Goldstein_7_17_26

Goldstein_7_17_26 - Homework 11: # 10.7 b, 10.17, 10.26...

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Homework 11: # 10.7 b, 10.17, 10.26 Michael Good Nov 2, 2004 10.7 A single particle moves in space under a conservative potential. Set up the Hamilton-Jacobi equation in ellipsoidal coordinates u , v , φ defined in terms of the usual cylindrical coordinates r , z , φ by the equations. r = a sinh v sin u z = a cosh v cos u For what forms of V ( u,v,φ ) is the equation separable. Use the results above to reduce to quadratures the problem of point parti- cle of mass m moving in the gravitational field of two unequal mass points fixed on the z axis a distance 2 a apart. Answer: Let’s obtain the Hamilton Jacobi equation. This will be used to reduce the problem to quadratures. This is an old usage of the word quadratures, and means to just get the problem into a form where the only thing left to do is take an integral. Here T = 1 2 m ˙ r 2 + 1 2 m ˙ z 2 + 1 2 mr 2 ˙ φ 2 r = a sinh v sin u ˙ r = a cosh v sin u ˙ v + a sinh v cos u ˙ u z = a cosh v cos u ˙ z = a sinh v cos u ˙ v - a cosh v sin u ˙ u Here ˙ r 2 + ˙ z 2 = a 2 (cosh 2 v sin 2 u +sinh 2 v cos 2 u )(˙ v 2 + ˙ u 2 ) = a 2 (sin 2 u +sinh 2 v )(˙ v 2 + ˙ u 2 ) 1
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To express in terms of momenta use p v = ∂L ˙ v = ma 2 (sin 2 u + sinh 2 v v p u = ∂L ˙ u = ma 2 (sin 2 u + sinh 2 v ) ˙ u because the potential does not depend on ˙ v or ˙ u . The cyclic coordinate φ yields a constant I’ll call α φ p φ = mr 2 ˙ φ = α φ So our Hamiltonian is H = p 2 v + p 2 u 2 ma 2 (sin 2 u + sinh 2 v ) + p 2 φ 2 ma 2 sinh 2 v sin 2 u + V To find our Hamilton Jacobi expression, the principle function applies S = W u + W v + α φ φ - Et So our Hamilton Jacobi equation is 1 2 ma 2 (sin 2 u + sinh 2 v ) [( ∂W u ∂u ) 2 +( ∂W v ∂v ) 2 ]+ 1 2 ma 2 sinh 2 v sin 2 u ( ∂W φ ∂φ ) 2 + V ( u,v,φ ) = E This is 1 2 ma 2 [( ∂W u ∂u ) 2 +( ∂W v ∂v ) 2 ]+ 1 2 ma 2 ( 1 sinh 2 v + 1 sin 2 u ) α 2 φ +(sin 2 u +sinh 2 v ) V ( u,v,φ ) = (sin 2 u +sinh 2 v ) E A little bit more work is necessary. Once we solve for V ( u,v,φ ) we can then
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Goldstein_7_17_26 - Homework 11: # 10.7 b, 10.17, 10.26...

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