Symplectic Form
A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M which is nondegenerate such that at every point m, the alternating bilinear form omega_m on the tangent space T_mM is nondegenerate.
A symplectic form on a vector space V over F_q is a function f(x,y) (defined for all x,y in V and taking values in F_q) which satisfies
| f(lambda_1x_1+lambda_2x_2,y)=lambda_1f(x_1,y)+lambda_2f(x_2,y) |
(1)
|
| f(y,x)=-f(x,y), |
(2)
|
and
| f(x,x)=0. |
(3)
|
f is called non-degenerate if f(x,y)=0 for all y implies that x=0. Symplectic forms can exist on M (or V) only if M (or V) is even-dimensional. An example of a symplectic form over a vector space is the complex Hilbert space with inner product <·,·> given by
| f(x,y)=I<x,y>. |
(4)
|
See also
Symplectic Space, Vector SpaceExplore with Wolfram|Alpha
More things to try:
Cite this as:
Weisstein, Eric W. "Symplectic Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymplecticForm.html