Poisson brackets
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Poisson Brackets. The dynamic variables can be assigned to a single set. q 1 , q 2 , …, q n , p 1 , p 2 , …, p n z 1 , z 2 , …, z 2 n Hamilton’s equations can be written in terms of z a Symplectic 2 n x 2 n matrix Return the Lagrangian. Matrix Form.

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Poisson Brackets

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Poisson brackets

Poisson Brackets


Matrix form

The dynamic variables can be assigned to a single set.

q1, q2, …, qn, p1, p2, …, pn

z1, z2, …, z2n

Hamilton’s equations can be written in terms of za

Symplectic 2n x2n matrix

Return the Lagrangian

Matrix Form


Dynamical variable

A dynamical variable F can be expanded in terms of the independent variables.

This can be expressed in terms of the Hamiltonian.

The Hamiltonian provides knowledge of F in phase space.

Dynamical Variable

S1


Angular momentum

Angular Momentum

Example

  • The two dimensional harmonic oscillator can be put in normalized coordinates.

    • m = k = 1

  • Find the change in angular momentum l.

    • It’s conserved


Poisson bracket

The time-independent part of the expansion is the Poisson bracket of F with H.

This can be generalized for any two dynamical variables.

Hamilton’s equations are the Poisson bracket of the coordinates with the Hamitonian.

Poisson Bracket

S1


Bracket properties

The Poisson bracket defines the Lie algebra for the coordinates q, p.

Bilinear

Antisymmetric

Jacobi identity

Bracket Properties

{A + B, C} ={A, C} + {B, C}

S1

{kA, B} = k{A, B}

{A, B} = -{B, A}

{A, {B, C}}+ {B, {C, A}}+ {C, {A, B}} = 0


Poisson properties

In addition to the Lie algebra properties there are two other properties.

Product rule

Chain rule

The Poisson bracket acts like a derivative.

Poisson Properties


Poisson bracket theorem

Let za(t) describe the time development of some system. This is generated by a Hamiltonian if and only if every pair of dynamical variables satisfies the following relation:

Poisson Bracket Theorem


Not hamiltonian

Equations of motion must follow standard form if they come from a Hamiltonian.

Consider a pair of equations in 1-dimension.

Not Hamiltonian

Not consistent with H

Not consistent with motion

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