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homework5 - Homework Assignment 5 Physics 302, Classical...

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Homework Assignment 5 Physics 302, Classical Mechanics Fall, 2010 A. V. Kotwal Handed out: Friday, October 1, 2010 Due in class on: Friday, October 8, 2010 Problems 1. Consider the Hamiltonian of a one-dimensional simple harmonic oscillator, H ( q,p ) = 1 2 p 2 + 1 2 ω 2 q 2 . (a) Use Hamilton’s equations of motion to solve for q ( t ) and p ( t ) in terms of initial values ( q 0 ,p 0 ). (b) Evaluate the Poisson bracket { q ( t ) ,p ( t ) } with respect to ( q 0 ,p 0 ). (c) Show that the relationship between ( q ( t ) ,p ( t )) and ( q 0 ,p 0 ) represent a canonical transformation. 2. Show that the function S ( q,P,t ) = 2 ( q 2 + P 2 ) cot ωt - mωq P csc ωt is a solution of the Hamilton-Jacobi for Hamilton’s principal function for the simple harmonic oscil- lator with H = 1 2 m ( p 2 + m 2 ω 2 q 2 ) . Show that this function generates a correct solution to the motion of the harmonic oscillator. 3. Suppose the potential in a problem of one degree of freedom is linearly dependent upon time, such
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