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Concept
Symplectic geometry
Summary
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures.
By Darboux's Theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, hence only have global (topological) invariants. "Symplectic topology," which studies global properties of symplectic manifolds, is often used interchangeably with "symplectic geometry."
A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic 2-form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas.
Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean plane R2. In this case, the symplectic form is
and is an area form that measures the area A of a region S in the plane through integration:
The area is important because as conservative dynamical systems evolve in time, this area is invariant.
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