Poisson Brackets

1
Poisson Brackets
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Matrix Form

• The dynamic variables can be assigned to a single
  set.
• q1, q2, , qn, p1, p2, , pn
• z1, z2, , z2n
• Hamiltons equations can be written in terms of
  za
• Symplectic 2n x 2n matrix
• Return the Lagrangian

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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.

S1
4
Angular Momentum

• Example
• The two dimensional harmonic oscillator can be
  put in normalized coordinates.
• m k 1
• Find the change in angular momentum l.
• Its conserved

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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.
• Hamiltons equations are the Poisson bracket of
  the coordinates with the Hamitonian.

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Bracket Properties

• 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 kA, B
A, B -B, A
A, B, C B, C, A C, A, B 0
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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.

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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

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Not Hamiltonian

• 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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