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The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex volume of any N+1 points. One constraint in our 3D world is that D(4)=0.

Give each point a mass (Mi) and dynamic interdistances (i.e., Rij(t) are functions of "time") according to inertia and non-relativistic "Newtonian gravity". Is there a simple mathematical way to express the resulting dynamic constraints in these terms?

I'm basically wondering if these interdistances are somehow superior to normal coordinates in multi-body problems, but gravity doesn't appear to fit nicely...sorry for the vague question!

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I really can't parse this question. – Igor Rivin Feb 17 '12 at 3:44
I think the OP is trying to ask whether it is better to do classical dynamical systems of $N$ interacting particles instead of on $\mathbb{R}^{6N}$, where the position and momentum of each of the particles are specified, but on something like $\mathbb{R}^{N(N-1)}$, where the relative distances and relative speeds of pairs of particles are specified. If that is the case, the answer is probably "no" for sufficiently large number of particles: $N(N-1)$ is bigger than $6N if $N > 7$, so you have too many degrees of freedom and requires certain constraints (triangle inequality and what not). – Willie Wong Feb 17 '12 at 11:27
Willie's right about my goal: I'm looking for the formula for an interdistance acceleration, $d^{2}R_{12}/dt^{2}=f( \lbrace M_{i},R_{ij},dR_{ij}/dt \rbrace )$, for some number of point masses, like eight. In principle, my only solution is to first numerically embed these eight points into space and then to calculate this interdistance acceleration, but I was hoping for an analytic method (which could avoid numerics altogether since I don't need the embedding in the end). Can this really only be done numerically? I would even be willing to relax some of the geometry constraints if it helped! – bobuhito Feb 21 '12 at 8:06

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