PROBLEM SET 10

Due Wednesday, November 25, 1998

Reading: Goldstein, Chap. 8 (skip pp. 358-361), Secs. 9.1-9.4

Problems: Goldstein, 8.13, 8.19, 8.22 (skip the last line), 8.32 (first part only); LD 18

Comments and hints for the Goldstein problems:

   8.13(a): Recall that p is the canonical momentum, not the kinetic momentum. The Hamiltonian must be expressed as a function of p and r. Use vector notation in (a) instead of breaking the expressions into rectangular components - vector notation is more compact and also reveals the structure and subtleties in the problem.

8.13(b): Transform the Lagrangian to the rotating system, then find H. The results of LD12 will be useful.

   8.19: Write the new Lagrangian as L' = L + dF/dt where F is a function of the q's and t only. Show that Hamilton's equations stated in terms of p', q', H' are equivalent, though not identical, to the original equations of motion in terms of p, q, H. Note that H' is a function of p', q', not p, q, and that derivatives with respect to one of the variables are calculated with the conjugate variable fixed.

   This problem shows that we can change the canonical momenta and the Hamiltonian without changing the form of Hamilton's equations of motion. The transformation is a canonical transformation with a generating function .

   8.22: The Hamiltonian is rather simple when written in terms of the canonical mometa and Euler angles, though this seems rather unlikely at the outset. To reduce the problem to quadratures, i.e., to the evaluations of integrals, use the constants of the motion to derive the relation between and t, and "orbit" integrals for and .

   8.32: Use a variational argument based on Hamilton's principle in the Hamiton-Weiss form, and introduce Lagrange multipliers to handle the differential form of the constraints. Recall Goldstein Sec. 2.4.

		LD18: Use the Hamiltonian from Prob. 8.22. Start by finding the conditions under which a steady precession is possible. A sketch of the effective potential in will help. Call its second derivative at the minimum B. It is not possible to solve explicitly for the angle at which the steady precession occurs, but you can determine the defining condition and use the resulting relation to make some simplifications elsewhere.
		Why do we not consider perturbations in the constant momenta? What would changes in these quantities do, if anything?

Send comments or questions to: ldurand@hep.wisc.edu

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		8.13(a): Recall that p is the canonical momentum, not the kinetic momentum. The Hamiltonian must be expressed as a function of p and r. Use vector notation in (a) instead of breaking the expressions into rectangular components - vector notation is more compact and also reveals the structure and subtleties in the problem. 8.13(b): Transform the Lagrangian to the rotating system, then find H. The results of LD12 will be useful.
		8.19: Write the new Lagrangian as L' = L + dF/dt where F is a function of the q's and t only. Show that Hamilton's equations stated in terms of p', q', H' are equivalent, though not identical, to the original equations of motion in terms of p, q, H. Note that H' is a function of p', q', not p, q, and that derivatives with respect to one of the variables are calculated with the conjugate variable fixed.
		This problem shows that we can change the canonical momenta and the Hamiltonian without changing the form of Hamilton's equations of motion. The transformation is a canonical transformation with a generating function .

		8.22: The Hamiltonian is rather simple when written in terms of the canonical mometa and Euler angles, though this seems rather unlikely at the outset. To reduce the problem to quadratures, i.e., to the evaluations of integrals, use the constants of the motion to derive the relation between and t, and "orbit" integrals for and .
		8.32: Use a variational argument based on Hamilton's principle in the Hamiton-Weiss form, and introduce Lagrange multipliers to handle the differential form of the constraints. Recall Goldstein Sec. 2.4.