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On the Number of Periodic Trajectories for a Hamiltonian Flow on a Convex Energy Surface

Ivar Ekeland and Jean-Michel Lasry
Annals of Mathematics
Second Series, Vol. 112, No. 2 (Sep., 1980), pp. 283-319
Published by: Annals of Mathematics
DOI: 10.2307/1971148
Stable URL: http://www.jstor.org/stable/1971148
Page Count: 37
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On the Number of Periodic Trajectories for a Hamiltonian Flow on a Convex Energy Surface
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Abstract

In this paper, we look for periodic solutions, with prescribed energy &h \epsilon R$, of Hamilton's equations: $(H) \dot{x} = \partial H \over \partial p (x, p), \dot{p} = - \partialH \over \partialx (x, p)$. It is assumed that the Hamiltonian $H is convex on R^n \times R^n, and that the origin (0, 0) is an isolated equilibrium. It is also assumed that some ball B around the origin can be found such that the energy surface H^{-1} (h) lies outside B but inside $\sqrt2 B$. Under these assumptions, we prove that there are at least n distinct periodic orbits of the Hamiltonian flow (H) with energy level h.

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