symplectomorphism

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symplectomorphism

Wikipedia

Blend of symplectic +‎ isomorphism.

symplectomorphism (plural symplectomorphisms)

1. (mathematics) An isomorphism of a symplectic manifold; a diffeomorphism which preserves symplectic structure.
   • 1977, Alan Weinstein, Lectures on Symplectic Manifolds, American Mathematical Society, page 34:

     In prequantizing symplectomorphisms of the type T^*g, the only special property which we used was the fact that they preserve canonical 1-forms.

   • 2001, A. Dzhamay, G. Wassermann (translators), V. I. Arnol'd, A. B. Givental' Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 39,

     Poincare's argument is based on the fact that the fixed points of a symplectomorphism of the annulus are precisely the critical points of the function ${\displaystyle \textstyle F(x,y)=\int {(fdv-gdu)}}$, where ${\displaystyle u=(X+x)/2}$, ${\displaystyle v=(Y+y)/2}$, true under the assumption that the Jacobian ${\displaystyle \partial (u,v)/\partial (x,y)}$ is different from zero.

   • 2008, Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2nd printing with corrections, page 63,

     The symplectomorphisms of a symplectic manifold ${\displaystyle (M,\omega )}$ form the group

     ${\displaystyle {\text{Sympl}}(M,\omega )=\lbrace f:M{\overset {\simeq }{\longrightarrow }}M\ \vert \ f^{*}\omega =\omega \rbrace }$.