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Poisson Brackets

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Bracket Properties. The Poisson bracket defines the Lie algebra for the coordinates q, p. ... Let za(t) describe the time development of some system. ... – PowerPoint PPT presentation

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Title: Poisson Brackets


1
Poisson Brackets
2
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

3
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

5
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.

S1
6
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
S1
kA, B kA, B
A, B -B, A
A, B, C B, C, A C, A, B 0
7
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.

8
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

9
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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