Symplectomorphism

In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.

A diffeomorphism between two symplectic manifolds ${\displaystyle f:(M,\omega )\rightarrow (N,\omega ')}$ is called a symplectomorphism if

where ${\displaystyle f^{*}}$ is the pullback of ${\displaystyle f}$. The symplectic diffeomorphisms from ${\displaystyle M}$ to ${\displaystyle N}$ are a (pseudo-)group, called the symplectomorphism group (see below).

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