Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems
Alexei A. Deriglazov
Abstract
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion.
Topics & Concepts
First class constraintJacobi identityPoisson bracketSymplectic geometrySymplectic manifoldPoisson algebraSuperintegrable Hamiltonian systemSymplectic integratorSymplectomorphismHamiltonian systemPoisson manifoldHamiltonian mechanicsPhysicsHamiltonian (control theory)Manifold (fluid mechanics)Algebra over a fieldDifferential geometryClassical mechanicsPure mathematicsCovariant Hamiltonian field theoryMathematicsSymplectic representationLie algebraQuantum mechanicsMechanical engineeringPhase spaceMathematical optimizationEngineeringHomotopy and Cohomology in Algebraic TopologyAdvanced Topics in AlgebraAdvanced Algebra and Geometry