INTRODUCTION TO SYMPLECTIC MECHANICS: LECTURE IV. Maurice de Gosson

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1 INTRODUCTION TO SYMPLECTIC MECHANICS: LECTURE IV Maurice e Gosson

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3 4 Hamiltonian Mechanics Physically speaking, Hamiltonian mechanics is a paraphrase (an generalization!) of Newton s secon law, popularly expresse as force equals mass times acceleration 1. The symplectic formulation of Hamiltonian mechanics can be retrace (in embryonic form) to the work of Lagrange between 1808 an 1811; what we toay call Hamilton s equations were in fact written own by Lagrange who use the letter H to enote the Hamiltonian to honor Huygens 2 not Hamilton, who was still in his early chilhoo at that time! It is however unoubtely Hamilton s great merit to have recognize the importance of these equations, an to use them with great e ciency in the stuy of planetary motion, an of light propagation. 4.1 Hamiltonian ows We will call Hamiltonian function (or simply Hamiltonian ) any real function H 2 C 1 (R 2n z R t ) (although most of the properties we will prove remain vali uner the assumption H 2 C k (Rz 2n R t ) with k 2: we leave to the reaer as an exercise in orinary i erential equations to state minimal smoothness assumptions for the valiity of our results). The Hamilton equations _x j (t) pj H(x(t); p(t); t), _p j (t) xj H(x(t); p(t); t); (1) associate with H are form a (generally non-autonomous) system of 2n ifferential equations. The conitions of existence of the solutions of Hamilton s equations, as well as for which initial points they are e ne, are etermine by the theory of orinary i erential equations (or ynamical systems, as it is now calle). The equations (1) can be written economically as _z = J@ z H(z; t) (2) where J is the stanar symplectic matrix. De ning the Hamilton vector el by X H = J@ z H = (@ p x H) (3) 1 This somewhat unfortunate formulation is ue to Kirchho. 2 See Lagrange s Mécanique Analytique, Vol. I, pp an

4 2 (the operator z is often calle the symplectic graient), Hamilton s equations are equivalent to (X H (z; t); ) + z H = 0: (4) In fact, for every z 0 2 R 2n z, (X H (z; t); z 0 ) = h@ x H(z; t); x 0 i h@ p H(z; t); p 0 i = h@ z H(z; t); z 0 i which is the same thing as (4). This formula is the gate to Hamiltonian mechanics on symplectic manifols. In fact, formula (4) can be rewritten concisely as i XH + H = 0 (5) where i XH (;t) is the contraction operator: i XH (;t)(z)(z 0 ) = (X H (z; t); z 0 ): The interest of formula (5) comes from the fact that it is intrinsic (i.e. inepenent of any choice of coorinates), an allows the e nition of Hamilton vector els on symplectic manifols: if (M;!) is a symplectic manifol an H 2 C 1 (M R t ) then, by e nition, the Hamiltonian vector el X H (; t) is the vector el e ne by (5). One shoul be careful to note that when the Hamiltonian function H is e ectively time-epenent (which is usually the case) then X H is not a true vector el, but rather a family of vector els on R 2n z epening smoothly on the parameter t. We can however e ne the notion of ow associate to X H : De nition 1 Let t 7! z t be the solution of Hamilton s equations for H passing through a point z at time t = 0, an let ft H be the mapping R 2n z! R 2n z e ne by ft H (z) = z t. The family (ft H ) = (ft H ) t2r is calle the ow etermine by the Hamiltonian function H or the ow etermine by the vector el X H. A caveat: the usual group property f H 0 = I, f H t f H t 0 = f H t+t 0, (f H t ) 1 = f H t (6) of ows only hols when H is time-inepenent; in general ft H ft H 6= f H 0 t+t an 0 (ft H ) 1 6= f H t (but of course we still have the ientity f0 H = I). For notational an expository simplicity we will implicitly assume (unless otherwise speci e) that for every z 0 2 R 2n z there exists a unique solution t 7! z t of the system (2) passing through z 0 at time t = 0. The moi cations to iverse statements when global existence (in time or space) oes not hol are rather obvious an are therefore left to the reaer. As we note in previous subsection the ow of a time-epenent Hamiltonian vector el is not a one-parameter group; this fact sometimes leas to technical complications when one wants to perform certain calculations. For this reason it is helpful to introuce two (relate) notions, those of suspene Hamilton

