(a) A single particle moves in space under a conservative potential. Set up the Hamilton–J...

(a) 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

 

For what forms of V(u, v, ϕ) is the equation separable?

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(b) Use the results of part (a) to reduce to quadratures the problem of a point particle of mass m moving in the gravitational field of two unequal mass points fixed on the z axis a distance 2a apart.