I would like to model the motion of the planets as point size masses using matrices. I am not including moons, maybe later...Any help is much appreciated as I work through this

The complete graph with 10 nodes shows 45 edges.

10nodes.jpg

This graph is directed if along every edge the orientation is taken to be X( na to nb < na) is negative.

Numbering the massive bodies, I number starting at 1 for the sun and give pluto node number 10.

Such that

X_{14}indicates the vector pointing from the sun to mars with the negative direction being toward the sun.

The position of each node in R3 is given by the vector Xi(t) = [Xi1, Xi2, Xi3] with initial positions Xio at t = to.

The force vector F, between any two nodes is F_{ij}and is proportional to the product of the masses of nodes i and j, divided by the ||Xij||^2.

Assume continuous and differentiable in all dimensions.

dx/dt = V

dV/dt = A = d2x/d2t

And Fij(t) = k*mi*mj * (xij hat) / (xij)^2

Even though this matrx Fij is 45 dimensions I can tell that it will be symmetrical and singular, since all terms Fii are zero and that the rank of the matrix will be 44 ( I think) leaving a nullspace of 1 dimension.

will stop here for now as I am having trouble seeing the equation in the form Ax= b. My first thought is to rearrange it in someway but the method is not clear to me. I am seeing exponents in a matrix equation so I am thinking this has to do with eigen values somehow, bu the matrix is singular.