Almost Poisson integration of rigid body systems. (English) Zbl 0782.70001

The authors discuss the numerical integration of Lie-Poisson systems using the mid-point rule. Such systems result from the reduction of Hamiltonian systems with symmetry by Lie group actions. They also present examples of reconstruction rules for the full dynamics. A primary motivation is to preserve in the integration process various conserved quantities of the original dynamics. A main result of this paper is an \(O(h^ 3)\) error estimate for the Lie-Poisson structure, where \(h\) is the integration step size. The authors note that Lie-Poisson systems appear naturally in many areas of physical science and engineering, including theoretical mechanics of fluids and plasmas, satellite dynamics, and polarization dynamics. They consider a progressively complicated series of examples related to rigid body systems. They also consider a dissipative example associated to a Lie-Poisson system. The behavior of the mid-point rule and an associated reconstruction rule is numerically explored.


70-08 Computational methods for problems pertaining to mechanics of particles and systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
65K10 Numerical optimization and variational techniques
Full Text: DOI Link