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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). 2 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). 4 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.