Problems: Goldstein, 9.37; LD 20, LD 21, LD 22, LD 23

9.37: Use standard spherical coordinates with the vertical. For the last question, it is useful to note that where we choose standard spherical coordinates with . The question is why = Lz and =Ly' have a vanishing bracket while Lz and Ly do not. What is different about Ly and Ly' or and ?

Work the problem out explicitly using the Poisson bracket relations rather than using Goldstein's vector notation, which simply summarizes the results of the calculation. Note that both A and B must transform as vectors under the Poisson bracket relations for A·B to be a scalar with respect to rotations. An expression such as A·b with b a constant unit vector, for example b = (1,0,0), simply picks out one component of A in a particular coordinate frame, and does not transform as a scalar. That is, b is not a true dynamical vector with respect to the Poisson brackets with L. This observation is relevant to the interpretation of the results in Goldstein 9.37.

		K, L, p, and H are the generators of the Poincaré group, the group of relativistic boosts, rotations, and translations in space and time. The expression for K given here can be derived using Noether's theorem.
		Note that while q is a cononical variable, t is not. The time simply appears as a parameter used to describe the development of the motion. In interpreting the result in (c), recall that the variations of the canonical variables used in Hamilton's principle and generated through the Poisson bracket relations are all at fixed time t, while a Lorentz transformation relates q'(t') to q(t). Reexpressing the transformed coordinate q'(t) obtained from the bracket relations as a function of t' should lead you to the "expected" result. The fact that one cannot construct a consistent set of generators for the Poincaré group in the presence of interactions written in terms of the instantaneous positions of the particles in spacetime is the basis of the "no interaction theorem" for relativistic particles. It does not preclude interactions of particles with fields.

		It is useful in problems involving changes of the coordinates q to start with the Lagrangian rather than the Hamiltonian formulation of the problem, change variables there, and then determine the new canonical momenta and Hamiltonian. That is the procedure followed here, with the new Lagrangian given. The alternative in a purely Hamiltonian description is to construct the generating function F(q,P) for the point transformation of the coordinates and use that to determine the new momenta, and then transform the original Hamiltonian.
		Parabolic coordinates are used to separate and solve the Schrödinger equation for the Coulomb scattering problem in quantum mechanics. The separation allows one to find analytic expressions for the quantum version of the Rutherford scattering amplitude and cross section. (See, for example, L. Schiff, Quantum Mechanics, Sec. 21 and the end of Sec. 16. Note, however, that Schiff uses a different definition of the parabolic coordinates with 2, 2 replaced by , .) In contrast, separation in spherical coodinates only gives the scattering amplitude as an infinite partial-wave series, not a very useful result.

Start by determining . The integral for can be evaluated by writing it as an integral between the turning points (the points qmin and qmax where dWq/dq vanishes) and using q2 as the integration variable. See Goldstein Eqs. 10-108c and 10-116 for an alternative evaluation of the resulting radial integral. This problem illustrates the derivation of action variables and the transformation of the Hamiltonian using the Hamilton-Jacobi method. Note that the transformation to "proper" action variables with no degenerate frequencies must be canonical.

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© 1997, 1998, Loyal Durand