Classical Dynamics of Particles and Systems · Problem 7-30

Central-Force Motion — Marion & Thornton

Physics — Mechanics · poisson-brackets

Problem

Consider any two continuous functions of the generalized coordinates and momenta g(qk,pk)g(q_k, p_k) and h(qk,pk)h(q_k, p_k). The Poisson brackets are defined by [g, h]≡∑k(∂g∂qk∂h∂pk−∂g∂pk∂h∂qk)[g,\,h] \equiv \sum_k \left(\frac{\partial g}{\partial q_k}\frac{\partial h}{\partial p_k} - \frac{\partial g}{\partial p_k}\frac{\partial h}{\partial q_k}\right) Verify the following properties of the Poisson brackets: (a) dgdt=[g, H]+∂g∂t\dfrac{dg}{dt} = [g,\,H] + \dfrac{\partial g}{\partial t} (b) q˙j=[qi, H]\dot{q}_j = [q_i,\,H],   p˙j=[pj, H]\;\dot{p}_j = [p_j,\,H] (c) [pl, pj]=0,  [ql, qj]=0[p_l,\,p_j] = 0,\;[q_l,\,q_j] = 0 (d) [ql, pj]=δlj[q_l,\,p_j] = \delta_{lj} where HH is the Hamiltonian. If the Poisson bracket of two quantities vanishes, the quantities are said to commute. If the Poisson bracket of two quantities equals unity, the quantities are said to be canonically conjugate. (e) Show that any quantity that does not depend explicitly on the time and that commutes with the Hamiltonian is a constant of the motion of the system. Poisson-bracket formalism is of considerable importance in quantum mechanics.

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