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Homework 8
1) A particle of mass m can slide freely along a wire AB whose perpendicular distance to the origin
O is h. The line OC rotates about the origin at constant angular velocity    . The position of
the particle can be described in terms of the angle  and distance q to the point C. If the particle
is subject to a gravitational force, and if the initial conditions are
 (0)  0,
q(0)  0,
q(0)  0
show that the time dependence of the coordinate q is
q(t ) 
g
2 2
(cosh(t )  cos(t ))
Sketch this result. Compute the Hamiltonian for the system, and compare with the total energy. Is
the total energy conserved?
2) A particle is constrained to move (without friction) on a circular wire rotating with constant
angular speed  about a vertical diameter. Find the equilibrium position of the particle, and
calculate the frequency of small oscillation around this position. Find and interpret physically a
critical angular velocity   c that divides the particle’s motion into two distinct types.
Construct phase diagram for the two cases   c and   c .
3) A particle of mass m moves under the influence of gravity along the helix z  k , r  const. ,
where k is a constant and z is vertical. Obtain the Hamiltonian equations of motion.
4) (a) Formulate the canonical equations including friction.
A hanger of mass m for a car with a spring-damping system is moving along the x-direction with
constant velocity v. It performs sinusoidal oscillations in y according to: ys  a cos(
masses of both tires can be ignored.
2
x) . The
l
(b) Formulate the equation of motion for the y-coordinate of the hanger using the canonical
equations.
(c) How does the spring constant D and the friction coefficient c have to be dimensioned to ensure
that the hanger is moving as quietly as possible?
5)
Proof the Jacobi identity:
[f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0
Homework #10
PHY523 Fall 2012
Due 12/6/2012
Reading Assignment: Please read Chapter 13
Problem #1
Pre-Quantum
In this question we “manipulate” the harmonic oscillator Hamiltonian. This is an illustration of the
manner in which classical mechanics was first formally introduced. Some of these things should look
familiar to you, if you have studied the quantum harmonic oscillator!
We consider a Hamiltonian of a 1D harmonic oscillator:
H=
1 2
p + m2 ω 2 q 2
2m
(a) Express H as a function of
r
mω p a=
q+i
,
2
mω
a∗ =
r
mω p q−i
.
2
mω
(1)
(2)
Make sure you simplify your answer.
(b) There is a mathematical operation referred to as “Poisson Brackets.” For two functions A(p,q),
and B(p,q), we define the poisson bracket as
[A, B] ≡
∂A ∂B
∂B ∂A
−
∂q ∂p
∂q ∂p
(3)
Calculate the Poisson brackets for: [a, a∗ ] , [H, a] , and [H, a∗ ]. You may find the following
relationship useful: [AB, C] = A[B, C] + B[A, C] for any functions A, B, and C.
(c) Show that for an arbitrary function f (q, p, t), the following relationship is true:
∂f
+ [f, H]
f˙ =
∂t
(4)
(d) Write down the equations of motion for a(p, q) and a∗ (p, q) and solve them for arbitrary initial
conditions.
Problem #2
Phase-Space
Problem #3
Liouville’s
Theorem
Find the hamiltonian, H for a mass m confined to the x axis and subject to a force F = −kx3 where
k > 0. Sketch and describe the phase-space orbits.
A beam of protons is moving along an accelerator pipe in the z-direction. The particles are uniformly
distributed in a cylindrical volume of length L0 (in the z direction) and radius R0 . The particles
have momenta uniformly distributed with pz in an interval p0 ± pz and the transverse (along x-y)
momentum p⊥ in a circle of radius ∆p⊥ . To increase the particles’ spatial density, the beam is
focused by electric and magnetic fields, so that the radius shrinks to a smaller value R. What does
Liouville’s theorem tell you about the spread in the transverse momentum p⊥ and the subsequent
behavior of the radius R? (Assume that the focusing mechanism does not affect the length of the
bunch of particles moving through the pipe, or the spread in pz ).