Convexity Methods in Hamiltonian Mechanics by Ivar Ekeland (auth.)

By Ivar Ekeland (auth.)

In the case of thoroughly integrable structures, periodic ideas are stumbled on by means of inspection. For nonintegrable structures, comparable to the three-body challenge in celestial mechanics, they're came upon via perturbation thought: there's a small parameter € within the challenge, the mass of the perturbing physique for example, and for € = zero the method turns into thoroughly integrable. One then attempts to teach that its periodic suggestions will subsist for € -# zero sufficiently small. Poincare additionally brought international tools, counting on the topological houses of the circulation, and the truth that it preserves the 2-form L~=l dPi 1\ dqi' the main celebrated end result he acquired during this path is his final geometric theorem, which states that an area-preserving map of the annulus which rotates the internal circle and the outer circle in contrary instructions should have fastened issues. And now one other historic subject matter look: the least motion precept. It states that the periodic options of a Hamiltonian procedure are extremals of an appropriate fundamental over closed curves. In different phrases, the matter is variational. This truth used to be recognized to Fermat, and Maupertuis positioned it within the Hamiltonian formalism. despite its nice aesthetic allure, the least motion precept has had little effect in Hamiltonian mechanics. there's, after all, one exception, Emmy Noether's theorem, which relates integrals ofthe movement to symmetries of the equations. yet until eventually lately, no periodic answer had ever been came across by means of variational methods.

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Now define in L~(O, kT) subspaces Mi and N j by (note the different range of the index) (11) (12) N j = rjpEo(O,T) Mi = ripE_ (0, T) for 0 ~ j ~ k - 2 for 0 ~ i ~ k - 1 . 5. The Iteration Formula 35 The subspaces are mutually orthogonal. Setting (13) M = E9Mi and N = E9Nj j we check easily that qkT is negative definite on M, and vanishes on N. In other words, N is a qkT-isotropic subspace, and any qkT-isotropic vector in M EB N must belong to N. Counting dimensions, we find: (14) dimME9N = (k -1)vT + kiT.

81), (82) and (83), (84), this gives: (86) which is precisely formula (77) for the Krein type (Po, 0). Lastly, we deal with the case of Krein-indefinite eigenvalues. Again, choose E > 0 so small that (81) and (82) hold, and there are no eigenvalues of R(T) of the form w o ei8 , with 0 < () :s; E. 5, if t =1= T but IT - tl is small enough, R(t) will have only Krein-definite eigenvalues on the unit circle. Take t < T for instance. Then there will be exactly p;; Krein-positive and q;; Krein-negative eigenvalues on the arc eie , -E < () < E (counting multiplicities).

Linear Hamiltonian Systems 30 (43), the ei are bounded in L~(O, 1). So the set {sei/Ails E 1} is bounded, and its image by the compact map lIt (see Lemma 1) must be precompact. In other words, there is a sequence sen) ---+ 0' with sen) > 0', and a vector w E L~(O, 1) such that: (48) Rewrite Eq. (44): ,s(n»)-lil1 esen) -_ s (n ) ( Ai i (49) Bl (s (n )) esen) i . Since the left-hand side converges to w, so does the right-hand side. But ---+ Bl(O') by continuity, and so: Bl (s(n)) (50) Since (Bl (s(n))e:(n),e:(n)) = 1, we get by continuity (Bl(O')ei,ei) = 1.

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