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You can find the lecture notes at this link
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Week 2 - Hamilton's Principle [1] [3]
Summary
Lagrange's Equations and the Principle of Least Action. Cyclic
coordinates and conserved quantities.
Reading List
- ``Lagrangian Dynamics''
REF: Wells, D. A. 1967, Chapter 17.
- ``Classical Mechanics''
REF: Goldstein, H. 1980, Chapter 2.
- ``Mechanics''
REF: Landau, L. D., Lifshitz, E. M. 1988, Chapter 1.
Problem Set - 28 September 2005 - Answers
Problem 1 - Great Circle
Show that the shortest (or longest) line connecting two points on a
sphere is a segment of a great circle
Problem 2 - Spring
A mass m attached to a coil spring having a constant k, oscillates along a smooth horizontal line with a motion given by
x = A sin ω t where ω = (k/m)1/2. Assuming a varied path represented by
x = A sin ω t + ε sin 2 ω t,
show that for the actual path taken over the interval t=0 to
t=π/2ω (one fourth of a complete oscillation),
/ t=π/2ω
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/ t=0
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δL dt = 0
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and that for the varied path this integral is equal to 3/8 mπωε2.
Problem 3 - Power Lines
Write an integral to calculate the total potential energy of a cable of
mass M and length L that follows a curve y(x). Find the
function y(x) that minimizes the potential energy.
Problem 4 - Fermat's Principle
Fermat's principle states that light takes the path that minimizing
the time to travel between two points. In a medium of index of
refraction n light travels at a velocity c/n in a
straight line. Use Fermat's principle to derive Snell's Law, i.e that
at an interface between materials with indicies of refraction
n1 and n2, the angle that the
light makes with respect to normal on each side of the interface is
n1 sin &theta1
= n2 sin &theta2
Last modified: Wednesday, 30 November 2005 12:14:25
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