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Show that if a one-parameter group of difeomorphisms of a symplectic manifold preserves the symplectic structure then it is a locally hamiltonian phase flow.

Note that

A locally hamiltonian vector field on a symplectic manifold $(M^{2n}, \omega^2)$ is the vector field $I \omega^1$, where $\omega^1$ is a closed 1-form on $M^{2n}$ and $\omega^1(\eta) = \omega^2(\eta, I\omega^1)$.

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Hints:

  1. Prove that a one-parameter group $(\Phi_t)_{t\in I}$ of diffeomorphisms $\Phi_t: M \to M$ is generated by a vector field $X\in\Gamma(M)$.

  2. Prove that if the one-parameter group $(\Phi_t)_{t\in I}$ preserves the a form $\omega$ then ${\cal L}_{X}\omega =0$.

  3. Prove that ${\cal L}_{X}\omega =0$ together with the fact that $\omega$ is a symplectic two-form (in particular the fact that $\omega$ is non-degenerate) imply that $X$ is locally a Hamiltonian vector field.

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  • $\begingroup$ Could you explain it further? Many thanks! $\endgroup$ – Eden Harder Mar 31 '14 at 0:34

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