5 3 ow an time-epenent Hamilton ow. We begin by noting that Hamilton s equations _z = J@ z H(z; t) can be rewritten as where t (z(t); t) = ~ X H (z(t); t) (7) ~X H = (J@ z H; 1) = (@ p x H; 1). (8) De nition 2 (i) The vector el ~ X H on the extene phase space R 2n+1 z;t R 2n z R t is calle the suspene Hamilton vector el ; its ow ( f ~ t H ) is calle the the suspene Hamilton ow etermine by H. (ii) The two-parameter family of mappings R 2n z! R 2n z e ne by the formula (ft;t H 0(z0 ); t) = f ~ t H t 0 (z0 ; t 0 ) (9) is calle the time-epenent ow etermine by H. Notice that by e nition ~ f H t thus satis es t ~ f H t = ~ X H ( ~ f H t ). (10) The point with introucing X ~ H is that it is a true vector el on extene phase-space the while X H is, as pointe out above, rather a family of vector els parametrize by t as soon as H is time-epenent. The system (10) being autonomous in its own right, the mappings f ~ t H satisfy the usual group properties: ~f t H f ~ t H = f ~ H 0 t+t, ( f ~ H 0 t ) 1 = f ~ H t, f ~ H 0 = I: (11) Notice that the time-epenent ow has the following immeiate interpretation: ft;t H is the mapping 0 R2n z! R 2n z which takes the point z 0 at time t 0 to the point z at time t, the motion occurring along the solution curve to Hamilton equations _z = J@ z H(z; t) passing through these two points. Formula (9) is equivalent to ~f H t (z 0 ; t 0 ) = (f H t+t 0 ;t 0(z0 ); t + t 0 ). (12) Note that it immeiately follows from the group properties (11) of the suspene ow that we have: f H t;t 0 = I, f H t;t 0 f H t 0 ;t 00 = f H t;t 00, (f H t;t 0) 1 = f H t 0 ;t (13) for all times t; t 0 an t 00. When H oes not epen on t we have ft;t H = f H 0 t t0; in particular ft;0 H = ft H. Let H be some (possibly time-epenent) Hamiltonian function an ft H = ft;0. H We say that ft H is a free symplectomorphism at a point z 0 2 R 2n z if Dft H (z 0 ) is a free symplectic matrix. Of course ft H is never free at t = 0 since f0 H is the ientity. In Proposition 4 we will give a necessary an su cient conition for the symplectomorphisms ft;t H 0 to be free. Let us rst prove the following Lemma, the proof of which makes use of the notion of generating function:

