^2)+a(t)(u_1^2+u_2^2)u_2-b^2(t)(u_1^2+u_2^2)u_1\=0$$ where $h$ is a function of $s_1=u_1\.u_1+u_2\.u_2,\, s_2=u_1\.u_2-\.u_1u_2+u_1^3+u_1u_2^2$ and where $a(t),\,b(t)$ are arbitrary functions of $t$; it is clearly not symmetric under the rotation operator $$X\=u_2{\pd\ov{\pd u_1}}-u_1{\pd\ov{\pd u_2}}$$ (unless $h\equiv 0$), however it turns out to be $\La$-symmetric (but not $\La$-invariant) under rotations if $\La=\la\,I$ with $\la=u_2$. Indeed, e.g., one has $X^{(1)}_\La\,h=~0$, $X^{(1)}_\La\,(\.u_1-u_1u_2)=(\.u_2+u_1^2)$, etc. Introducing \sy-adapted coordinates, which are $w_1=r=(u_1^2+u_2^2)^{1/2},\, \eta=t,\, z=\th$, with obvious notations, the system becomes %\vfill\eject $$\hskip-1.7cm \.r\, h(r\.r,r^2(\.\th+r\cos\th))+a(t)r^3\=0$$ $$(\.\th+r\cos\th)\,h(r\.r,r^2(\.\th+r\cos\th))-b^2(t)r^2\=0$$ which turns to be $\La$-invariant under $X=\pd/\pd\th$ with $\la=r\sin\th$. As expected, thanks to Theorem 1, this system contains only the four quantities $r,t$ and $\ze_1=\.r,\ze_2=\.\th+r\cos\th$. If, e.g., $h=s_2$, the system can be also put in the explicit form of a DS: $$\.r=\pm(a(t)/b(t))r \q\q\ \.\th\=\pm b(t)-r\cos\th$$ and -- as a consequence -- according to Theorems 2 and 3, one (and only one) of the above \eq s does not contain $z$ (here: $\th$). Then the system can be easily solved. \sk\ni {\bf Example 3}. Consider any DS for $u_a=u_a(t), \, a=1,2,3$, of the form $$\.u_1\=h_1(t,w_1,w_2)+(a-3b)u_2u_3+b\,u_3^3+h_2(t,w_1,w_2)\,u_3+h_3(t,w_1,w_2)\,u_3^2$$ $$\hskip-4.9cm \.u_2\=h_2(t,w_1,w_2)+2u_3h_3 (t,w_1,w_2)+au_3^2$$ $$\hskip-7.6cm \.u_3\=cu_3+2h_3(t,w_1,w_2)$$ where $a,b,c$ are constants and $h_a$ are functions of $t,\, w_1=2u_2-u_3^2,$ $w_2=3u_1-3u_2u_3+u_3^3$. Systems of this form are $\La$-invariant under the \vf $$X\=u_2{\pd\ov{\pd u_1}}+u_3{\pd\ov{\pd u_2}}+{\pd\ov{\pd u_3}}\q\q {\rm with} \q\q \La= \pmatrix{0 & 0 & (a-3b)u_2\cr 0&0&(2a-c)u_3\cr 0&0&c} \ .$$ The $X$-invariant quantities are just $w_1,w_2$, together with $\eta=t$. The coefficients $M_\a,M_q$ (see eq.s (\ref{ma},\ref{mq})), with $z=u_3$, are $$M_1\=4(a-c)z\q\q, \q\q M_3=c\q\q,$$ $$M_2\=3(a-3b-c)u_2-6(a-c)u_3^2=3(a-3b-c)(w_1+z^2)/2-6(a-c)z^2\ . $$ The characteristic \eq s (\ref{fodi}) can then be easily solved to obtain the three first-order differential $\La$-invariant quantities $\ze_1=\.w_1-2(a-c)z^2,\,\zeta_2=\.w_2-(3/2)(a-3b-c)w_1z-(3/2)(a+b-c)z^3,\,\zeta_3=\.z-c\,z$. Direct calculation shows that this system becomes $$\ze_a-g_a(t,w_1,w_2)\=0$$ where $g_1=2h_1,\,g_2=3h_1-3h_3w_1,\,g_3=2h_3$, and then contains only the quantities $t,w_1,w_2,\ze_a$, in agreement with Theorem 1. If instead one prefers to write the system as an explicit DS, then it is $$\hskip-4.1cm \.w_1\=2(a-c)z^2+g_1(t,w_1,w_2)$$ $$\.w_2\={3\ov 2}(a-3b-c)w_1z+{3\ov 2}(-a-b+c)z^3+g_2(t,w_1,w_2)$$ $$\hskip-4.8cm\.z\,\=c z+g_3(t,w_1,w_2)\ .