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PHY 504
Problem Set #3
due 17 September 2010
ሺݔሻ ൌ
భ
మ
sin
ଶ
ሺݔሻ
.
1.
Derivation 2.8.
2.
A particle of mass
m
moves in a potential
ܸ
݉߱
ଶ
(a)
Write down a Lagrangian and obtain Lagrange’s equation from it.
(b)
Solve the equation numerically using Mathematica.
Make three plots, for
qualitatively different sets of initial conditions. (One plot should show small
oscillations about
ݔൌ0
.)
Here is a sample code to numerically solve an ordinary
differential equation and plot the solution:
(c)
Use conservation of energy to obtain an equivalent firstorder equation.
Express
the solution
ݐሺݔሻ
as an integral.
Can Mathematica do the integral?
3.
Two particles of equal mass move in the
xy
plane and are connected by a spring of force
constant
k.
One particle is constrained to move on a circle of radius
a
.
(a)
Write down a Lagrangian for the two particles in rectangular coordinates.
Use a
Lagrange multiplier to incorporate the constraint.
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This note was uploaded on 11/16/2010 for the course PHY 504 taught by Professor Staff during the Fall '08 term at Kentucky.
 Fall '08
 Staff
 mechanics, Mass

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