Symplectic Geometry

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Symplectic Geometry is the study of symplectic manifolds and symplectic forms. It is a branch of geometry that has its origins in classical mechanics, where it describes the phase space underlying Hamiltonian mechanics.

A key concept in symplectic geometry is the symplectic form, which is a closed, non-degenerate differential 2-form. On a symplectic manifold, the symplectic form gives rise to the symplectic structure, a fundamental concept that allows for the study of dynamics and conservation laws in a geometric context.

Key Properties of Symplectic Forms:

• Closedness: A symplectic form $\omega$ is closed, meaning its exterior derivative is zero: $d\omega = 0$.
• Non-degeneracy: At any point, the symplectic form is non-degenerate, meaning that if $\omega(v,w) = 0$ for all $w$, then $v = 0$.

See also

• Symplectic manifold
• Cotangent bundle

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