Symplectic Topology
Introduction
Symplectic topology is a branch of differential geometry and topology that studies symplectic manifolds. These are a special kind of manifold that have a symplectic form, a non-degenerate, closed, differential 2-form. Symplectic topology has significant applications in the study of Hamiltonian mechanics, where it provides a geometric interpretation of the dynamics of non-dissipative systems.
History
Symplectic topology has its roots in the mathematical formulation of classical mechanics. The term "symplectic" was first introduced by Hermann Weyl in the 1930s, derived from the Greek word "συμπλέκτικος" (symplektikos), which means "weaving together". The field has since grown and diversified, with significant contributions from many mathematicians including Vladimir Arnold, Mikhail Gromov, and Alan Weinstein.
Symplectic Manifolds
A symplectic manifold is a smooth manifold equipped with a closed, non-degenerate differential 2-form, known as the symplectic form. This form allows for the definition of symplectic vector fields and symplectic diffeomorphisms, which preserve the symplectic form. Symplectic manifolds are even-dimensional, as the symplectic form is a 2-form.
Symplectic Form
The symplectic form on a manifold is a closed, non-degenerate 2-form. This means that its exterior derivative is zero, and at every point, it provides a non-degenerate skew-symmetric bilinear form on the tangent space. The symplectic form allows for the definition of symplectic vector fields, which are the vector fields whose flow preserves the symplectic form.
Symplectic Vector Fields and Diffeomorphisms
Symplectic vector fields are the vector fields that preserve the symplectic form under their flow. In other words, if φ_t is the flow of a symplectic vector field X, then φ_t is a symplectomorphism for all t. Symplectomorphisms are diffeomorphisms of a symplectic manifold that preserve the symplectic form. They form a group under composition, known as the symplectomorphism group of the manifold.
Hamiltonian Mechanics
Symplectic topology has deep connections with Hamiltonian mechanics, a reformulation of classical mechanics that describes systems in terms of generalized coordinates and momenta. The phase space of a Hamiltonian system is a symplectic manifold, and the dynamics of the system can be described by a Hamiltonian vector field, a special kind of symplectic vector field.
Hamiltonian Vector Fields
A Hamiltonian vector field on a symplectic manifold is defined by a function, known as the Hamiltonian of the system. The flow of a Hamiltonian vector field preserves the symplectic form, and thus describes the evolution of the system in phase space. Hamilton's equations of motion, which describe the dynamics of the system, can be derived from the properties of the Hamiltonian vector field.
Symplectic Integration
Symplectic integration is a numerical method for approximating the solutions of Hamilton's equations of motion. Unlike other numerical methods, symplectic integrators preserve the symplectic form, and thus the geometric structure of the phase space. This makes them particularly well-suited for long-term integration of Hamiltonian systems, such as planetary orbits.
J-Holomorphic Curves
J-holomorphic curves, introduced by Mikhail Gromov, are a central tool in symplectic topology. They are maps from Riemann surfaces to symplectic manifolds that are holomorphic with respect to an almost complex structure compatible with the symplectic form. J-holomorphic curves have been used to define invariants of symplectic manifolds, such as Gromov-Witten invariants and quantum cohomology.