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
Qi = Qi (q, p) , Pi = Pi (q, p) ,
then by ﬁnding Hamilton’s equations in the new variables show that the transformation is
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
H= + mgq ,
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
H= − ,
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
A = −p × L + mk ,
is also a constant of the motion. List all constants of motion and determine an involutive
set for the Kepler problem.