What is the original purpose of the Laplace-Runge Lenz vector?

[BobG]

What was the original purpose of the Laplace Runge Lenz vector?

I understand the components (comparing centrifugal force to gravitational force) and, more importantly, that if you multiply the LRL by the reciprical of the geocentric gravitational constant (or helio..., etc), you get a vector that points towards perigee in a magnitude that tells you your eccentricity.

But, as is (before you convert to the eccentricity vector), it seems to be incomplete? If it was designed to calculate an orbit's eccentricity, you would think he would have divided out the geo/helio/etc centric constant right off the bat. The only thing I can think of is that a different method, independent of a geocentric gravitational constant, used to be used to find the eccentricity and that, if the eccentricity were already known, the LRL vector might have been used to calculate the geocentric gravitational constant?

 

[styler]

The idea is that central force problems have a deeepr symmetry than SO3.
The LRL vector shows that this is so.
It was noted that since it commuted with the hamiltonian (possion brackets there i guess, classically) there was a larger symmetry algebra and another conserved quantity. Though there was no good physical interpretation of this quantity it ws considred valauble form the standpoint of symmetry

There's actually a very cool way to derive the modes of spherical problems using it and Pauli used it to look at the Schrödinger eq for spinless non rel hydrogen. I did an undergrad project that devloped this method in modern language and showed that subgroups of SO4 (interesting ones) abound in the simple Schrödinger problem despite the fact that QM books say nothing about it.


Sternberg makes some lame comments about this in his otherwise pretty cool book "Group Theory and Physics" but if you want more details let me know and i can point you to some obscure papers that i once dug up.

 

[JohnDubYa]

I suggest reading the work by Delande and Gay, as well as Dieter Wintgen. My dissertation has a very-readable appendix devoted to the symmetry of the hydrogen atom with respect to the Runge-Lenz vector, and I can email anyone interested a copy.

 

[styler]

Cool.
Does it include discussion of representations of S0(4,2) ~ S0(2,1) X S0(4)?
I believe the related work of Demeyer, Vanden Berge and Fack is collected in a book of papaers presented at the 15th internatrional Coloquim on Group Theoretical Methods in Physcis.

 

[JohnDubYa]

Yes, to your question. It also includes a complete primer on Lie algebra.

 

[konndanley]

Hello John,

I have a colleague who is interested in your thesis, and trying to track it down.

I hope this can find it's way...