Lie Group Integrators for Cotangent Bundles of Lie Groups and Their Application to Systems of Dipolar Soft Spheres
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The objective of this thesis is to study numerical integrators and their application to solving ordinary differential equations arising from mechanical systems. Many mechanical problems are naturally formulated on Lie groups or on the groups tangent or cotangent bundle, especially those where the equations of motion are derived from variational principles. The bundles inherit the structure of the original Lie group and can be considered Lie groups themselves. Often these systems are highly complicated and one will need to use numerical methods for solving them. One class of numerical methods which is particularly suitable for such problems is Lie group integrators. We shall apply the Runge-Kutta-Munthe-Kaas (RKMK) methods in such a setting. We will give an introduction to Lie group theory, which we utilize to identify a Lie group structure on the cotangent bundles of Lie groups. Next, we provide a short overview of Lagrangian and Hamiltonian mechanics. Thereafter, we will conduct numerical experiments on two Hamiltonian systems using the RKMK methods. The first system concerns the motion of a free rigid body, and the second system the motion of a system of identical dipolar soft spheres in a molecular dynamics setting.