# How to deal with the singular reduction of the Hamiltonian n body problem?

I would like to consider the reduced Hamiltonian $n$ body problem, but am struggling with the angular momentum reduction seeing as the $SO(3)$ action is not free and the reduction is singular.

## Background

Starting with the 'translation-reduced system', seeing as the $R^3$ action is easy to deal with (e.g. explicitly via Jacobi vectors), I have been following Littlejohn and Reinsch [1], and considering the action of $SO(3)$ on the (translation-reduced) configuration space $Q \cong R^{3(n-1)}$, which feel natural for this mechanical system.
[Is it better to consider the lifted $SO(3)$ action on $T^* Q$, the momentum map $J$ and use symplectic reduction? and if so why?]
However, $SO(3)$ acts properly but not freely on $Q$, so we get a stratified (by orbit type) fibration of configuration space over shape space $Q/SO(3)$. The principal stratum consists of non-collinear configurations, then we have the two singular strata of collinear configurations and the $n$ particle collision [2].
Littlejohn and Reinsch [1] only consider the non-collinear fibration, which gives a principal $SO(3)$ bundle over the non-collinear stratum of shape space. Iwai and Yamaoka [2] also consider collinear configurations, but separately.

## Question

I feel that it should be possible to consider both non-collinear and collinear configurations simultaneously, probably staying away from $n$ body collisions, but don't know how to go about this.
Is it possible to talk of such as a reduced Hamiltonian system $(M, \omega, H)$, say if I ensure that the angular momentum is not parallel to the line of syzygy?,
i.e. what is the topology of the reduced phase space $M$? and what about the reduced sympectic form $\omega$ and Hamiltonian $H$?
Also are there particularly well suited coordinates\charts for the reduction and reduced space that would include collinear configurations?

Finally are there any good references discussing these issues? I can't seem to find them.

[1] Littlejohn and Reinsch 1997 - Gauge fields in the separation of rotations and internal motions in the n-body problem
[2] Iwai and Yamaoka 2005 - Stratified reduction of classical many-body systems with symmetry

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My favorite paper on singular reduction is "Stratified symplectic spaces and reduction". Admittedly it does not have much by way of examples, but "Examples of singular reduction" does. Section 5 may be of particular interest. You may also want to look at this old preprint.

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Dear Eugene Lerman, the links don't work properly. If I am not wrong, then your linked papers should be: "Stratified symplectic spaces and reduction"(math.cornell.edu/~sjamaar/papers/stratified.pdf), "Examples of singular reduction"(math.cornell.edu/~sjamaar/papers/lms.pdf) and "Stability of symmetric tops via one variable calculus"(xxx.lanl.gov/abs/dg-ga/9608010). –  Giuseppe Tortorella Aug 25 '12 at 7:14
I think I have fixed the problem –  Eugene Lerman Aug 29 '12 at 19:55
Eugene, thanks for your reply! Unfortunately, I have not had the time this summer to properly look at the papers you mention. Actually, I was hoping to reduce the system via an explicit change of coordinates, so that I could then have the Hamiltonian and symplectic form explicitly, and also not get lost in the proper (symplectic) reduction via momentum maps, which I still don't fully understand. But I will have a good look at what you linked and try to understand whether it gives me enough information about the system. –  Dayal C Strub Oct 8 '12 at 7:27

I am not completely sure I understand your question. There seems to be at least two parts, one concerning singular reduction and the other dealing with regularizing collisions. There is a fairly extensive literature in each area. For the first, you might start with http://www.math.cornell.edu/~sjamaar/papers/lms.pdf. You might also look at McGehee's work on regularizing collisions - this is older work from the '70's and 80's.

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I upvoted your answer. I should have mentioned McGehee's work on my answer. –  Eugene Lerman Aug 30 '12 at 13:43
thanks for your reply. I was actually mainly concerned with the singular reduction and hoping I could forget about the collissions and blow up to start with, but will have a look at McGehee's work when I get there. –  Dayal C Strub Oct 8 '12 at 7:27