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homework4 - Homework Assignment 4 Physics 302 Classical...

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Homework Assignment 4 Physics 302, Classical Mechanics Fall, 2010 A. V. Kotwal Handed out: Friday, September 24, 2010 Due in class on: Friday, October 1, 2010 Problems (Ten points for each problem unless noted otherwise.) 1. (15 points ) Consider the following first-order differential equations of motion of a particle in an external field: dx dt = p x - ay 2 dp x dt = - kx + 2 ay ( p y - 2 axy ) dy dt = p y - 2 axy dp y dt = ky + 2 ax ( p y - 2 axy ) + 2 ay ( p x - ay 2 ) , where a and k are nonzero constants. (a) Show that this system is Hamiltonian without constructing the Hamiltonian. (b) Construct the Hamiltonian for this Hamiltonian flow. 2. (20 points ) Consider the following Hamiltonian H = p 1 - p 2 2 q 1 2 + p 2 + ( q 1 + q 2 ) 2 . To solve this problem, we will make a canonical transformation so that new coordinates are related to the old as Q 1 = q 2 1 , Q 2 = q 1 + q 2 . (a) Construct a generic generating function for this transformation; (b) Find a particular generating function which will transform this Hamiltonian to a new Hamilto- nian which will depend only on P 1 and P 2 , i.e. Q 1
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