MATH 402 - Assignment #3
Due on Monday February 21, 2011
Name —————————————–
Student number —————————
1
Problem 1:
Consider a pendulum consisting of a bob of mass m and negligible physical extension on the end of a massless rigid rod of length l, suspended
below a fixed pivot.
(a) Write down the lagrangian in terms of a single generalized coordinate, the angle θ the rod makes with the vertical.
(b) Drive the equation of motion for θ(t).
(c) Assume the pendulum has a small oscillation, solve the differential
equation from part (b).
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Problem 2:
Introduce a suitable set of generalized coordinates and derive the equations of motion for a single particle of mass m constrained to move on
the surface of a given sphere of radius R; V = mgz with g > 0 constant (spherical pendulum). Find and simplify the differential equation
which needs to be satisfied depending on only one of the variables but
do not solve the equation.
3
Problem 3:
(a) Write down and solve the Hamilton-Jacobi differential equation for
a system of one particle constrained to lie on a given straight line;
V = kx2 /2 with k > 0 constant and x the displacement from a fixed
point on the line (harmonic oscillator).
(b) Find out the generating function S(∂H/∂q, q, t).
(c) Find the solution of the hamilton equations of motion (use K ≡ 0).
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Problem 4:
Let ϕ(x) and ψ(c) be two distinct solutions of τ φ00 (x) + λσ(x)φ(x) = 0
for the the same value of λ.
(a) Prove that the wronskian w = ψϕ0 − ϕψ 0 is constant.
(b) Use the end point conditions to prove that w is constant zero.
(c) Prove that ϕ and ψ are linearly dependant.