Poisson Brackets

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## Poisson Brackets

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**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**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**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**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**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**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**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**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