6 4 Lemma 3 The symplectomorphism f : R 2n z! R 2n z is free in a neighborhoo U of z 0 2 R 2n z if an only if Df(z 0 ) is a free symplectic matrix for z 0 2 U, that is, if an only if et(@x=@p 0 ) 6= 0. Proof. Set z = f(z 0 ); we have Df(z 0 6 ) (z0 0 (z0 ) 0 (z0 ) an the symplectic matrix Df(z 0 ) is thus free for z 0 2 U if an only 0 (z0 ) 6= 0: We next make the following crucial observation: since f is a symplectomorphism we have p ^ x = p 0 ^ x 0 an this is equivalent, by Poincaré s lemma to the existence of a function G 2 C 1 (R 2n z ) such that px = p 0 x 0 + G(x 0 ; p 0 ): Assume now that Df(z 0 ) is free for z 0 2 U; then the conition et(@x=@p 0 ) 6= 0 implies, by the implicit function theorem, that we can locally solve the equation x = x(x 0 ; p 0 ) in p 0, so that p 0 = p 0 (x; x 0 ) an hence G(x 0 ; p 0 ) is, for (x 0 ; p 0 ) 2 U, a function of x; x 0 only: G(x 0 ; p 0 ) = G(x 0 ; p 0 (x; x 0 )). Calling this function W : we thus have W (x; x 0 ) = G(x 0 ; p 0 (x; x 0 )) px = p 0 x 0 + W (x; x 0 ) = p 0 x 0 x W (x; x 0 )x x 0W (x; x 0 )x 0 which requires p x W (x; x 0 ) an p 0 x 0W (x; x 0 ) an f is hence free in U. The proof of the converse goes along the same lines an is therefore left to the reaer. We will use the notations H pp ; H xp, an H xx for the matrices of secon erivatives of H in the corresponing variables. Proposition 4 There exists " > 0 such that ft H is free at z 0 2 R 2n z for 0 < jt t 0 j " if an only if et H pp (z 0 ; t 0 ) 6= 0. In particular there exists " > 0 such that ft H (z 0 ) is free for 0 < jtj " if an only if et H pp (z 0 ; 0) 6= 0. Proof. Let t 7! z(t) = (x(t); p(t)) be the solution to Hamilton s equations _x p H(z; t), _p x H(z; t) with initial conition z(t 0 ) = z 0. A secon orer Taylor expansion in t yiels z(t) = z 0 + (t t 0 )X H (z 0 ; t 0 ) + O((t t 0 ) 2 );

7 5 an hence It follows that x(t) = x 0 + (t t 0 )@ p H(z 0 ; t 0 ) + O((t t 0 ) = (t t 0)H pp (z 0 ; t 0 ) + O((t t 0 ) 2 ). hence there exists " > 0 such is invertible in [t 0 "; t 0 [\]t 0 ; t 0 + "] if an only if H pp (z 0 ; t 0 ) is invertible; in view of Lemma 3 this is equivalent to saying that f H t is free at z 0. Example 5 The result above applies when the Hamiltonian H is of the physical type nx 1 H(z; t) = p 2 j + U(x; t) 2m j since we have In this case f H t j=1 H pp (z 0 ; t 0 ) = iag[ 1 2m 1 ; :::; 1 2m n ]. is free for small non-zero t near each z 0 where it is e ne. 4.2 The variational equation An essential feature of Hamiltonian ows is that they consist of symplectomorphisms. We are going to give an elementary proof of this property; it relies on the fact that the mapping t 7! Df H t;t 0(z) is, for xe t0, the solution of a ifferential equation, the variational equation, an which plays an important role in many aspects of Hamiltonian mechanics (in particular the stuy of perioic Hamiltonian orbits). Proposition 6 For xe z set S H t;t 0(z) = Df H t;t 0(z). (i) The function t 7! S t;t 0(z) satis es the variational equation t SH t;t 0(z) = JD2 H(f H t;t 0(z); t)sh t;t 0(z), SH t;t(z) = I (14) where D 2 H(ft;t H 0(z)) is the Hessian matrix of H calculate at f t;t H 0(z); (ii) We have St;t H 0(z) 2 Sp(n) for every z an t; t0 for which it is e ne, hence ft;t H is a symplectomorphism. 0 Proof. Proof of (i). It is su cient to consier the case t 0 = 0. Set ft;0 H = ft H an St;t H = S t. Taking Hamilton s equation into account the time-erivative of 0 the Jacobian matrix S t (z) is t S t(z) = t (Df H t (z)) = D t f t H (z)

