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Poisson Brackets
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1. Poisson Brackets

2. 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

3. 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

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. • It’s conserved

5. 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

6. 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

7. 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

8. 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

9. 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 next