Description
Question 1 (2 marks)
Consider a harmonic oscillator with Hamiltonian,
H =
p
2
2m
+
mω2
2
q
2
. (1)
(a) Use the Hamilton-Jacobi equation to show that Hamilton’s principal function for this problem is,
S(q, α, t) = mω
2
(q
2 + α
2
)cot(ωt) − mωαqcosec(ωt). (2)
(b) Demonstrate that S(q, α, t) furnishes the expected solution for the dynamics of the oscillator.
Question 2 (3 marks)
Consider a projectile whose motion is restricted to a 2D plane (defined by x and y co-ordinates). Use the
Hamilton-Jacobi equation to confirm that the path of the projectile under the influence of gravity is:
x(t) = vxt + x(0),
y(t) = −
1
2
gt2 + vyt + y(0).
(3)
Make sure that your solution includes an explicit form for the Hamilton principal function.
Question 3 (2 marks)
Consider a particle of mass m in a one-dimensional box potential,
V (x) = (
V0 if |x| ≤ a0
∞ if x > a0
. (4)
Use the action-angle formalism to compute the frequency of the particle’s periodic motion. Make sure
your expression is given in terms of the particle’s initial condition. Is your answer consistent with your
expectations?
Question 4 (3 marks)
Consider a particle of mass m that is constrained to move along a wire parameterized by
x(φ) = l [φ + sin(φ)] ,
y(φ) = l [1 − cos(φ)] ,
(5)
where l is taken to be a constant. The particle is subject to gravity along y (pointing down).
(a) Compute a Lagrangian describing the particle’s motion.
(b) Use your answer to (a) to compute the associated Hamiltonian. Justify that the Hamiltonian is equal
to the energy and is conserved (but do not assume H = T + V to initially compute the Hamiltonian).
(c) Obtain an expression for the canonical momentum pφ in terms of φ and the energy E.
(d) The motion described by (a) and (b) is periodic. Use action-angle variables to show that the frequency
of the motion is identical for any initial condition with φ ≤ π.
Hints: You might find the substitution
u =
p
2mgl/E sin(φ/2) handy to solve an integral that arises.