(a) Starting from the Jacobi identity , show that if and are both constants of the…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Poisson brackets, phase flow and Liouville's theorem

Problem

(a) Starting from the Jacobi identity {f,{g,H}}+{g,{H,f}}+{H,{f,g}}=0\{f,\{g,H\}\}+\{g,\{H,f\}\}+\{H,\{f,g\}\}=0, show that if f(q,p)f(q,p) and g(q,p)g(q,p) are both constants of the motion with no explicit time dependence, then {f,g}\{f,g\} is also a constant of the motion (Poisson's theorem). (b) A particle of mass mm moves under a central force, H=px2+py2+pz22m+V(r)H=\dfrac{p_x^2+p_y^2+p_z^2}{2m}+V(r), and all three components of L\mathbf{L} are known (from Chapter 4) to be conserved. Using {Lx,Ly}=Lz\{L_x,L_y\}=L_z, say what Poisson's theorem adds in this case, and whether it produces a genuinely new constant. (c) Consider nn degrees of freedom with a Hamiltonian H(q,p,t)H(q,p,t) that is twice continuously differentiable in qq and pp, possibly with explicit time dependence. Show that the Jacobian of the map z→z+v δt\mathbf z\to\mathbf z+\mathbf v\,\delta t generated by Hamilton's equations equals 1+O(δt2)1+O(\delta t^2) for any such HH, and state what this implies for a comoving region of phase space. (d) A phase-space density ρ(q,p,t)\rho(q,p,t) is found to satisfy ∂ρ/∂t=0\partial\rho/\partial t=0 at every point, for all time. Using the Liouville equation, explain why ρ\rho must be a function only of the constants of the motion, and why this does not mean ρ\rho takes the same value everywhere.

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