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Symplectic Manifold

A symplectic manifold is a pair (M,omega), where M is a manifold and omega is a symplectic form on M. The phase space R^(2n)=R^n×R^n is a symplectic manifold. Near every point on a symplectic manifold, it is possible to find a set of local "Darboux coordinates" in which the symplectic form has the simple form

omega=sum_(k)dq_k ^ dp_k

(Sjamaar 1996), where dq_k ^ dp_k is a wedge product.

See also

Arnold Conjecture, Gromov Width, Hamiltonian Diffeomorphism, Lagrangian Submanifold, Manifold, Moment Map, Nakajima Quiver Variety, Symplectic Ball Packing, Symplectic Diffeomorphism, Symplectic Form, Symplectic Taming

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References

Sjamaar, R. "Symplectic Reduction and Riemann-Roch Formulas for Multiplicities." Bull. Amer. Math. Soc. 33, 327-338, 1996.

Referenced on Wolfram|Alpha

Symplectic Manifold

Cite this as:

Weisstein, Eric W. "Symplectic Manifold." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymplecticManifold.html

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