Homework #4 — PHYS 601 — Fall 2014
Due on Monday, October 13, 2014 online
Professor Victor Yakovenko
Office: 2115 Physics
Web page: http://physics.umd.edu/~yakovenk/teaching/
GPS: Goldstein, Poole, Safko, Classical Mechanics, 3rd edition, 2002, ISBN 0-201-65702-3
LL: Landau and Lifshitz, Mechanics, 3rd edition, 1976, ISBN 978-0-7506-2896-9
Total score is 36 points.
LL Ch. 3, GPS Ch. 3.1–3.9, The Central Force Problem
1. Elliptic orbits in the Kepler problem, 12 points.
(a) Sketch an ellipse. Select the x axis along the line connecting the foci and the y
axis in the perpendicular direction.
(b) The Kepler orbit for the gravitational potential U (r) = −α/r is given by the
following equation in the polar coordinates, see GPS Eq. (3.85) and LL Eq. (15.5):
r0
= 1 + e cos φ,
r
r0 =
L2
,
mα
(1)
where 0 ≤ e < 1 is the eccentricity of the orbit. Using Eq. (1) calculate the
maximal and minimal distances rmax and rmin and their combinations (rmax +
rmin )/2 and (rmax − rmin )/2 in terms of r0 and e. Show in your sketch that
(rmax + rmin )/2 is the major axis of the ellipse, and (rmax − rmin )/2 is the distance
from the center of the ellipse to a focus.
(c) In the coordinate system with the origin at the focus, the Cartesian coordinates
of the particle are x0 = r cos φ and y 0 = r sin φ. Now let us shift the origin of the
coordinate system to the center of the ellipse, so that x = x0 + (rmax − rmin )/2 and
y = y 0 . Show that in the new coordinate system the orbit satisfies the following
equation and find expressions for a and b in terms of r0 and e:
x2 y 2
+ 2 = 1.
a2
b
(2)
Hint: Write Eq. (1) as r = r0 − ex0 and square it x02 + y 2 = (r0 − ex0 )2 , then
substitute x0 in terms of x and obtain Eq. (2).
(d) Calculate the area πab of the ellipse in terms of the energy |E|, angular momentum
L, and other parameters of the problem.
Hint: The turning points rmax and rmin are solutions of the following equation:
−|E| =
L2
α
−
2
2mr
r
⇐⇒
r2 −
α
L2
r+
= 0.
|E|
2m|E|
(3)
For this quadratic equation, the sum of the roots is rmax + rmin = α/|E| = 2a.
√
Using your previous results, show find that b = ar0 and eliminate r0 using the
second Eq. (1).
2
Homework #4, Phys601, Fall 2014, Prof. Yakovenko
(e) Calculate the period T of the orbit using the area of the ellipse and the areal or
sectorial velocity given in LL Eq. (14.3) and GPS Eq. (3.8). Confirm the third
law of Kepler T ∝ a3/2 (see LL Ch. 2.10).
2. Problem 3.19 from GPS, 12 points. Yukawa potential.
A particle move in the Yukawa potential
k
V (r) = − e−r/a ,
r
(4)
where k and a are positive constants.
(a) Reduce the problem to the equivalent one-dimensional problem and obtain effective potential for radial motion. Discuss qualitative nature of orbits for different
values of energy and angular momentum.
(b) Consider an orbit that is nearly circular and calculate the angular advance of the
apsides per revolution in terms of the parameter ρ/a, where ρ is the radius of the
circular orbit. Assume that ρ a, so ρ/a 1 is a small parameter, and perform
calculations to the lowest non-vanishing order in ρ/a.
Directions:
Obtain the radial potential energy V 0 (r) as in Eq. (3.22’). Use the equation
dV 0 /dr = 0 to find the radius of a circular orbit and use the second derivative
d2 V 0 /dr2 to find the frequency of radial oscillations. Find the difference between
the radial frequency and the angular velocity φ̇.
(c) Show that the nearly circular orbit can be approximated by a precessing ellipse.
Is precession in the same or the opposite direction to the orbital angular velocity?
Caveats: Some editions of GPS give a formula for F (r) instead of V (r), which is not
equivalent to Eq. (4). Be sure to use Eq. (4). Do not trust the answer to Part (b)
given in some editions of GPS.
3. Problem 3.28 from GPS, 12 points. Magnetic monopole.
The Lorentz force in the presence of a magnetic field B is F = (q/c) v × B in the
Gaussian system of units, where q is the electric charge, c is the speed of light, and v
is velocity. In the SI system of units, there is no c in the denominator.
Part (a) is valid for any central potential V (r), not necessarily the Kepler potential.
So, do not assume that V (r) = −k/r in this Part.
In Part (b), you need to construct the central force f (r) and the corresponding potential
V (r) such that there is a conserved vector. So, do not assume that V (r) = −k/r in
this Part, but construct V (r), which would be similar to −k/r, but with an additional
term.
October 3, 2014