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b(con\014ning)g(the)g(motion)h(in)e(a)i(giv)m(en)g(region) f(of)h(the)g(phase{)212 4719 y(space.)212 4832 y(In)28 b Fl(x)p Fm(3)h(w)m(e)g(will)c(b)s(e)j(concerned)g(with)f(the)i (existence)g(of)f(p)s(erio)s(dic)e(orbits,)i(whic)m(h)f(corresp)s(ond)g (to)i(commen-)212 4945 y(surabilities)j(b)s(et)m(w)m(een)k(the)g (rotational)g(and)f(rev)m(olutional)f(p)s(erio)s(ds)f(of)j(the)g (satellite.)f(In)g(particular,)f(w)m(e)212 5058 y(concen)m(trate)d(on)e (the)g(sync)m(hronous)f(resonance)h(\(i.e.,)h(when)e(the)h(t)m(w)m(o)h (p)s(erio)s(ds)d(coincide\))h(and)g(on)h(the)g(3:2)212 5170 y(resonance)i(\(suc)m(h)f(that)h(t)m(wice)g(the)f(p)s(erio)s(d)e (of)j(rev)m(olution)e(equals)h(three)h(times)e(the)i(p)s(erio)s(d)d(of) i(rotation\).)212 5283 y(Most)i(of)f(the)g(satellites)f(are)h(observ)m (ed)g(to)g(mo)m(v)m(e)h(in)e(the)h(sync)m(hronous)e(resonance,)j(while) d(only)h(Mercury)212 5396 y(has)i(b)s(een)f(observ)m(ed)h(in)e(the)i (3:2)h(spin{orbit)d(resonance.)i(The)g(p)s(eculiarit)m(y)d(of)j (Mercury)g(ma)m(y)g(b)s(e)f(due)h(to)2056 5915 y(1)p eop %%Page: 2 2 2 1 bop 212 337 a Fm(its)30 b(large)h(eccen)m(tricit)m(y)h(in)d (comparison)h(to)h(the)g(other)g(satellites)f(of)g(the)h(solar)f (system)h(\(see)h([2],)f([6)q(]\).)g(P)m(e-)212 450 y(rio)s(dic)e (orbits)i(are)g(obtained)g(through)g(an)g(implemen)m(tation)f(of)h(a)h (constructiv)m(e)g(v)m(ersion)e(of)i(the)f(implicit)212 562 y(function)36 b(theorem)i(com)m(bined)e(with)g(p)s(erturbation)f (theory)-8 b(.)38 b(The)f(existence)h(of)f(the)h(sync)m(hronous)e(and) 212 675 y(3:2)c(resonances)e(are)h(obtained)f(for)g(ph)m(ysically)e (relev)-5 b(an)m(t)31 b(v)-5 b(alues)30 b(of)g(the)h(parameters)f (\(see)i([5]\).)212 788 y(In)j Fl(x)p Fm(4)i(w)m(e)f(concen)m(trate)i (on)e(the)g(in)m(v)-5 b(arian)m(t)35 b(rotational)g(tori,)h(with)f(the) h(aim)f(to)h(pro)m(v)m(e)h(the)f(existence)g(of)212 901 y(a)h(pair)f(of)g(surfaces)h(whic)m(h)e(trap)i(a)g(giv)m(en)f(p)s(erio) 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b(consider)e(a)h(triaxial)e(satellite)i Fk(S)5 b Fm(,)31 b(for)f(example)h(the)g(Mo)s(on,)g(rotating)g(ab)s(out)g(an)g(in)m (ternal)e(spin{axis)212 2181 y(and)i(orbiting)e(around)h(a)h(cen)m (tral)h(planet)e Fk(P)13 b Fm(,)31 b(the)h(Earth.)f(As)g(is)f(w)m(ell)g (kno)m(wn,)g(the)i(Mo)s(on)f(alw)m(a)m(ys)g(p)s(oin)m(ts)212 2294 y(the)c(same)g(face)h(to)f(the)g(Earth,)f(due)g(to)i(the)f(fact)g (that)h(its)e(p)s(erio)s(d)e(of)j(rotation)g(coincides)e(with)h(the)h (p)s(erio)s(d)212 2407 y(of)38 b(rev)m(olution)f(around)g(the)h(Earth:) g(one)g(sa)m(ys)h(that)f(the)h(Mo)s(on)f(is)f(trapp)s(ed)f(in)h(a)h (1:1)h(or)f(sync)m(hronous)212 2520 y(spin-orbit)27 b(resonance.)j (More)g(generally)-8 b(,)29 b(denoting)f(b)m(y)i Fk(T)2253 2534 y Fj(r)r(ev)2389 2520 y Fm(and)f Fk(T)2618 2534 y Fj(r)r(ot)2745 2520 y Fm(the)g(satellite's)g(rev)m(olutional)f(and) 212 2633 y(rotational)j(p)s(erio)s(ds,)d(one)i(has)h(the)f(follo)m (wing)212 2805 y Fi(De\014nition.)35 b Fm(A)c(spin{orbit)d(resonance)j 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y Fh(1)538 4926 y Fk(;)15 b(F)636 4940 y Fh(2)676 4926 y Fm(\))31 b(as)955 5176 y Fk(F)1013 5190 y Fh(1)1053 5176 y Fm(\()p Fk(X)r(;)15 b(Y)21 b Fm(;)15 b Fk(")1361 5138 y Fh(2)1401 5176 y Fm(\))83 b Fl(\021)1673 5061 y Fe(Z)1756 5087 y Fh(2)p Fj(\031)r(q)1719 5249 y Fh(0)1872 5176 y Fm(\()p Fk(Y)41 b Fm(+)20 b Fk("y)2179 5190 y Fh(1)2218 5176 y Fm(\()p Fk(t)p Fm(\))h(+)f Fk(")2475 5138 y Fh(2)2515 5176 y Fk(\021)s Fm(\()p Fk(t)p Fm(\)\))p Fk(dt)h Fl(\000)f Fm(2)p Fk(\031)s(p)955 5414 y(F)1013 5428 y Fh(2)1053 5414 y Fm(\()p Fk(X)r(;)15 b(Y)21 b Fm(;)15 b Fk(")1361 5377 y Fh(2)1401 5414 y Fm(\))83 b Fl(\021)1673 5299 y Fe(Z)1756 5325 y Fh(2)p Fj(\031)r(q)1719 5488 y Fh(0)1887 5414 y Fk(g)s Fm(\()p Fk(X)29 b Fm(+)20 b Fk(Y)g(t)g Fm(+)g Fk("x)2474 5428 y Fh(1)2514 5414 y Fm(\()p Fk(t)p Fm(\))g(+)g Fk(")2770 5377 y Fh(2)2810 5414 y Fk(\030)t Fm(\()p Fk(t)p Fm(\))p Fk(;)15 b(t)p Fm(\))p Fk(dt)32 b(:)2056 5915 y Fm(5)p eop %%Page: 6 6 6 5 bop 212 337 a Fm(By)38 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Fm(cos)22 b(~)-50 b Fk(x)20 b Fl(\000)g Fk("e)2584 5116 y Fm(~)2564 5140 y Fk(f)10 b Fm(\()c(~)-51 b Fk(x;)15 b(t)p Fm(\))212 5329 y(for)23 b(some)h(function)934 5305 y(~)914 5329 y Fk(f)35 b Fl(\021)1109 5305 y Fm(~)1090 5329 y Fk(f)9 b Fm(\()d(~)-51 b Fk(x;)15 b(t)p Fm(\).)25 b(Next,)f(w)m(e)g(expand)f(in)f(T)-8 b(a)m(ylor)23 b(series)g(around)g (the)g(unp)s(erturb)s(ed)d(equilib-)212 5442 y(rium)25 b(p)s(osition)g(and)h(w)m(e)h(diagonalize)f(the)h(time{indep)s(enden)m (t)e(quadratic)h(terms,)h(obtaining)e(a)i(harmonic)2056 5915 y(8)p eop %%Page: 9 9 9 8 bop 212 337 a Fm(oscillator)34 b(p)s(erturb)s(ed)f(b)m(y)i(higher)f (order)g(terms.)h(W)-8 b(e)37 b(compute)e(action{angle)h(v)-5 b(ariables)34 b(\()p Fk(I)7 b(;)15 b(')p Fm(\))36 b(for)f(the)212 450 y(harmonic)29 b(oscillator,)h(obtaining)f(the)i(Hamiltonian)1358 654 y Fk(H)7 b Fm(\()p Fk(I)g(;)15 b(';)g(t)p Fm(\))27 b Fl(\021)e Fk(!)s(I)j Fm(+)19 b Fk(")p 2113 581 53 4 v(h)q Fm(\()p Fk(I)7 b Fm(\))21 b(+)f Fk("R)q Fm(\()p Fk(I)7 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b(by)e(A.)g(N.)g(Kolmo)-5 b(gor)g(ov)35 b(on)e(the)f(invarianc)-5 b(e)33 b(of)g(quasi-p)-5 b(erio)g(dic)354 2546 y(motions)39 b(under)g(smal)5 b(l)39 b(p)-5 b(erturb)g(ations)41 b(of)e(the)g(Hamiltonian)p Fm(,)f(Russian)d(Math.)j(Surv)m(ey)e Fi(18)h Fm(\(1963\),)354 2659 y(13-40.)212 2847 y([2])47 b(A.)36 b(Celletti,)g Fa(A)n(nalysis)j(of)f(r)-5 b(esonanc)g(es)40 b(in)f(the)f(spin-orbit)h (pr)-5 b(oblem)40 b(in)f(Celestial)g(Me)-5 b(chanics:)38 b(The)354 2960 y(synchr)-5 b(onous)33 b(r)-5 b(esonanc)g(e)33 b(\(Part)g(I\))p Fm(,)c(Journal)f(of)h(Applied)d(Mathematics)31 b(and)d(Ph)m(ysics)g(\(ZAMP\))i Fi(41)354 3073 y Fm(\(1990\),)j (174-204.)212 3260 y([3])47 b(A.)32 b(Celletti,)g Fa(Construction)k(of) f(libr)-5 b(ational)36 b(invariant)g(tori)f(in)g(the)g(spin-orbit)g(pr) -5 b(oblem)p Fm(,)34 b(Journal)d(of)354 3373 y(Applied)c(Mathematics)32 b(and)e(Ph)m(ysics)f(\(ZAMP\))i Fi(45)g Fm(\(1993\),)i(61-80.)212 3561 y([4])47 b(A.)37 b(Celletti,)e(and)h(L.)h(Chierc)m(hia,)e Fa(A)j(Constructive)h(The)-5 b(ory)40 b(of)f(L)-5 b(agr)g(angian)40 b(T)-7 b(ori)39 b(and)g(Computer-)354 3674 y(assiste)-5 b(d)33 b(Applic)-5 b(ations)p Fm(,)32 b(Dynamics)d(Rep)s(orted)g (\(C.K.R.T.)h(Jones,)g(U.)g(Kirc)m(hgrab)s(er,)e(H.O.)i(W)-8 b(alther)354 3787 y(Managing)30 b(Editors\),)g(Springer-V)-8 b(erlag,)30 b Fi(4)g Fm(\(New)h(Series\))f(\(1995\),)j(60-129.)212 3974 y([5])47 b(A.)27 b(Celletti,)g(and)g(C.)g(Chierc)m(hia,)f Fa(Construction)31 b(of)f(stable)h(p)-5 b(erio)g(dic)31 b(orbits)g(for)g(the)f(spin{orbit)h(pr)-5 b(ob-)354 4087 y(lem)32 b(of)h(Celestial)h(Me)-5 b(chanics)p Fm(,)31 b(Reg.)g(Chaotic)f(Dynamics)g Fi(3)p Fm(,)h(n.)f(3)h(\(1998\),)i (107-121.)212 4275 y([6])47 b(A.)37 b(Celletti,)f(and)g(C.)g(F)-8 b(alcolini,)36 b Fa(Construction)k(of)f(invariant)h(tori)f(for)g(the)g (spin-orbit)g(pr)-5 b(oblem)40 b(in)354 4388 y(the)33 b(Mer)-5 b(cury-Sun)32 b(system)p Fm(,)f(Celestial)e(Mec)m(h.)j(Dynam.) f(Astronom.)f Fi(53)h Fm(\(1992\),)i(113-127.)212 4575 y([7])47 b(A.)29 b(N.)h(Kolmogoro)m(v,)g Fa(On)i(the)g(c)-5 b(onservation)33 b(of)f(c)-5 b(onditional)5 b(ly)34 b(p)-5 b(erio)g(dic)33 b(motions)g(under)f(smal)5 b(l)33 b(p)-5 b(er-)354 4688 y(turb)g(ation)34 b(of)f(the)g(Hamiltonian)p Fm(,)f(Dokl.)e(Ak)-5 b(ad.)31 b(Nauk.)g(SSR)e Fi(98)i Fm(\(1954\),)i(527-530.)212 4876 y([8])47 b(J.)41 b(Moser,)i Fa(On)g(invariant)i(curves)e(of)g(ar)-5 b(e)g(a-pr)g(eserving)46 b(mappings)f(of)e(an)h(annulus)p Fm(,)f(Nac)m(h.)g(Ak)-5 b(ad.)354 4989 y(Wiss.)29 b(G\177)-45 b(ottingen,)32 b(Math.)f(Ph)m(ys.)f(Kl.)g(I)s(I)f Fi(1)i Fm(\(1962\),)i(1-20.)2056 5915 y(9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF