Example sheet IV: 2.3.09 Hamiltonian Dynamics S2 2008/09 Exercise 1 ( Criteria for canonical transformations (da capo) ) Show that if the fundamental Poisson brackets are invariant under the time-independent transformation Qi = Qi (q, p) , Pi = Pi (q, p) , then by ﬁnding Hamilton’s equations in the new variables show that the transformation is canonical. Find the values of α, β such that the transformation Q = q α cos βp , P = q α sin βp , is canonical and obtain the form of the generating function F3 (p, Q) for this transformation. Exercise 2* ( Poisson’s Theorem ) Given that u(q, p, t) and v(q, p, t) are two constants of motion, proof that the Poissonbracket w = [u, v] is also a constant of motion. Exercise 3 ( HJE for free particle ) Find the Hamilton–Jacobi equation for a free particle in 3-dimensions. Solve this equation for the characteristic function W . Exercise 4 ( HJ method for freely falling particle ) Use the Hamilton–Jacobi method to ﬁnd a solution for the motion of a particle of mass m falling in a uniform gravity ﬁeld whose Hamiltonian is given by p2 H= + mgq , 2m where the initial values of q and p are q0 and p0 respectively. Exercise 5 ( HJE for d = 2 oscillator ) Find the Hamilton–Jacobi equation for a two dimensional harmonic oscillator in cartesian co-ordinates. Solve this equation for the characteristic function W . Find x(t) and y(t). Determine the frequencies directly using action-angle variables. Exercise 6 ( Runge-Lenz vector ) The Hamiltonian for the Kepler problem (inverse square law attractive central force) is p2 k H= − , 2m r where r2 = ri ri , p2 = pi pi and k > 0. Show that the angular momentum vector, L = r × p, is a constant of the motion. Furthermore show that the eccentricity (Runge–Lenz) vector A deﬁned by r A = −p × L + mk , r is also a constant of the motion. List all constants of motion and determine an involutive set for the Kepler problem.
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