8 6 that is t S t(z) = D(X H (ft H (z))). Using the fact that X H = J@ z H together with the chain rule, we have D(X H (f H t (z))) = D(J@ z H)(f H t (z); t) = JD(@ z H)(f H t (z); t) = J(D 2 H)(f H t (z); t)df H t (z) hence S t (z) satis es the variational equation (14), proving (i). Proof of (ii). Set S t = S t (z) an A t = (S t ) T JS t ; using the prouct rule together with (14) we have A t t = (S t) T JS t + (S t ) T J S t t t = (S t ) T D 2 H(z; t)s t (S t ) T D 2 H(z; t)s t = 0. It follows that the matrix A t = (S t ) T JS t is constant in t, hence A t (z) = A 0 (z) = J so that (S t ) T JS t = J proving that S t 2 Sp(n). A relate result is: Proposition 7 Let t 7! X t be a C 1 mapping R solution of the i erential system We have S t 2 Sp(n) for every t 2 R: t S t = X t S t, S 0 = I.! sp(n) an t 7! S t a Proof. The conition X t 2 sp(n) is equivalent to JX t being symmetric. Hence t (ST t JS t ) = S T t X T t JS t + S T t JX t S t = 0 so that St T JS t = S0 T JS 0 = J an S t 2 Sp(n) as claime. Hamilton s equations are covariant (i.e., they retain their form) uner symplectomorphisms. Let us begin by proving the following general result about vector els which we will use several times. If X is a vector el an f a i eomorphism we enote by Y = f X the vector el e ne by Y (u) = D(f 1 )(f(u))x(f(u)) = [Df(u)] 1 X(f(u)). (15) (f X is calle the pull-back of the vector el X by the i eomorphism f.) Lemma 8 Let X be a vector el on R m an (' X t ) its ow. Let f be a i eomorphism R m! R m. The family (' Y t ) of i eomorphisms e ne by is the ow of the vector el Y = (Df) 1 (X f). ' Y t = f 1 ' X t f (16)

9 7 Proof. We obviously have ' Y 0 = I; in view of the chain rule t 'Y t (x) = D(f 1 )(' X t (f(x)))x(' X t (f(x)) = (Df) 1 (' Y t (x))x(f(' Y t (x))) an hence t 'Y t (x) = Y (' Y t (x)) which we set out to prove. Specializing to the Hamiltonian case, this lemma yiels: Proposition 9 Let f be a symplectomorphism. (i) We have X Hf (z) = [Df(z)] 1 (X H f)(z). (17) (ii) The ows (f H t ) an (f Hf t ) are conjugate by f: an thus f X H = X Hf when f is symplectic. f Hf t = f 1 f H t f (18) Proof. Let us prove (i); the assertion (ii) will follow in view of Lemma 8 above. Set K = H f. By the chain z K(z) = [Df(z)] T (@ z H)(f(z)) hence the vector el X K = J@ z K is given by X K (z) = J[Df(z)] z H(f(z)). Since Df(z) is symplectic we have J[Df(z)] T = [Df(z)] 1 J an thus which is (17). X K (z) = [Df(z)] 1 J@ z H(f(z)) Remark 10 Proposition 9 can be restate as follows: set (x 0 ; p 0 ) = f(x; p) an K = H f; if f is a symplectomorphism then we have the equivalence _x 0 p 0K(x 0 ; p 0 ) an _p 0 x 0K(x 0 ; p 0 ) () (19) _x p H(x; p) an _p x H(x; p): Example 11 For instance, the change of variables (x; p) 7! (I; ) e ne by x = p 2I cos, p = p 2I sin is symplectic. An interesting fact is that there is a wie class of Hamiltonian functions whose time-epenent ows (ft H ) consist of free symplectomorphisms if t is not too large (an i erent from zero). This is the case for instance when H is of the physical type nx 1 H(z; t) = (p j A j (x; t)) 2 + U(x; t) (20) 2m j j=1 (m j > 0, A j an U smooth). More generally:

