Numerical Methods for Nonholonomic Rigid Body Dynamics
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We discuss general nonholonomic systems on manifolds in the setting of both continousand discrete mechanics, before focusing on systems with symmetry that enable a reduction of the equations of motion to a quotient space of the conguration manifold. In particular systems where the conguration manifold is a Lie group are considered, and among those we focus on a rigid body that rotates under a nonholonomic constraint.Both methods for the nonholonomic rigid body that originate from discretizing the reduced Lagrange-d'Alembert principle and a reduced discrete Lagrange-d'Alembert principle are considered. The methods from the rst approach are s-stage Gauss methodsof order 2s that combine a Gauss SRK-DAE2 method with a GaussMagnus method. Among the second kind we look at a reduced nonholonomic integratorfor the rigid body by McLachlan and Perlmutter in and a modied version witha symmetric projection. The methods are compared in terms of order, error growth,reversibility, energy conservation and how well they satisfy the nonholonomic constraint. Our results for this system indicate that preservation of energy is strongly connected to the geometric properties of the solutions. In long time integration energy-conservation is seen to be the dominating factor in determining accuracy. Our results indicate that all methods, with the exception of the unmodied nonholonomic integrator preserve energy well. In the test problem we have considered reversibility and the preservation of the nonholonomic constraint important for energy conservation. However it has been observed that the symmetry of the method alone can not always guarantee good energy behavior, so a satisfactory explanation of what can cause good geometric performance for an integrator applied to a nonholonomic system is still missing.