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HW6 - PHYS 3202 Classical Mechanics II Homework#6 Due at...

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PHYS 3202 Classical Mechanics II - Homework #6 Due at 12:05pm, Friday March 11, 2011 in the class In homework, quiz, and final exam, please show intermediate steps of your calculations. Whenever appropriate, you may draw diagram(s). Problem 1 (10 points): Find the symplectic area A enclosed by a clockwise loop in the 2 dimensional phase plane: q ( θ ) = c 1 cos θ, p ( θ ) = - c 2 sin θ, where c 1 and c 2 are positive constants, and the parameter θ increases from 0 to 2 π . If we parameterize the loop as q = c 1 cos( λ 2 ) , p = - c 2 sin( λ 2 ) , where the parameter λ increases from 0 to 2 π , what is the symplectic area enclosed by the loop? If every point on the plane moves according to Hamilton’s equations, governed by a particular Hamiltonian function H ( q, p, t ), the above loop will in general be transformed into different shapes at different locations in the phase plane at later times. If ∂H ∂t > 0 for all values of ( q, p, t ), how does the symplectic area of the transformed loop depend on the time t ? Is H conserved?
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