10 8 Lemma 12 There exists " > 0 such that ft H is free at z 0 2 R 2n z for 0 < jtj " if an only if et H pp (z 0 ; 0) 6= 0, an hence, in particular, when H is of the type (20). Proof. Let t 7! z(t) be the solution to Hamilton s equations _x p H(z; t), _p x H(z; t) with initial conition z(0) = z 0. A secon orer Taylor expansion at time t = 0 yiels z(t) = z 0 + tx H (z 0 ; 0) + O((t) 2 ) where X H = J@ z H is the Hamiltonian vector el of H; in particular an hence x(t) = x 0 + t@ p H(z 0 ; 0) + O(t = th pp(z 0 ; 0) + O(t 2 ) where H pp enotes the matrix of erivatives of H in the variables p j. It follows that there exists " > 0 such is invertible in [ "; 0[\]0; "] if an only if H pp (z 0 ; 0) is invertible; in view of Lemma 3 this is equivalent to saying that ft H is free at z The group Ham(n) The group Ham(n) is the connecte component of the group Symp(n) of all symplectomorphisms of (R 2n z ; ). Each of its points is the value of a Hamiltonian ow at some time t. The stuy of the various algebraic an topological properties of the group Ham(n) is a very active area of current research. We will say that a symplectomorphism f of the stanar symplectic space (R 2n z ; ) is Hamiltonian if there exists a function H 2 C 1 (R 2n+1 z;t ; R) an a number a 2 R such that f = fa H. Taking a = 0 it is clear that the ientity is a Hamiltonian symplectomorphism. The set of all Hamiltonian symplectomorphisms is enote by Ham(n). We are going to see that it is a connecte an normal subgroup of Symp(n); let us rst prove a preparatory result which is interesting in its own right: Proposition 13 Let (f H t ) an (f K t ) be Hamiltonian ows. Then: f H t f K t = f H#K t with H#K(z; t) = H(z; t) + K((f H t ) 1 (z); t): (21) (f H t ) 1 = f H t with H(z; t) = H(f H t (z); t): (22) Proof. Let us rst prove (21). By the prouct an chain rules we have t (f t H ft K ) = ( t f t H )ft K + (Dft H )ft K t f t K = X H (ft H ft K ) + (Dft H )ft K X K (ft K )

11 9 an it thus su ces to show that Writing (Df H t )f K t X K (f K t ) = X K(f H t ) 1(f K t ). (23) (Df H t )f K t X K (f K t ) = (Df H t )((f H t ) 1 f H t f K t ) X K ((f H t ) 1 f H t f K t ) the equality (23) follows from the transformation formula (17) in Proposition 9. Formula (22) is now an easy consequence of (21), noting that (f H t f H t ) is the ow etermine by the Hamiltonian K(z; t) = H(z; t) + H((f H t ) 1 (z); t) = 0; f H t f H t is thus the ientity, so that (f H t ) 1 = f H t as claime. Let us now show that Ham(n) is a group, as claime: Proposition 14 Ham(n) is a normal an connecte subgroup of the group Symp(n) of all symplectomorphisms of (R 2n z ; ). Proof. Let us show that if f; g 2 Ham(n) then fg 1 2 Ham(n). We begin by remarking that if f = fa H for some a 6= 0 then we also have f = f1 Ha where H a (z; t) = ah(z; at). In fact, setting t a = at we have z a t = J@ zh a (z a ; t) () za t a = J@ zh(z a ; t a ) an hence f Ha t = f H at. We may thus assume that f = f H 1 an g = f K 1 for some Hamiltonians H an K. Now, using successively (21) an (22) we have fg 1 = f H 1 (f K 1 ) 1 = f H# K 1 hence fg 1 2 Ham(n). That Ham(n) is a normal subgroup of Symp(n) immeiately follows from formula (18) in Proposition 9: if g is a symplectomorphism an f 2 Ham(n) then f Hg 1 = g 1 f H 1 g 2 Ham(n) (24) so we are one. The result above motivates the following e nition: De nition 15 The set Ham(n) of all Hamiltonian symplectomorphisms equippe with the law fg = f g is calle the group of Hamiltonian symplectomorphisms of the stanar symplectic space (R 2n z ; ). The topology of Symp(n) is e ne by specifying the convergent sequences: we will say that lim j!1 f j = f in Symp(n) if an only if for every compact set K in R 2n z the sequences (f jjk ) an (D(f jjk )) converge uniformly towars f jk

