Classical Mechanics P601
Homework 3
1. A surface of revolution has two given parallel rings as it’s boundaries. What should
the shape of the surface be so that it has minimal possible area?
2. Assume that the speed of light in a given slab of material is proportional to the
height above the base of the slab. Show that light moves in circular arcs in this material.
You can use the Fermat’s principle of least time i.e. light taken the path of extremum
time between two points.
3. As mentioned in class, one significant advantage of Lagrangian mechanics is that
one can work in arbitrary coordinate systems, even non-inertial ones. Consider free
particle with respect to a frame rotating with angular velocity ~ω = (0,0, ω) with respect
to inertial frame. Write down the Lagrangian in terms of coordinates of the rotating
frame and derive equations of motion.
4. A particle slides on the inside surface of a frictionless cone. The cone is fixed
with its tip on the ground and it’s axis vertical.The half-angle at the tip is α. Find the
equation of motion for the motion under gravity.
5. A pendulum consists of mass mand a massless stick of length l. The pendulum
support oscillates horizontally with the position given by x(t) = Acos(ωt). What is the
general solution for the angle of the pendulum as a function of time?
6. In the class, we derived conserved charge for rotation about z-axis. Derive the
conserved charge for the case of rotation about some arbitrary direction ˆn.
7. Find the conserved quantity corresponding to Galilean transformations for a system
of Newtonian particles with potential energy depending on relative position only.
8. Extremize the functional
E[φ] = Zd3x1
8G|∇φ|2+ρ(x, y, z)φ(x, y , z)(0.1)
9. Find the energy function hand corresponding equations of motion for a relativistic
particle L=−m√1−v2(in units c= 1).
10. For a force of the form Fi=f(r)xi, find the potential. Show that lij =xjdxi
dt −
xidxj
dt is a conserved quantity.Interpret this.
11. Consider an electron electrostatically bound in an atom. Using virial theorem,
show that when electron loses potential energy (e.g. when making a transition between
en- ergy levels), half the energy goes in increased kinetic energy (rest goes in emission,
for example).
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