PS8 - Phys601/F11/Problem Set 8 Due Oct 31, 2011 (Upgrades...

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Phys601/F11/Problem Set 8 Due Oct 31, 2011 (Upgrades in red) 8.1H A particle has the Hamiltonian H = p 2 /2 – q 2 /2. 1. Sketch the contours of constant H for all values of H. Sketch contours for the function Q’ = pq. Show that contour lines of H and Q’ are orthogonal to each other by computing the dot product of the phase space gradients of H and Q’. 2. Make a canonical transformation to coordinates P and Q where P = H. Find Q, then p(P,Q) and q(P,Q) and their inverses (be careful with the sign of P and any square roots). Make a sketch of the contour lines of Q. This is obviously not orthogonal. 8.2H Consider a mass m moving vertically in the Earths gravitational field, g . Write down the Hamiltonian and Hamilton’s equations. Sketch the contours of constant H. Make a canonical transformation to coordinates P and Q where P = H. Find Q, then p(P,Q) and q(P,Q) and their inverses. Make a sketch of the contour lines of Q. What are Hamilton’s equations in P(t) and Q(t)? Compare with the H equations derived earlier. 8.3H
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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PS8 - Phys601/F11/Problem Set 8 Due Oct 31, 2011 (Upgrades...

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