Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems

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 the proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of reduction of a Hamiltonian system with known integrals of motion.

arXiv comment35 pages, matches with published version, typos fixed

References in corpus (21)

• Gravitational Faraday effect from on-shell amplitudes
• