Classical Mechanics (Kibble & Berkshire) · Problem 14.14

The general theory of small oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · poisson-brackets

Problem

Poisson brackets Suppose that u(q,p)u(\boldsymbol{q}, \boldsymbol{p}) and v(q,p)v(\boldsymbol{q}, \boldsymbol{p}) are any two functions of position in the phase space (q,p)(\boldsymbol{q}, \boldsymbol{p}) of a mechanical system S\mathcal{S}. Then the Poisson bracket [u,v][u,v] of uu and vv is defined by [u,v]=gradq u⋅gradp v−gradp u⋅gradq v=∑j=1n(∂u∂qj∂v∂pj−∂u∂pj∂v∂qj).[u,v] = \mathrm{grad}_{\boldsymbol{q}}\, u \cdot \mathrm{grad}_{\boldsymbol{p}}\, v - \mathrm{grad}_{\boldsymbol{p}}\, u \cdot \mathrm{grad}_{\boldsymbol{q}}\, v = \sum_{j=1}^{n}\left(\frac{\partial u}{\partial q_j}\frac{\partial v}{\partial p_j} - \frac{\partial u}{\partial p_j}\frac{\partial v}{\partial q_j}\right). Prove the following properties of Poisson brackets. Algebraic properties [u,u]=0,[v,u]=−[u,v],[λ1u1+λ2u2,v]=λ1[u1,v]+λ2[u2,v][u,u] = 0, \quad [v,u] = -[u,v], \quad [\lambda_1 u_1 + \lambda_2 u_2, v] = \lambda_1[u_1,v] + \lambda_2[u_2,v] [[u,v],w]+[[w,u],v]+[[v,w],u]=0.[[u,v],w] + [[w,u],v] + [[v,w],u] = 0. This last formula is called Jacobi's identity. It is quite important, but there seems to be no way of proving it apart from crashing it out, which is very tedious. Unless you can invent a smart method, leave this one alone. Fundamental Poisson brackets [qj,qk]=0,[pj,pk]=0,[qj,pk]=δjk,[q_j, q_k] = 0, \quad [p_j, p_k] = 0, \quad [q_j, p_k] = \delta_{jk}, where δjk\delta_{jk} is the Kronecker delta. Hamilton's equations Show that Hamilton's equations for S\mathcal{S} can be written in the form q˙j=[qj,H],p˙j=[pj,H],(1≤j≤n).\dot{q}_j = [q_j, H], \quad \dot{p}_j = [p_j, H], \qquad (1 \leq j \leq n). Constants of the motion ((i)) Show that the total time derivative of u(q,p)u(\boldsymbol{q}, \boldsymbol{p}) is given by dudt=[u,H]\frac{du}{dt} = [u, H] and deduce that uu is a constant of the motion of S\mathcal{S} if, and only if, [u,H]=0[u, H] = 0. ((ii)) If uu and vv are constants of the motion of S\mathcal{S}, show that the Poisson bracket [u,v][u,v] is another constant of the motion. [Use Jacobi's identity.] Does this mean that you can keep on finding more and more constants of the motion?

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