PH6246, Section 3916, Fall 2015, Homework 10
Due at the start of class on Friday, November 20.
Answer all questions. Please write neatly and include your name on the front page of your
answers. To gain maximum credit you should explain your reasoning and show all working.
1. Using Cartesian coordinates, consider motion in three dimensions that is governed by the
following Hamiltonian:
1
H=
n
where r =
√
r · r and p =
√
n
p
b
− n ,
a
r
p · p and {r i , pj }PB = δji .
a) Find the equations of motion.
b) Show that there is a constant of the motion given by:
D=
p·r
− Ht.
n
c) The transformation Q = λr, p = λP is canonical. If we also use t′ = λn t, show
that the equations of motion are invariant. Note: The constant D is associated
with this invariance.
2. Consider the following point transformation in a system with two degrees freedom:
Q1 = q12
and
Q2 = q1 + q2 .
a) Find the most general transformation equations for P1 and P2 consistent with the
overall transformation being canonical.
b) Show that with a particular choice for P1 and P2 the Hamiltonian:
a p1 − p2 2
+ bp2 + c(q1 + q2 )2
H=
2
2q1
where a, b and c are constants, can be transformed into one in which both Q1 and
Q2 are ignorable.
c) Hence solve the problem and obtain expressions for q1 , q2 , p1 and p2 in terms of
their initial values and t.
3. Consider the following transformation:
√
2P1 sin Q1 + P2 ,
x = α1
√
y = α1
2P1 cos Q1 + Q2 ,
px = α2
√
2P1 cos Q1 − Q2 ,
√
py = − α2
2P1 sin Q1 − P2 ,
where α is some fixed parameter.
a) Show that it is canonical.
b) Apply this transformation to the problem of a non-relativistic particle of charge
q moving in an electromagnetic field for which A = 21 (−By, Bx, 0). With an
appropriate choice of α express the Hamiltonian for this problem in terms of the
(Qi , Pi ) coordinates.
c) From this Hamiltonian, find the motion of the particle as a function of time.
4. The Hamiltonian for a system has the form:
1 1
2
4
+p q .
H=
2 q2
a) Find the equations of motion for q.
b) Now find a canonical transformation that reduces H to the form of an harmonic
oscillator.
c) Show that the solution for the transformed variables is such that the equation of
motion in a) is satisfied.
5. Consider a system with at least two constants of motion.
a) Prove that the Poisson brake of two constants of motion is itself a constant of the
motion, even when the constants depend explicitly on time.
b) If the Hamiltonian and a quantity F are constants of the motion, show that the
nth partial derivative of F with respect to t must also be a constant of the motion.
c) Consider a free particle of mass m, for which the Hamiltonian is certainly conserved. Show that there is a constant of the motion given by the quantity:
F =x−
pt
.
m
d) By direct computation, show that the partial derivative of F with t agrees with
{H, F }, both of which are constants of the motion by parts a) and b).
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