Poisson Bracket

DEFINITION

The Poisson bracket is a Bilinear map turning two Differentiable Function s on a Symplectic Manifold into a Function on that Symplectic Manifold . In particular, if we have two Function s, ''f'' and ''g'', then the Poisson bracket

:

\{f,g\}=	ilde{\omega}(df,dg)

where ω is the Symplectic Form ,

ilde{\omega}

is the Two-vector such that if ω is viewed as a map from Vectors to 1-forms ,

ilde{\omega}

is the linear map from 1-forms to Vectors satisfying

\omega(	ilde{\omega}(\alpha))= \alpha

for all 1-forms α and d is the Exterior Derivative . The bivector

ilde{\omega}

is sometimes called a Poisson structure on the symplectic manifold.

CANONICAL COORDINATES

In Canonical Coordinates

(q^i,p_j)

on the Phase Space , the Poisson bracket takes the form

:

\{f,g\} = \sum_{i=1}^{N} \left[ 
rac{\partial f}{\partial q^{i}} rac{\partial g}{\partial p_{i}} -
rac{\partial f}{\partial p_{i}} rac{\partial g}{\partial q^{i}}

ight]

.

LIE ALGEBRA

The Poisson brackets are Anticommutative . Note also that they satisfy the Jacobi Identity . This makes the space of Smooth Function s on a Symplectic Manifold an infinite-dimensional Lie Algebra with the Poisson bracket acting as the Lie Bracket . The corresponding Lie Group is the group of Symplectomorphisms of the symplectic manifold (also known as Canonical Transformation s).

Given a differentiable Vector Field ''X'' on the Tangent Bundle , let

P_X

be its Conjugate Momentum . The conjugate momentum mapping is a Lie Algebra anti-homomorphism from the Poisson bracket to the Lie Bracket :

:

\{P_X,P_Y\}=-P_{

{Link without Title} }.

This important result is worth a short proof. Write a vector field ''X'' at point ''q'' in the Configuration Space as

:

X_q=\sum_i X^i(q) rac{\partial}{\partial q^i}

where the

\partial /\partial q^i

is the local coordinate frame. The conjugate momentum to ''X'' has the expression

:

P_X(q,p)=\sum_i X^i(q) \;p_i

where the

p_i

are the momentum functions conjugate to the coordinates. One then has, for a point

(q,p)

in the Phase Space ,

:

\{P_X,P_Y\}(q,p)= \sum_i \sum_j \{X^i(q) \;p_i, Y^j(q)\;p_j \}

:::

=\sum_{ij} 
p_i Y^j(q) rac {\partial X^i}{\partial q^j} - 
p_j X^i(q) rac {\partial Y^j}{\partial q^i}

:::

= - \sum_i p_i \;

{Link without Title} ^i(q)
:::

= - P_{

{Link without Title} }(q,p) \,

The above holds for all

(q,p)

, giving the desired result.

TIME EVOLUTION

The time evolution of a function ''f'' on the symplectic manifold can be given as a one-parameter family of Symplectomorphism s, with the time ''t'' being the parameter. The total time derivative can be written as

:

rac{d}{dt} f=
rac{\partial }{\partial t} f + \{\,f,H\,\} =
rac{\partial }{\partial t} f - \{\,H,f\,\} =
\left(rac{\partial }{\partial t}  - \{\,H, \cdot\,\}
ight)f.

	CATEGORIES ABOUT POISSON BRACKET

	symplectic geometry

	hamiltonian mechanics

	binary operations

	fundamental physics concepts

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Information About

Poisson Bracket