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Poisson Brackets

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Matrix Form The dynamic variables can be assigned to a single set. q 1, q 2, …, q n, p 1, p 2, …, p nq 1, q 2, …, q n, p 1, p 2, …, p n z 1, z 2, …, z 2nz 1, z 2, …, z 2n Hamilton’s equations can be written in terms of z Symplectic 2 n x 2 n matrixSymplectic 2 n x 2 n matrix Return the LagrangianReturn 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. S1S1

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Angular Momentum Example The two dimensional harmonic oscillator can be put in normalized coordinates. m = k = 1m = k = 1 Find the change in angular momentum l. It’s conservedIt’s 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. Hamilton’s equations are the Poisson bracket of the coordinates with the Hamitonian. S1S1

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Bracket Properties The Poisson bracket defines the Lie algebra for the coordinates q, p. BilinearBilinear AntisymmetricAntisymmetric Jacobi identityJacobi identity S1S1 {A + B, C} ={A, C} + {B, C} { A, B} = {A, 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 ruleProduct rule Chain ruleChain rule The Poisson bracket acts like a derivative.

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Poisson Bracket Theorem Let z (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 motion Not consistent with H next

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Hamiltonian. Generalized Momentum The generalized momentum was defined from the Lagrangian. Euler-Lagrange equations can be written in terms of p.

Hamiltonian. Generalized Momentum The generalized momentum was defined from the Lagrangian. Euler-Lagrange equations can be written in terms of p.

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