Computationally solving bodies that push and pull Is there a way to find a solution for the positions (as a function of time) of multiple free bodies that push and pull on each other?
Say for instance I have a collection of cells which can individually contract or expand, and are joined to one another. For simplicity, I'm assuming two dimensions; no gravity, friction, or other external forces; and the cells are circular, and therefore connect at a point. When a cell contracts or expands, it simply reduces or increases its radius, respectively. When it contracts, it pulls on any cells that are attached to it. When it expands, it pushes on any attached cells, and in both cases it does so with a particular force. The mass of each cell is known.
I suspect this may be related to an N-body problem (perhaps even more difficult) and therefore cannot be feasibly solved exactly. If that's the case, can anyone suggest any approach for reasonably approximating this system?
 A: You need to formulate the mathematical model for this system. Once you have differential equations describing the behavior of these objects (cells) then you can time-integrate these equations on a computer; the only question is how many objects you want to include in the calculation - the more the bigger computer you'll need. My guess is that a few tens of cells can be handled today by a laptop, if you need hundreds or more then probably a large parallel machine is needed to make it a practical calculation. This problem is not a bad as the N-body problem because here it sounds like only forces between nearest neighbors are important, this makes it easier. Anyway, one thing that we know is the Newton law - if you have forces acting on an object then you know its acceleration hence you can find its velocity and position. However, what you need to specify is what is the force between two objects, if you know their positions and radii. And, for this problem you need to describe the time-evolution of the cell radius - how it depends on the forces acting on the cell etc. All this needs to be formulated as differential equations; but once this is done then solving it numerically is relatively straightforward. To get some feel what this calculation may look like I suggest keeping initially the radii of the cells constant, and use some simple potential to approximate the force between cells that are touching each other, and use some simple time-integration algorithm like the leapfrog method http://en.wikipedia.org/wiki/Leapfrog_integration
