Homework 9: # 8.19, 8.24, 8.25
Michael Good
Nov 2, 2004
8.19
The point of suspension of a simple pendulum of length l and mass m is constrained to move on a parabola z = ax2 in the vertical plane. Derive a Hamiltonian governing the motion of the pendulum
Homework 5: # 3.31, 3.32, 3.7a
Michael Good
Sept 27, 2004
3.7a Show that the angle of recoil of the target particle relative to the incident
1
direction of the scattered particle is simply = 2 ( ).
Answer:
It helps to draw a gure for this problem. I dont
Homework 7: # 4.22, 5.15, 5.21, 5.23, Foucault
pendulum
Michael Good
Oct 9, 2004
4.22
A projectile is red horizontally along Earths surface. Show that to a rst
approximation the angular deviation from the direction of re resulting from
the Coriolis eect v
Homework 1: # 1.21, 2.7, 2.12
Michael Good
Sept 3, 2004
1.21. Two mass points of mass m1 and m2 are connected by a string passing
through a hole in a smooth table so that m1 rests on the table surface and
m2 hangs suspended. Assuming m2 moves only in a ve
Homework 4: # 2.18, 2.21, 3.13, 3.14, 3.20
Michael Good
Sept 20, 2004
2.18 A point mass is constrained to move on a massless hoop of radius a xed
in a vertical plane that rotates about its vertical symmetry axis with constant
angular speed . Obtain the La
Homework 12: # 10.13, 10.27, Cylinder
Michael Good
Nov 28, 2004
10.13
A particle moves in periodic motion in one dimension under the inuence of a
potential V (x) = F |x|, where F is a constant. Using action-angle variables, nd
the period of the motion as
Homework 3: # 2.13, 2.14
Michael Good
Sept 10, 2004
2.13 A heavy particle is placed at the top of a vertical hoop. Calculate the
reaction of the hoop on the particle by means of the Lagranges undetermined
multipliers and Lagranges equations. Find the heig
Homework 3: # 2.13, 2.14
Michael Good
Sept 10, 2004
2.13 A heavy particle is placed at the top of a vertical hoop. Calculate the
reaction of the hoop on the particle by means of the Lagranges undetermined
multipliers and Lagranges equations. Find the heig
Homework 11: # 10.7 b, 10.17, 10.26
Michael Good
Nov 2, 2004
10.7
A single particle moves in space under a conservative potential. Set up
the Hamilton-Jacobi equation in ellipsoidal coordinates u, v , dened in
terms of the usual cylindrical coordinates r
Homework 8: # 5.4, 5.6, 5.7, 5.26
Michael Good
Oct 21, 2004
5.4
Derive Eulers equations of motion, Eq. (5.39), from the Lagrange equation of
motion, in the form of Eq. (1.53), for the generalized coordinate .
Answer:
Eulers equations of motion for a rigid
Homework 10: # 9.2, 9.6, 9.16, 9.31
Michael Good
Nov 2, 2004
9.2
Show that the transformation for a system of one degree of freedom,
Q = q cos p sin
P = q sin + p cos
satises the symplectic condition for any value of the parameter . Find a
generating fu
Homework 1: # 1, 2, 6, 8, 14, 20
Michael Good
August 22, 2004
1. Show that for a single particle with constant mass the equation of motion
implies the follwing dierential equation for the kinetic energy:
dT
=Fv
dt
while if the mass varies with time the co
General Relativity Exam
Michael Good
April 25, 2006
Problem 3
Taking Robertson-Walker metric
ds2 = dt2 + R(t)2
dr2
+ r 2 d 2
1 kr2
with k = +1, 0, 1 and R(t) the cosmological scale. Use G = 8T and
T ; = 0 to obtain the Friedmann equation
8 2
R2 + k =
R
3
Goldstein Chapter 1 Exercises
Michael Good
July 17, 2004
1
Exercises
11. Consider a uniform thin disk that rolls without slipping on a horizontal
plane. A horizontal force is applied to the center of the disk and in a direction
parallel to the plane of th
Goldstein Chapter 1 Derivations
Michael Good
June 27, 2004
1
Derivations
1. Show that for a single particle with constant mass the equation of motion
implies the follwing dierential equation for the kinetic energy:
dT
=Fv
dt
while if the mass varies with