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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 first-order 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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