Symplectic Manifold


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In Symplectic Geometry, a Symplectic Manifold is a smooth manifold equipped with a symplectic form, a special type of structure that plays a central role in Hamiltonian mechanics.

Formally, a symplectic manifold is a pair $(M, \omega)$, where $M$ is a smooth manifold, and $\omega$ is a closed, non-degenerate 2-form on $M$.


  • Closedness: The symplectic form $\omega$ is closed, meaning $d\omega = 0$.
  • Non-degeneracy: The symplectic form $\omega$ is non-degenerate at each point in $M$.
  • Even Dimension: A symplectic manifold must have even dimension.


  • Phase Space in Classical Mechanics: The cotangent bundle of a configuration space can be given a natural symplectic structure.
  • Complex Projective Spaces: These can be equipped with a Fubini-Study symplectic form.


  • Hamiltonian Dynamics: Symplectic manifolds provide the geometric framework for Hamiltonian mechanics, where the flow of a Hamiltonian function corresponds to the motion of a mechanical system.
  • Quantum Mechanics: The mathematical structure of quantum mechanics can be described using symplectic manifolds.

Symplectic manifolds serve as a unifying geometric structure in both classical and quantum physics and continue to be a rich area of research in mathematics and physics.


Topics in Statistical Mechanics and Thermodynamics


Topics in Topological Data Analysis

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