Report abstract: This report describes investigations into applying a symplectic, second order integration method to the Lennard-Jones potential. In this report, the dynamics of one, and two particles is studied. The integrator used was the Leapfrog method, due to the fact that it is the simplest second order symplectic method available. For the two-body system, the problem is reduced to the one body problem by a symmetry reduction. Using a symplectic integrator has the advantage of preserving energy of a nearby Hamiltonian so that orbits are confined. Motivation for this study draws from understanding the application of the Leapfrog Method on the Lennard-Jones potential, and in particular, understanding the limits on the timestep. Further studies can be conducted to extend the findings of the one and two-body problems described in this report to understand the reliability of the Leapfrog method on the N-body case.

Geometric integration

Geometric integrators are numerical methods used to find algorithms for computing solutions to differential equations. Geometric integrators are used frequently in real world problems, such as modelling planetary motion and quantum mechanics. These integrators have the advantage of preserving certain properties up to round off error. Standard integrating methods are unsuitable for calculating solutions to most real world problems of that cannot be solved explicitly; in this instance geometric integrators provide a necessary means to calculating such solutions. During my studies I conducted two research papers with Prof. Robert McLachlan in the Applied Dynamics group of the Institute of Fundamental Sciences at Massey University.

"Geometric Integrator for Poincaré Maps of Hamiltonian Dynamical Systems"

Report abstract: This report describes our investigations into making a change of variables to integrate a Hamiltonian dynamical system so that a Poincaré mapping will land exactly on the section. In this report, we focus mainly on the two degree of freedom Hénon-Heiles system. The integrator used was the midpoint rule rather than the previously used Leapfrog method due to the implicit form of the new Hamiltonian. The method used was a symplectic change of variables that transformed the system so that each iteration landed exactly on the section under the stopping condition. The method preserves invariant two tori in this example as it confines the energy space and the invariant circles do not have the problem of smearing from interpolation error.

"A study of the leapfrog method applied to the Lennard-Jones potential"