Let $(M,\omega = d\alpha)$ be an exact symplectic manifold. Then a symplectomorphism $\varphi \colon M \to M$ is said to be exact, iff $\varphi^*\alpha - \alpha$ is exact. Is there a terminology for the special case when $\varphi^*\alpha = \alpha$? I thought of something like the symplectomorphism preserves the exact symplectic form.
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In his classic book on classical mechanics Whittaker calls these transformations Mathieu transformations. The term appears in Wikipedia.
