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Consider a autonomous Hamiltonian $h:W\rightarrow \mathbb{R}$, where $W$ is a symplectic manifold. Let $\mathrm{sgrad}(h)$ denote the vector field on $W$ that is dual to the differential $Dh$ using the symplectic form.

Then it seems to be often stated that if $h$ is $C^2$ small then the only 1-periodic orbits of $\mathrm{sgrad}(h)$ are constants. Why is this?

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up vote 5 down vote accepted

See Hofer and Zehnder's book "Symplectic invariants and Hamiltonian dynamics" p184-185 for a proof.

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thanks! that was very quick! –  user12998 Feb 15 '11 at 19:34
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