Hi,
(reposting this from sci.math)
For calculating gravity in software for a multibody system, one method is to do a gravity calculation between each pair of bodies per timestep, so for a large number of masses the number of required calculations goes up quickly. One method to reduce the required calculations is to lump areas of the space so that all masses in that area are considered a single mass. This is effective when there are groups of masses far apart from each other.
I was thinking of an alternative way to calculate simulated gravity that generates a space curvature field from all the combined bodies gravitational fields, and then keeps this curvature field up to date as the bodies change or move. Each timestep the curvature field is used to adjust the motions of each mass.
The calculations required for adjusting the masses motions scales linearly with the number of bodies.
The tricky part is to make an easy to update curvature field that can have its curvature efficiently adjusted due to the motion of the masses.
I think doing it fully mathematical with "reversible" polynomials or some equivalent idea might work..
This is a shot in the dark but maybe something like this:
"A class of reversible quadratic polynomial vector fields on S2"
If the overall equation describing the total gravitational field can be adjusted when a mass is moved without having to recalculate the total equation, then it might be an efficient way to simulate multibody gravity.
For example a complicated overall gravitational field equation could be adjusted when a mass translates past a certain amount that it is determined that an update to the overall gravitational field equation is required to maintain a specified accuracy.
The math for this would need to adjust the overall equation without recalculating it which is the hard part I think, and seems like it would be quite advanced math if it is possible. Each masses gravity curvature formula impact on the overall gravitational field formula needs to be able to be added, removed, and readded to the overall formula efficiently.
Even if it works it would still be a lot of calculations required to update the overall gravitational field equation.
Any idea if this is possible or practical?
cheers, Jamie