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.


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.


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


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.

  • $\begingroup$ 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). $\endgroup$ – agtortorella Aug 25 '12 at 7:14
  • $\begingroup$ I think I have fixed the problem $\endgroup$ – Eugene Lerman Aug 29 '12 at 19:55
  • $\begingroup$ 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. $\endgroup$ – 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.

  • $\begingroup$ I upvoted your answer. I should have mentioned McGehee's work on my answer. $\endgroup$ – Eugene Lerman Aug 30 '12 at 13:43
  • $\begingroup$ 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. $\endgroup$ – Dayal C Strub Oct 8 '12 at 7:27

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