How to deal with the singular reduction of the Hamiltonian n body problem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:43:10Z http://mathoverflow.net/feeds/question/97391 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97391/how-to-deal-with-the-singular-reduction-of-the-hamiltonian-n-body-problem How to deal with the singular reduction of the Hamiltonian n body problem? Dayal C Strub 2012-05-19T12:56:02Z 2012-08-29T19:54:51Z <p>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.</p> <h2>Background</h2> <p>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. <br/> [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?]<br/> However, $SO(3)$ acts properly but not freely on $Q$, so we get a stratified (by orbit type) fibration of configuration space over <em>shape space</em> $Q/SO(3)$. The principal stratum consists of non-collinear configurations, then we have the two singular strata of collinear configurations and the $n$ particle collision [2].<br/> 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. </p> <h2>Question</h2> <p>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. <br/> 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 <em>syzygy</em>?, <br/> i.e. what is the topology of the reduced phase space $M$? and what about the reduced sympectic form $\omega$ and Hamiltonian $H$? <br/> Also are there particularly well suited coordinates\charts for the reduction and reduced space that would include collinear configurations?</p> <p>Finally are there any good references discussing these issues? I can't seem to find them.</p> <hr> <p>[1] Littlejohn and Reinsch 1997 - Gauge fields in the separation of rotations and internal motions in the n-body problem<br/> [2] Iwai and Yamaoka 2005 - Stratified reduction of classical many-body systems with symmetry</p> http://mathoverflow.net/questions/97391/how-to-deal-with-the-singular-reduction-of-the-hamiltonian-n-body-problem/103797#103797 Answer by Eugene Lerman for How to deal with the singular reduction of the Hamiltonian n body problem? Eugene Lerman 2012-08-02T16:26:16Z 2012-08-29T19:54:51Z <p>My favorite paper on singular reduction is <a href="http://www.math.cornell.edu/~sjamaar/papers/stratified.pdf" rel="nofollow">"Stratified symplectic spaces and reduction"</a>. Admittedly it does not have much by way of examples, but <a href="http://www.math.cornell.edu/~sjamaar/papers/lms.pdf" rel="nofollow">"Examples of singular reduction"</a> does. Section 5 may be of particular interest. You may also want to look at <a href="http://xxx.lanl.gov/abs/dg-ga/9608010" rel="nofollow">this</a> old preprint. </p> http://mathoverflow.net/questions/97391/how-to-deal-with-the-singular-reduction-of-the-hamiltonian-n-body-problem/104916#104916 Answer by Bill Satzer for How to deal with the singular reduction of the Hamiltonian n body problem? Bill Satzer 2012-08-17T14:04:43Z 2012-08-17T14:04:43Z <p>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 <a href="http://www.math.cornell.edu/~sjamaar/papers/lms.pdf" rel="nofollow">http://www.math.cornell.edu/~sjamaar/papers/lms.pdf</a>. You might also look at McGehee's work on regularizing collisions - this is older work from the '70's and 80's.</p>