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