By Diana Nguyen, The University of Sydney
This project studies the classic problem in physics: the n-body problem. This problem is concerned with how n particles (or extended bodies) move relative to each other under the influence of only gravity. Complications arise as each body attracts the others within the system and so their motions are highly entangled. Thus, the number of variables in this problem grows very rapidly as more bodies are introduced. In fact, an exact solution for this problem cannot be found for a system with more than 2 bodies (the 2 body problem is also known as the Kepler problem). The best we can do is to find accurate numerical solutions that can then be used to simulate and hence understand the interactions of these bodies.
Numerical solutions usually give accurate approximation. However, large errors occur when any of these bodies collide with one another. This is because of Newton’s law of universal gravitation: the gravitational attraction between two bodies is inversely proportional to the square of their separations. This is not so much of a problem with extended bodies but since we usually model an extended object as a point particle with all its mass concentrated at its centre of gravity, this gravitational attraction can “blow-up” during collisions. To rectify these errors, we must regularise the problem. The process of regularisation involves enlarging the system by adding a redundant degree of freedom to each particle so that during collisions, the usual separation in 3-dimensional space may be zero but the new “separation” in this extended space remains non-zero. There are many ways of performing such an extension, but in our project, we study a particular method of regularisation that involves the use of 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 the complex numbers, whereby we induce two extra imaginary units j and k that also are square roots of -1. There are both pros and cons when it comes to working with quaternions: even though quaternion multiplication is associative and they obey the usual laws of calculus like product and quotient rules, it is not commutative.
Using regularisation also has its downside. While it does make the equations ‘nicer’ and eliminate singularities during binary collisions (those that involve only 2 bodies), it does not remove blow-ups due to collisions of more than 2 bodies. The problem also introduces extra variables as originally each body has three co-ordinates. After regularization, however, 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. So while we have solved the problem with accuracy, we have sacrificed quite a bit of efficiency. A way of recovering some of this efficiency is to use a technique known as symmetry reduction.
Symmetry pervades all of physics and is the most elegant and useful tool when it comes to simplifying physical problems. Noether’s theorem allows us to establish a conservation law whenever our system has a continuous symmetry. For example, conservation of momentum arises as a consequent of translational symmetry in space, and conservation of energy comes from time translational symmetry. We try to do the same by making the equations symmetric in terms of rotations and translations of the co-ordinates (with a focus on the 3-body problem).
This turned out to be possible. 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.
Diana Nguyen was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.