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.
Etymology The term "symplectic", as adopted into mathematics by Hermann Weyl, is a neo-Greek 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" represents the corresponding Greek sym-plektikos (συμπλεκτικός "twining or plaiting together, copulative"). In both cases, the stems come from the Indo-European root *pleḱ-, expressing the concept of folding or weaving, and the prefixes suggest "togetherness". The name reflects the deep connections between complex and symplectic structures. By Darboux's theorem, symplectic manifolds are locally isomorphic to the standard symplectic vector space. Hence they have only global (topological) invariants. The term "symplectic topology" is often used interchangeably with "symplectic geometry".
Overview
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 R 2 {\displaystyle \mathbb {R} ^{2}} . In this case, the symplectic form is
ω = d p ∧ d q {\displaystyle \omega =dp\wedge dq}
and is an area form that measures the area A of a region S in the plane through integration:
A = ∫ S ω . {\displaystyle A=\int _{S}\omega .}
The area is important because as conservative dynamical systems evolve in time, this area is invariant. Higher dimensional symplectic geometries are defined analogously. A 2n-dimensional symplectic geometry is formed of pairs of directions
( ( x 1 , x 2 ) , ( x 3 , x 4 ) , … ( x 2 n − 1 , x 2 n ) ) {\displaystyle ((x_{1},x_{2}),(x_{3},x_{4}),\ldots (x_{2n-1},x_{2n}))}
in a 2n-dimensional manifold along with a symplectic form
ω = d x 1 ∧ d x 2 + d x 3 ∧ d x 4 + ⋯ + d x 2 n − 1 ∧ d x 2 n . {\displaystyle \omega =dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4}+\cdots +dx_{2n-1}\wedge dx_{2n}.}
This symplectic form yields the size of a 2n-dimensional region V in the space as the sum of the areas of the projections of V onto each of the planes formed by the pairs of directions
A = ∫ V ω = ∫ V d x 1 ∧ d x 2 + ∫ V d x 3 ∧ d x 4 + ⋯ + ∫ V d x 2 n − 1 ∧ d x 2 n . {\displaystyle A=\int _{V}\omega =\int _{V}dx_{1}\wedge dx_{2}+\int _{V}dx_{3}\wedge dx_{4}+\cdots +\int _{V}dx_{2n-1}\wedge dx_{2n}.}
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