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On the hypotheses of Rabinowitz’ periodic orbit theorems. (English) Zbl 0388.58020
Summary: In recent work of Rabinowitz, Ekeland, Clarke, and the author, the existence of periodic orbits with prescribed energy for Hamiltonian systems has been proven under the assumption of convexity or some kind of starshapedness of the energy surface. The ideas described in the present paper arose in an attempt to extend these existence theorems to more general systems by canonical transformations of the Hamiltonian. It is shown that the presence of a contact structure is a common feature of all the theorems, and it is conjectured that the contact structure is sufficient to imply the existence of a periodic orbit. An example is given of an embedded sphere which cannot be canonically transformed to become starshaped.

MSC:
53D35 Global theory of symplectic and contact manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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