# Homework Help: Need some problems (Goldstein)

1. Oct 7, 2006

### dicerandom

Hi All,

I'm hoping that someone will be kind enough to grab their copy of Goldstein's Classical Mechanics (3rd edition) and write out the text of problems 2.4, 2.6, and 2.9 for me.

I ordered the book online a month ago but half.com screwed up my order and I still haven't gotten the textbook I canceled my order there, got a refund, and ordered the book from a different site so it should be here by Tuesday. Unfortunately my assignment is due Monday and the library on campus only has the 2nd edition, where the problems are quite different :grumpy:

So yeah, any help would be greatly appreciated :)

2. Oct 7, 2006

The exercises are numerated different in my book, so I assumed it's chapter 2, and exercises 4, 6 and 9. So, here they go:

4. Show that the geodesics of a spherical surface are great circles, i.e., circles whose centers lie at the center of the sphere.

6. Find the Euler-Lagrange equation describing the brachistochrone curve for a particle moving inside a spherical Earth of uniform mass density. Obtain a first integral for this differential equation by analogy to the Jacobi integral h. With the help of this integral, show that the desired curve is a hypocycloid (the curve described by a point on a circle rolling on the inside of a larger circle). Obtain an expression for the time of travel along the brachistochrone between two points on Earth's surface. How long would it take to go from New York to Los Angeles (assumed to be 4800 km apart on the surface) along a brachistochrone tunnel (assuming no friction) and how far below the surface would the deepest point of the tunnel be?

9. A chain or rope of indefinite length passes freely over pulleys at heights y1 and y2 above the plane surface of Earth, with a horizontal distance x2 - x1 between them. If the chain or rope has a uniform linear mass density, show that the problem of finding the curve assumed between the pulleys is identical with that of the problem of minimum surface of revolution.

You're lucky I'm bored.

3. Oct 7, 2006

### dicerandom

Judging from my notes and discussion in class those definitely look like the right problems.

Thanks a lot