PHY 3221 Classical Mechanics, Homework #8, due 8:30 am, 04/02/12
1 Consider a periodic function, y(t), with period 2π, such that in the interval [−π, π) it is defined
by:
{
− sin2 (t), −π ≤ t < 0;
y(t) =
sin2 (t),
0 ≤ t < π.
a) Find the Fourier representation of y(t).
b) Integrate the Fourier representation of y(t) (term by term) and plot the sum of the first five terms
of the result. Compare it with the exact integral of y(t).
c) Differentiate the Fourier representation of y(t) (term by term) and plot the sum of the first five
terms of the result. Compare it with the exact derivative of y(t).
d) What is the degree of differentiability of i) y(t), ii) its integral, and iii) its derivative?
e) Explain which of the integral and the derivative you have calculated in parts (b) and (c) is better
represented by its Fourier representation, and give a reason for your observation.
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2 A cylinder of radius ρ has points on its surface labelled by the cylindrical coordinates (z, ϕ).
a) Write out an expression for the distance along a curve between
P1 and P2 , ending up
∫ 2 two points,
′
with an integral to be performed over z, i.e., of the form 1 f (z, ϕ, ϕ )dz.
b) Find ∂f /∂ϕ′ , and use the Euler Lagrange equation for the integral to show that this derivative
must be a constant, say a.
c) Rearrange your result to solve for ϕ′ as a function of {z, a, ρ}.
d) Integrate this equation to find ϕ as a function of {z, a, ρ}.
e) If the path makes an angle θ with the z-direction, find an expression for sin(θ) in terms of {a, ρ}.
Hence show that a cannot be completely arbitrary.
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3 Using spherical polar coordinates, consider a sphere of radius R. Imagine that a curve on the
sphere is parameterized by a parameter u.
a) Write our an integral for the length of the curve on the surface of the sphere, ending up with an
integral to be performed over u.
b) Find ∂f /∂ϕ′ , where ′ represents a derivative with respect to u, and√use the Euler Lagrange
equation related to ϕ to show that ∂f /∂ϕ′ , must be a constant, say 1/ 1 + a2 .
c) Rearrange your equation so as to be able to solve for ϕ′ . Note: eliminating ϕ′ in favor of θ′ would
not provide any new information in the Euler Lagrange equation related to θ. Instead, rewrite
the equation you have obtained here in terms of dϕ/dθ.
d) Using the substitution cot(θ) = av, integrate the equation to find ϕ as a function of v, and hence
as a function of θ.
e) Explain why the solution you have found has appropriate properties to represent a great circle.
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