Symplectic Map
Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of (p_i,q_i) planes. It is the generalization of an area-preserving map.
Formally, a symplectic map is a real-linear map T that preserves a symplectic form f, i.e., for which
| f(Tx,Ty)=f(x,y) |
for all x, y. Every symplectic map T on a complex Hilbert space H may be written as U(coshS+JsinhS), where U is unitary, S is positive, and J is an anti-linear involution (i.e., complex conjugation).
See also
Area-Preserving Map, Liouville's Phase Space TheoremExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Symplectic Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymplecticMap.html