Geodesic motion on the symplectic leaf of $$SO(3)$$ with distorted e(3) algebra and Liouville integrability of a free rigid body
Alexei A. Deriglazov
Abstract
Abstract The solutions to the Euler–Poisson equations are geodesic lines of SO (3) manifold with the metric determined by inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra e (3). The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler–Poisson equations is written in terms of exponential of the Hamiltonian vector field.
Topics & Concepts
Poisson algebraPoisson bracketMathematicsSymplectic geometryGeodesicSymplectomorphismEuler's formulaPoisson manifoldTensor fieldVector fieldPure mathematicsSymplectic manifoldMathematical analysisHamiltonian (control theory)Mathematical physicsLie algebraExact solutions in general relativityGeometryMathematical optimizationElasticity and Wave PropagationDynamics and Control of Mechanical SystemsQuantum chaos and dynamical systems