[Phys-L] Re: Help on a problem from Goldstein

[Leigh Palmer <palmer@SFU.CA>]

My thanks to those who responded.

On 13-Apr-05, at 2:00 AM, Automatic digest processor wrote:

I asked:

> Problem 13 from Chapter 6 of Goldstein's "Classical Mechanics" reads:
>
>    Show that the relativistic motion of a particle in an attractive
>    inverse square law of force is a precessing ellipse. Compute the
>    precession of the perihelion of Mercury resulting from this
>    effect. ( The answer, about 7" per century, is much smaller than
>    the actual precession of 40" per century which can be accounted
>    for correctly only by general relativity.)
>
> Can someone kickstart my brain with a hint as to what approach I should
> consider to solve this problem?

Brian Whatcott replied:

> A somewhat similar problem is sometimes offered:   [Phys 4222 UFlorida]
>
> "Suppose that the sun were surrounded by a dust cloud of uniform
> density  which extended at least as far as the orbital radius of the
> Earth. The effect of the dust cloud is to modify the gravitational
> force experienced by the Earth, so that the potential energy of the
> Earth is (neglecting the effects of the planets)
> U(r) = -GMm/r + 1/2 kr^2
> where M is the mass of the sun, m is the mass of the Earth, G is the
> gravitational constant, and k = 4 pi rho mG/3 (note that k > 0, so
> this additional term is attractive). The effect of the dust cloud is
> to cause elliptical orbits about the sun to precess slowly."
>
>
> Does this provide any traction, I wonder?

It certainly relates to the problem I'm working on, but the precession
regresses the apsides rather than advancing them. I'm looking for
another way to describe Mercury's anomalous motion, which has the
apsides advancing.

Joel Rauber contributed:

> Leigh,
>
> You're showing your age, that is the problem in the first edition.   It
> is now problem 26 in chapter 7.

I haven't seen a newer edition. Herbert Goldstein is probably deceased
by now. Since his book sold for less than ten bucks in the US and was
pirated in the third world, he probably starved to death.

Unethical behavior is killing physicists.

Thanks to your note, Joel, I was introduced to cramster.com. No one has
submitted a solution to this problem yet. I don't think I would buy one
even if I could.

Unethical behavior is killing physics.

John Mallinckrodt writes:

> Leigh,
>
> Apparently the assignment is to apply special, but not general
> relativity.  I suppose I'd try to solve the equation
>
>    -G M r_vec/r^2 = d(gamma v_vec)/dt
>
> separating out the radial and azimuthal components and performing
> something like the usual small amplitude radial oscillation analysis
> to find the apsidal angle.
>
> But maybe that isn't saying anything you hadn't already thought of
> and the stumbling block lies further along that path.

Thanks. That sounds like a sound approach. I hadn't appreciated the
applicability of that approach, but I'm ultimately looking for a more
intuitive description based on special relativistic effects like mass
increase with velocity*. I will try it with an inverse cubic term on
the left. (John is acquainted with the problem I'm working on.)

and finally Bernard Cleyet chimed in:

> Oh good that's the ed. I can't find. Cost < $10.
>
> My first thought was somewhere I'd heard one could expand and use the
> first and second? terms.  Thereby, adding a cubic term to the force.
> (quad potential, naturellement).  Some theorem, I think, that only the
> linear and quad central force result in no precession.  That's why some
> early on suggested gravity was not exactly inverse square.

Yes. I hit on that technique myself. It is applicable to my problem,
though an unphysical term, one that seems to depend on the radius of
the central mass (!), also shows up. I can ignore it, of course, but it
is larger than the inverse r-cube term I'm keeping, so I feel I must be
on the wrong track. While the problem I'm working on is really
unphysical, it should have a reasonable solution lacking such terms.

Many thanks for the responses, fellows. It is nice to have colleagues
on my desktop who will indulge my questions somewhat less jusgmantally
than the ones I see face to face at school.

Leigh

* I guess I should bite my tongue. I do understand that mass is now an
invariant. I'm quite sure it wasn't when Goldstein wrote that problem,
and the old ways die hard.

N. B. Anyone who is as upset about cramster.com as I am should start a
new thread with a different subject line. I do have something to say
about that.
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