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Une idée du type ”géodesiques brisées” pour les systèmes hamiltoniens. (A ”broken geodesic” type suggestion for Hamiltonian systems). (French) Zbl 0576.58010
Let M denote the n-dimensional torus, \(\lambda\) the canonical 1-form of its cotangent bundle \(T^*M\) and \(\Sigma\) the zero section of \(T^*M\). The paper deals with the following theorem: if \((j_ t)_{0\leq t\leq 1}\) is a smooth family of \(C^{\infty}\) imbeddings of M into \(T^*M\) satisfying (i) \(j_ 0(M)=\Sigma\), (ii) \(j^*_ t \lambda\) is closed for all t and exact for \(t=1\), then \(j_ 1(M)\cap \Sigma\) contains at least \(n+1\) points and at least \(2^ n\) points whenever \(j_ 1\) is transversal to \(\Sigma\).
This is a conjecture by Arnol’d, which the author recently proved using an intermediate infinite dimensional variational problem. The proof presented in the present paper avoids the detour over infinite dimensions and is believed to be applicable also in a more general context.
Reviewer: W.Sarlet

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E30 Variational principles in infinite-dimensional spaces