12 10 an D(f jk ), respectively. The topology of Ham(n) is the topology inuce by Symp(n). We are now gong to prove a eep an beautiful result ue to Banyaga. It essentially says that a path of time-one Hamiltonian symplectomorphisms passing through the ientity at time zero is itself Hamiltonian. It will follow that Ham(n) is a connecte group. Let t 7! f t be a path in Ham(n), e ne for 0 t 1 an starting at the ientity: f 0 = I. We will call such a a path a one-parameter family of Hamiltonian symplectomorphisms. Thus, each f t is equal to some symplectomorphism f Ht 1. A striking an not immeiately obvious! fact is that each path t 7! f t is itself is the ow of a Hamiltonian function! Theorem 16 Let (f t ) be a one-parameter family in Ham(n). Then (f t ) = (f H t ) where the Hamilton function H is given by H(z; t) = Z 1 0 (X(uz; t); z)u with X = ( t f t) (f t ) 1. (25) Proof. By e nition of X we have t f t = Xf t so that all we have to o is to prove that X is a (time-epenent) Hamiltonian el. For this it su ces to show that the contraction i X of the symplectic form with X is an exact i erential one-form, for then i X = H where H is given by (25). The f t being symplectomorphisms, they preserve the symplectic form an hence L X = 0. In view of Cartan s homotopy formula we have L X = i X + (i X ) = (i X ) = 0 so that i X is close; by Poincaré s lemma it is also exact. Let (ft H ) 0t1 an (ft K ) 0t1 be two arbitrary paths in Ham(n). The paths (ft H ft K ) 0t1 an (f t ) 1t1 where ( f2t K when 0 t 1 2 f t = f2t H 1f1 K when 1 2 t 1 are homotopic with xe enpoints. Let us construct explicitly a homotopy of the rst path on the secon, that is, a continuous mapping h : [0; 1] [0; 1]! Ham(n) such that h(t; 0) = ft H ft K an h(t; 1) = f t. De ne h by h(t; s) = a(t; s)b(t; s) where a an b are functions ( I for 0 t s 2 a(t; s) = f(2t H s s)=(2 s) for 2 t 1 ( f K s 2t=(2 s) for 0 t 1 2 b(t; s) = f1 K s for 2 t 1.

13 11 We have a(t; 0) = f H t, b(t; 0) = f K t hence h(t; 0) = f H t f K t ; similarly that is h(t; 1) = f t. ( f K 2t for 0 t 1 2 h(t; 1) = f2t H 1f1 K 1 for 2 t Hamiltonian perioic orbits Let H 2 C 1 (R 2n z ) be a time-inepenent Hamiltonian function, an (f H t ) the ow etermine by the associate vector el X H = J@ z. De nition 17 Let z 0 2 R 2n z ; the mapping : R t! R 2n z, (t) = f H t (z 0 ) is calle (Hamiltonian) orbit through z 0. If there exists T > 0 such that ft+t H (z 0) = ft H (z 0 ) for all t 2 R one says that the orbit through z 0 is perioic with perio T. [The smallest possible perio is calle primitive perio ]. The following properties are obvious: Let, 0 be two orbits of H. Then the ranges Im an Im 0 are either isjoint or ientical. The value of H along any orbit is constant ( theorem of conservation of energy ).. The rst property follows from the uniqueness of the solutions of Hamilton s equations, an the secon from the chain rule, setting (t) = (x(t); p(t)): t H((t)) = h@ xh((t)); _x(t)i + h@ p H((t)); _p(t)i = h _p(t); _x(t)i + h _x(t); _p(t)i = 0 where we have taken into account Hamilton s equations. Assume now that is a perioic orbit through z 0. We will use the notation S t (z 0 ) = Df H t (z 0 ). De nition 18 Let : t 7! ft H (z 0 ) be a a perioic orbit with perio T. The symplectic matrix S T (z 0 ) = DfT H(z 0) is calle monoromy matrix. The eigenvalues of S T (z 0 ) are calle the Floquet multipliers of. The following property is well-known in Floquet theory:

