Physics 210 Classical Mechanics Autumn, 2003
Homework Assignment #1
Due Tuesday, October 7
1. Derivation 4, Chapter 1. Nonholonomic constraints and integrating factors.
2. Derivation 9, Chapter 1. Gauge transformation of the electromagnetic field.
Compare the effect on the Lagrangian with Equation 1.57′.
3. Derivation 10, Chapter 1. Invariance of the Lagrange Equations under point
transformations. Careful, you need to be quite rigorous and careful with your
derivatives and partial derivatives. This is good practice.
4. Exercise 14, Chapter 1. Example of kinetic energy in generalized coordinates. Do
not assume that the two points remain in the plane of the circle!
5. Exercise 22, Chapter 1. Double pendulum. Use the generalized coordinates shown in
the figure. Be careful—they are not orthogonal coordinates.
6. Calculus of variations:
a) Find the extremal of ∫+
2
12
11dty
t&
(
)
01 =y,
(
)
12 =y
b) Find the geodesics on a sphere. Use spherical coordinates with constant
a
r
=
.
Choose your integration variable so that you can immediately write a first integral
of the Euler equation. The second integration then is a simple trig integral. To
recognize your result as a great circle, find in terms of
φ
and
θ
the equation of
intersection of a sphere with a plane that passes through its center.
7. Exercise 14, Chapter 2. Hoop rolling on a cylinder. Note that we are working a
simplified problem in which we assume that the hoops never slip until they lose
contact (not realistic, of course). Therefore, you do not need to employ a Lagrange
multiplier for the rolling constraint, but only for the radial constraint.
8. Exercise 19, Chapter 2. Symmetry and conserved quantities. All but the last are
pretty obvious. For the last case (helical solenoid) work it out from first principles.
Call the slope of the solenoid α and its radius R. What two generalized coordinates
are appropriate for specifying a point on the solenoid? Now extend this to a set of 3
coordinates specifying an arbitrary point in space. Which coordinate is cyclic, given
the stated symmetry? What is its conjugate momentum?
9. Show that the effect of a uniform distribution of dust of density ρ about the sun is to
add to the gravitational attraction of the sun on a planet of mass m an additional
attractive central force mkrF
−
=
′
, with 3/4Gk
πρ
≡
(Hint: think back to
applications of Gauss’ law in integral form in E&M for spherically symmetric charge
distributions). If the mass of the sun is
M
, find the angular velocity of revolution of
the planet in a circular orbit of radius 0
r, and find the angular frequency of small
radial oscillations. (Hint: do this by expanding the effective radial potential in a
Taylor series and finding the coefficient of the quadratic term.) You should find that
the radial frequency is different from the rotational frequency, so that the orbit is not
closed. Hence show that if
F
′
is much less than the attraction due to the sun, a
