By Somasuntharam Arunasalam
My project investigated a famous physics problem known as the n-body problem. This refers to the problem of finding how n bodies move with relation to each other when under the influence of gravity. The main issue with this problem is that each body will be attracting every other body by gravity. Thus, the number of variables in this problem grows quite rapidly as we introduce more and more bodies. In fact, an exact solution for this problem cannot be found for a system with more than 2 bodies. Thus, we can only attempt to simplify the problem to obtain accurate numerical solutions, i.e. solutions generated by computers.
One issue with obtaining a numerical solution is that around collisions there can be large errors. This can be seen as follows: the gravitational attraction between two bodies increases, as they get closer to each other. However, we model these bodies as points in space. Thus, a collision implies that the distance between them is zero. This amounts to the force being ‘infinite’. To avoid such a scenario, we use a tool known as regularisation. This method involves objects known as quaternions. Those who are familiar with the notion of complex numbers will know of the imaginary unit, i, which is the square root of -1. Quaternions are essentially an extension of this concept. They introduce three units, i, j and k that are all square roots of -1. Using quaternions allows us to change the variables that we work with in such a way that the equations remain the same at all points except that they do not ‘blow up’ when two bodies collide.
Using regularisation works like a double-edged sword. While it does make the equations ‘nicer’, it actually increases the number of variables in the problem. This is seen by the fact that originally, each body has three co-ordinates. Now, after regularization, the bodies have four co-ordinates each since a quaternion has a real part as well as three imaginary parts corresponding to i, j and k. Hence, we also need to simplify the problem. To do this, we use a technique known as symmetry reduction. Symmetries are very commonly used in physics. For example in simple mechanics we use laws like the conservation of momentum and energy to easily solve the problem. We try to do the same by making the equations symmetric in terms of rotations and translations of the co-ordinates. This turned out to be possible and we were able to write everything in a symmetric way.
However, this did not yield a more accurate way of solving this problem, as it did not simplify the equations enough. Despite this, the symmetry reduction allowed for the discovery of a rich structure to this problem.
Somasuntharam Arunasalam was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.