Symplectic diversity

A symplectic manifold is a manifold with a given symplectic form , that is, a closed non-degenerate differential 2-form .

The symplectic manifold allows us to introduce Hamiltonian mechanics in a natural geometric way and gives a visual interpretation of many of its properties.

Content

Definition

Differential 2-form $\omega$ ${\ displaystyle \ omega}$ is called a symplectic structure if it is non-degenerate and closed , that is, its external derivative is equal to zero:

d\omega =0

{\ displaystyle d \ omega = 0}

and for any nonzero tangent vector $v\in T_{x}M$ ${\ displaystyle v \ in T_ {x} M}$

\imath _{v}\omega \neq 0

{\ displaystyle \ imath _ {v} \ omega \ neq 0}

Where $\imath _{v}$ ${\ displaystyle \ imath _ {v}}$ - vector substitution operation $v$ ${\ displaystyle v}$ .

Manifold $M$ ${\ displaystyle M}$ is called symplectic if a symplectic structure is given on it.

Hamiltonian Vector Fields

Let be $H\colon M\to \mathbb {R}$ ${\ displaystyle H \ colon M \ to \ mathbb {R}}$ Is an arbitrary function on a symplectic manifold. The symplectic structure maps 1-forms to $M$ ${\ displaystyle M}$ a special class of vector fields called Hamiltonian by the rule

dH=\imath _{v}\omega .

{\ displaystyle dH = \ imath _ {v} \ omega.}

Due to the non-degeneracy of the form $\omega$ ${\ displaystyle \ omega}$ vector field $v$ ${\ displaystyle v}$ defined uniquely, we denote it $I d H$ ${\ displaystyle IdH}$ . In canonical coordinates, this mapping takes the form

{\dot {\mathbf {q} }}={\frac {\partial H}{\partial \mathbf {p} }},\quad {\dot {\mathbf {p} }}=-{\frac {\partial H}{\partial \mathbf {q} }},

{\ displaystyle {\ dot {\ mathbf {q}}} = {\ frac {\ partial H} {\ partial \ mathbf {p}}}, \ quad {\ dot {\ mathbf {p}}} = - { \ frac {\ partial H} {\ partial \ mathbf {q}}},}

corresponding to the Hamilton equations , while $H$ ${\ displaystyle H}$ called the Hamilton function or the Hamiltonian . The Poisson brackets turn many Hamiltonians into $M$ ${\ displaystyle M}$ into Lie algebra and are defined by the rule

[F,G]=\omega (IdF,IdG).

{\ displaystyle [F, G] = \ omega (IdF, IdG).}

Related Definitions

Diffeomorphism of symplectic manifolds $f\colon M\to N$ ${\ displaystyle f \ colon M \ to N}$ is called a symplectomorphism if it preserves a symplectic structure.

Properties

Darboux theorem : all symplectic manifolds are locally symplectomorphic. Thus, in a neighborhood of any point of the manifold, one can choose canonical coordinates , also called Darboux coordinates , in which the symplectic structure takes the form

\omega =d\mathbf {p} \wedge d\mathbf {q}

{\ displaystyle \ omega = d \ mathbf {p} \ wedge d \ mathbf {q}}

Moreover, the Darboux basis is found in the tangent space of each point in the neighborhood under consideration.

The Hamiltonian phase flow retains a symplectic structure:

\ L_{IdH}\,\omega =0

{\ displaystyle \ L_ {IdH} \, \ omega = 0}

Here

L_{v}

{\ displaystyle L_ {v}}

Is the Lie derivative with respect to the vector field

v

{\ displaystyle v}

. Thus, the Hamiltonian phase flow is a symplectomorphism.

Contact structure

With each symplectic 2n-dimensional manifold, a (2n + 1) -dimensional contact manifold , called its contacting , is canonically connected. Conversely, for any (2n + 1) -dimensional contact manifold, there exists its symplectization , which is a (2n + 2) -dimensional manifold.

Variations and generalizations

Variety is called multisymplectic degree $k$ ${\ displaystyle k}$ if it is given a closed non-degenerate differential k- form.

Links

D.V. Anosov . "On the development of the theory of dynamical systems." Symplectic geometry.

Literature

Arnold V.I. Mathematical methods of classical mechanics. - 5th ed., Stereotyped. - M .: URSS editorial, 2003 .-- 416 p. - 1,500 copies - ISBN 5-354-00341-5 .
Arnold V.I., Givental A. B. Symplectic geometry. 2nd ed. - Izhevsk: RHD, 2000. - 168s.
Thirring V. Course in mathematical and theoretical physics. - K .: TIMPANI, 2004 .-- 1040 s.
Fomenko A.T. Symplectic geometry. Methods and applications. - M .: Publishing House Moscow State University, 1988 .-- 414 p.