Existence of Multiple Periodic Orbits on Star-Shaped Hamiltonian Surfaces
نویسندگان
چکیده
Consider the Hamiltonian system i = 1 , . . . , N. Here, H E C2(R2N, R). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface X = { z E W Z N ; H ( z) = c} ( c > 0 is a constant). The surface I: is required to verify certain geometric assumptions: B bounds a star-shaped compact region B and u 8 c B c pS for some ellipsoid %‘c RZN, 0 < (Y < p. We exhibit a constant S > 0 (depending in an explicit fashion on the lengths of the main axes of Lf and one other geometrical parameter of I) such that if furthermore p2 /a2< I + 8, then (HS) has at least N distinct geometric orbits on P. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S’-action, from which we derive abstract critical point theorems for S’-invariant functionals. We also derive an estimate for the minimal period of solutions to differential equations.
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