# Preservation Hamiltonian vector field

Suppose I have the Hamiltonian vector field $X_H$ on the symplectic manifold $(M, \omega)$. Consider the symplectic transformation $P: M \rightarrow M$. Will the linear terms of $X_H$ be preserved (up to symplectic linear transformation) under any symplectic transformation $P$?

I am not sure, but I would think so.

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If Q is the symplectic transformation generated by $X_H$, then Jacobian of Q, $DQ$ is also symplectic. – nonlinearism Mar 8 '13 at 20:44