Goldstein_13_27 - Homework 12: # 10.13, 10.27, Cylinder...

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Homework 12: # 10.13, 10.27, Cylinder Michael Good Nov 28, 2004 10.13 A particle moves in periodic motion in one dimension under the influence of a potential V ( x ) = F | x | , where F is a constant. Using action-angle variables, find the period of the motion as a function of the particle’s energy. Solution: Define the Hamiltonian of the particle H E = p 2 2 m + F | q | Using the action variable definition, which is Goldstein’s (10.82): J = I p dq we have J = I p 2 m ( E - Fq ) dq For F is greater than zero, we have only the first quadrant, integrated from q = 0 to q = E/F (where p = 0). Multiply this by 4 for all of phase space and our action variable J becomes J = 4 Z E/F 0 2 m p E - Fq dq A lovely u-substitution helps out nicely here. u = E - Fq du = - F dq J = 4 Z 0 E 2 mu 1 / 2 1 - F du J = 4 2 m F Z E 0 u 1 / 2 du = 8 2 m 3 F E 3 / 2 1
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Goldstein’s (10.95) may help us remember that ∂H ∂J = ν and because E = H and τ = 1 , τ = ∂J ∂E This is τ = ∂E [ 8 2 m 3 F E 3 / 2 ] And our period is τ = 4 2 mE F 10.27 Describe the phenomenon of small radial oscillations about steady circular mo- tion in a central force potential as a one-dimensional problem in the action-angle formalism. With a suitable Taylor series expansion of the potential, find the pe- riod of the small oscillations. Express the motion in terms of J and its conjugate angle variable.
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This note was uploaded on 03/17/2012 for the course PHYS 202 taught by Professor Atkin during the Spring '12 term at Amity University.

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Goldstein_13_27 - Homework 12: # 10.13, 10.27, Cylinder...

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