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Poisson Brackets

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Poisson Brackets

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

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.

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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

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.

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

Bilinear

Antisymmetric

Jacobi identity

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

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

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

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

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

Product rule

Chain rule

The Poisson bracket acts like a derivative.

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:

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

Consider a pair of equations in 1-dimension.

Not consistent with H

Not consistent with motion

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