Symplectic manifold

The symplectic manifold is the central object of symplectic geometry, a branch of differential geometry. The symplectic manifolds have a very strong relationship to theoretical physics.

Definition

A symplectic manifold is a smooth manifold with a symplectic form, ie a global, smooth and closed 2- form which is pointwise non-degenerate (see also symplectic space ). "Closed" means the external dissipation of the differential form disappears.

Symplectic manifolds must have an even dimension, since antisymmetric matrices can not be inverted in odd dimensions.

Poisson bracket

Since the mold is not degenerated, it defines with its inverse at each point, a bilinear map of one forms and

And the Poisson bracket of the functions and,

Lagrangian submanifold

A Lagrangian submanifold of a 2n -dimensional symplectic manifold is an n- dimensional submanifold with

That is, the restriction of the symplectic form vanishes on the tangent space of L. ( Submanifolds of arbitrary dimension that satisfy the latter condition, called isotropic. One can show that isotropic manifolds are at most n -dimensional. Lagrangian manifolds are therefore isotropic submanifolds maximum dimension. )

Lagrange diversity plays a key role in physics. A Lagrangian manifold with a real germ can be defined as follows: Consider a Lagrangian manifold with dimension k

Hamiltonian flow

In a Euclidean space of the gradient of a function, is that the vector field, the scalar product matches for each given vector field with the use of in,

In a symplectic manifold belongs to a given f and a given arbitrary function the vector field

The functions along an integral curve of to (interpreted as so-called Hamiltonian function of the system) associated Hamiltonian equations derived. The role of w is therefore adopted here by h, and it is used for h the Symplectic geometry and the Hamiltonian dynamics.

The vector field is therefore the symplectic gradient of or the infinitesimal Hamiltonian flow of.

Set of Darboux

The set of Darboux named after the mathematician Jean Gaston Darboux says:

In the neighborhood of each point of a symplectic manifold, there are local coordinate pairs with

The coordinate pairs thus defined are referred to as canonical conjugate.

Relationship to the Hamiltonian mechanics

In the Hamiltonian mechanics of the phase space is a symplectic manifold with the closed, symplectic form

This is not a special case, since by the theorem of Darboux can always write as in local coordinates. For symplectic manifolds are phase spaces of Hamiltonian mechanics.

The mathematical statement regarding, in fact equivalent to the so-called canonical equations of theoretical physics, in particular in analytical mechanics.

In this context it is also the Liouville theorem of importance, which plays a fundamental role in statistical physics. It essentially says that in Hamiltonian flows the phase space volume remains constant, which is crucial for determining the probability measures of this theory.