$$ Now, if $a=c$, the first \eq\ does not contain $z$; if $a=c$ and $b=0$, only the third \eq\ contains $z$, in agreement with Theorem 2. If $a=c=0$, then only the second \eq\ contains $z$ and ``plays the role'' of the $q-$th \eq\ in our notation. The case $a=b=c=0$ is of course the case of exact \sy\ $\La=0$. \sk\ni {\bf Example 4}. This is an example with non-autonomous DS and \vf\ $X$ with $\tau\not=0$ and therefore $\eta\not=t$. Consider the DS for $u_a=u_a(t), \, a=1,2,3$, $$\hskip-4.5cm\.u_1=t+h_1(s,w_1,w_2)\,\exp(-\la_1 t)$$ $$\hskip-4.5cm\.u_2=1+h_2(s,w_1,w_2)\,\exp(-\la_2 t)$$ $$\.u_3=u_2+h_2\exp(-\la_2 t){1-\exp(u_2\big(\la_2-\la_3)\big)\ov{\la_3-\la_2}}+h_3\exp(-\la_3 u_2)$$ where $h_a$ are nonvanishing functions of $s=u_2-t,\, w_1=u_1-t^2/2,\, w_2=u_3-u_2^2/2$. This system is $\La$-invariant under $$X\={\pd\ov{\pd t}}+t{\pd\ov{\pd u_1}}+{\pd\ov{\pd u_2}}+u_2{\pd\ov{\pd u_3}}$$ with $\La={\rm diagonal} (\la_1,\la_2,\la_3)$. We can choose as invariants under $X$ just $w_1,w_2$ and $\eta=u_2-t$, with $z=u_2$. It is now more useful to rewrite the system in these coordinates preserving its form of explicit DS, we get then $$\hskip-4.5cm w'_1\=(h_1/h_2)\exp\((\la_2-\la_1)(z-\eta)\)$$ $$w'_2\=\Big({1-\exp(\la_2-\la_3)z\ov{\la_3-\la_2}}-z\Big)+(h_3/h_2)\exp\big(z(\la_2-\la_3)-\la_2\eta)\big)$$ $$\hskip-5.4cm z'\=(1/h_2)\exp\big(\la_2(z-\eta)\big) \ .$$ We see that if $\la_1=\la_2$, or $\la_2=\la_3$ (notice that the case $\la_2=\la_3$ is well defined), one of the above \eq s does not contain $z$ in agreement with Theorem 2; whereas if $\la_1=\la_2=\la_3$ only one \eq\ contains $z$, as stated by Theorem 3; the same happens also if $\la_1=\la_2=0$, according to Theorem 2. \section{Concluding remarks} An interesting property which relates invariance with \lss\ is the following. Consider the case of an autonomous DS $\.u_a=f_a(u)$ which is $\La$-invariant under a \vf\ $X$ of the form $X=\phi_a(u)\pd/\pd u_a$. If $w=w(u)$ is any invariant under $X$, i.e. $\phi_a\pd w/\pd u_a$, then its {\it Lie derivative along} $f_a$, i.e. $$D_t^{(f)}\,w\equiv f_a{\pd w\ov{\pd u_a}}$$ satisfies $$X\,\(D_t^{(f)}\,w\)\=\Big(\phi_a{\pd\ov{\pd u_a}}\Big)\Big(f_b{\pd\ov{\pd u_b}}\Big)\,w\=\La_{ab}\phi_b{\pd w\ov{\pd u_a}} $$ hence $X\(D_t^{(f)}\,w\)=0$ if $\La=\la\,I$, having used the commutation rule (\ref{LPTI}) and the invariance property of $w$. On the other hand, eq. (\ref {reob}) gives directly, for $X$ of the above form, $$X^{(1)}_\La\,(D_t\,w)\=(\La\phi)_a{\pd w\ov{\pd u_a}} \ .$$ The strong difference is that the latter result is purely algebraic, being a consequence of the relation (\ref{XD}), and expresses a property of the \vf\ $X$ which holds {\it independently} of the presence of any DS (i.e., of any choice of the functions $f_a$). The former result, instead, states that the time evolution under the dynamics described by the DS $\.u_a=f_a$ of a quantity $w(u)$ which is invariant under a \vf\ $X$ {\it preserves} this invariance {\it even if $X$ is not} a (standard) \sy\ of the DS; it is enough to require that $X$ is a $\la$-\sy\ of the DS. It can be noticed that the present statement, concerning Lie derivatives, can be suitably extended to the case of several \vf s $X$ for the given DS (see \cite{HW}, Prop. 2.1). \sk Several other aspects of \lss\ (and of all their generalizations as well) could be further investigated. Apart from their geometrical interpretation (see the papers quoted in the Introduction), their action on changes of coordinates should be better understood, as well as their general role in finding solutions of differential \eq s which do not admit standard \sys : see e.g. \cite{MR1,MR2,GMM,PS} and the references therein; compare also, for instance, with \cite{IF}, for what concerns the problem of finding integrating factors for ODE's and its relationship with \sy\ properties. \sk It can be observed, finally, that any ODE $\De(t,u,\.u,\ddot u,\ldots)=0$ of arbitrary order~$>1$ can be transformed into a system of first-order ODE's, and therefore our results could be applied also to this case. This is true in principle: the only nearly obvious remark is that one has to consider no longer \vf s of the form $X=\tau(\pd/\pd t)+\phi(\pd/\pd u)$ involving only the two variables $t$ and $u$, but also extended \vf s $X=\tau(\pd/\pd t)+\phi_{(0)}(\pd/\pd u)+\phi_{(1)}(\pd/\pd \.u)+ \phi_{(2)}(\pd/\pd \ddot u)+\ldots$. It is ``conceptually'' different to look for \vf s of the former or of the latter form; on the other hand, the ``concrete effect'' of the existence of a \sy\ is different in the two contexts (i.e., lowering the order in the case of the ODE's, and respectively reducing the presence of the involved variables in the case of first-order systems, as shown). This holds in particular for $\la$- and $\La$-\sys , where also the prolongation rules of the \vf s are markedly different in the two cases. To emphasize this different role of $\La$-\sys\ in the context of first-order systems, it should be perhaps more appropriate to call them $\rho$-\sys\ (where $\rho$ stands for ``reducing'', in contrast with $\la$, which could stand for ``lowering''). \section*{Acknowledgments} It is a pleasure to thank Giuseppe Gaeta and Diego Catalano Ferraioli for detailed discussions and useful comments. Thanks are also due to one of the referees for his/her accurate and appropriate comments and suggestions, which helped me to improve the presentation of these results. %\vfill\eject \sk\sk %\section{} \begin{thebibliography}{99} %\small \bibitem{Ol} P.J. Olver, Application of Lie groups to Differential Equations, Springer, Berlin 1986 \bibitem{MR1} C. Muriel, J.L. Romero, IMA J. Appl. Math. 66 (2001) 111, and J. Appl. Math. 66 (2001) 477 \bibitem{MR2} C. Muriel, J.L. Romero, Theor. Math. Phys. 133 (2002) 1565; and J. Lie theory 13 (2003) 167 \bibitem{ASG} B. Abraham-Shrauner, A. 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