Reminder: Show the details of your argument, explain what you are doing in the course of your solutions, and give appropriate interpretations of the results! All are looked for in the grading.

Comments and hints:

		2.2 is more difficult than most if tackled in general. The general solution is not expected, but is worth studying. It can be described in terms of the intersection of a sphere with a plane through the center of the sphere provided you choose the integration constants properly (and remember your analytic geometry from advanced calculus!)
		To get an easy solution, use spherical coordinates with the initial point on the polar axes. This choice is always possible - why?
		2.3: Don't forget the constants of integration in the solution. How are they to be determined? And don't assume from the beginning that the motion takes place on a cycloid - that is what is to be shown!
		Motion on a cycloid has the interesting property that it is isochronous: the length of time which it takes for a particle to slide to the bottom starting at rest is independent of the height at which it is released. You may wish to show this. The time is a minimum only for the conditions which give the brachistochrone. There are various interesting applications of this result.

Comment:

This is an inverse of the classical isoperimetric problem of maximizing the area enclosed by a rope of fixed length. Remember the constants of integration, but determine them as soon as you can!

Comments and hints:

		LD3: (a) Use the constraint to eliminate . The algebra here can get messy so think before you calculate! Part (a) illustrates the appearance of an unexpected Lagrangian constraint through the Euler-Lagrange equations, and the elimination of this holonomic constraint by elimination of one variable.
		Part (b) involves a change to a simpler set of coordinates (spherical coordinates), and illustrates the method of changing variables in the Lagrangian. What is the "velocity" in the new coordinates? We will see the equations for free motion on a sphere again later.
		Part (c) gives a useful alternative to using the constrained Lagrangian. The potential term in L vanishes when the constraint is imposed, but the constraint still exists as shown in (a). By keeping the original form of the kinetic energy, but imposing the constraint in the differential form d(constraint) = 0 using the Lagrange multiplier method given in Goldstein, Sec. 2.4, one gets a simply solvable set of equations which display the motion in easily interpreted form. Note: Be sure to carry through the solution of the final equations, apply the constraint, and interpret the result!

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

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© 1997, 1998, Loyal Durand

Remark: The Higgs model for symmetry breaking in the standard electroweak theory is of this general type at low energies and large values of the Higgs-boson coupling lambda, with the complication that there are separate variables and at each point in space, and that the variables at different points are coupled.