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# HW7 - PHYS 3202 Classical Mechanics II Homework#7 Due at...

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PHYS 3202 Classical Mechanics II - Homework #7 Due at 12:05pm, Friday March 18, 2011 in the class Please show intermediate steps of your calculations. Whenever appro- priate, you may draw diagram(s). Problem 1 (5 points): a particle undergoing periodic motion in a conservative force field traces out a symplectic area A = Z τ 0 p · d r in each period τ . The particle’s kinetic energy is T . If the particle’s speed is much smaller than the speed of light in vacuum, show that A = 2 h T i τ, where h T i is the average kinetic energy in each period: h T i ≡ 1 τ R τ 0 Tdt . (To prevent confusion with the kinetic energy, we use symbol τ for the period.) Problem 2 (15 points): For each of the following 3 systems, compute the energy E as a function of the symplectic area A = R τ 0 p · d r within each period τ , and check whether E A β (if yes, give the value of β ). For each system, also compute the frequency of the motion ν 1 , and then verify the relation ν = E 0 ( A ). Omit gravity. For system 2, please eliminate the radius r in favor of A in your results.

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HW7 - PHYS 3202 Classical Mechanics II Homework#7 Due at...

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