14 12 Lemma 19 (i) Let S T (z 0 ) be the monoromy matrix of the perioic orbit. We have S t+t (z 0 ) = S t (z 0 )S T (z 0 ) (26) for all t 2 R. In particular S NT (z 0 ) = S T (z 0 ) N for every integer N. (ii) Monoromy matrices corresponing to the choice of i erent origins on the perioic orbit are conjugate of each other in Sp(n) hence the Floquet multipliers o not epen on the choice of origin on the perioic orbit; (iii) Each perioic orbit has an even number > 0 of Floquet multiplier equal to one. Proof. Proof of (i). The mappings ft H the equality ft H(z 0) = z 0 : form a group, hence, taking into account f H t+t (z 0 ) = f H t (f H T (z 0 )) so that by the chain rule, Df H t+t (z 0 ) = Df H t (f H T (z 0 ))Df H T (z 0 ) that is (26) since f H T (z 0) = z 0. Proof of (ii).eviently the orbit through any point z(t) of the perioic orbit is also perioic. We begin by noting that if z 0 an z 1 are points on the same orbit then there exists t 0 such that z 0 = f H t 0 (z 1 ). We have f H t (f H t 0 (z 1 )) = f H t 0 (f H t (z 1 )) hence, applying the chain rule of both sies of this equality, Df H t (f H t 0 (z 1 ))Df H t 0 (z 1 ) = Df H t 0 (f H t (z 1 ))Df H t (z 1 ). Choosing t = T we have f H t (z 0 ) = z 0 an hence that is, since f H t 0 (z 1 ) = z 0, S T (f H t 0 (z 1 ))S t0 (z 1 ) = S t0 (z 1 )S T (z 1 ) S T (z 0 )S t0 (z 1 ) = S t0 (z 1 )S T (z 1 ). It follows that the monoromy matrices S T (z 0 ) an S T (z 1 ) are conjugate an thus have the same eigenvalues. Proof of (iii). We have, using the chain rule together with the relation f H t f H t 0 = f H t+t 0 t 0 f t H (ft H (z 0)) 0 = Df t (z 0 )X H (z 0 ) = X H (ft H (z 0 )) t 0 =0 hence S T0 (z 0 )X H (z 0 ) = X H (z 0 ) setting t = T ; X H (z 0 ) is thus an eigenvector of S T (z 0 ) with eigenvalue one; the Lemma follows the eigenvalues of a symplectic matrix occurring in quaruples (; 1=; ; 1= ). The following theorem is essentially ue to Poincaré:

15 13 Theorem 20 Let E 0 = H(z 0 ) be the value of H along a perioic orbit 0. Assume that 0 has exactly two Floquet multipliers equal to one. Then there exists a unique smooth 1-parameter family ( E ) of perioic orbits of E with perio T parametrize by the energy E, an each E is isolate on the hypersurface E = fz : H(z) = Eg among those perioic orbits having perios close to the perio T 0 of 0 Moreover lim E!E0 T = T 0. One shows, using normal form techniques that when the conitions of the theorem above are ful lle, the monoromy matrix of 0 can be written as U 0 S T (z 0 ) = S0 T 0 S(z0 ~ S ) 0 with S 0 2 Sp(n), S(z ~ 1 0 ) 2 Sp(n 1) an U is of the type for some real 0 1 number ; the 2(n 1) 2(n 1) symplectic matrix S(z ~ 0 ) is calle the stability matrix of the isolate perioic orbit 0. It plays a funamental role not only in the stuy of perioic orbits, but also in semiclassical mechanics.