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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

It is not true. There is a matrix associated to the linear terms, and you should compute the eigenvalues of the matrix, all the eigenvalues of the matrix are independent of coordinates transformations, which are, by means of your symplectic transformations.

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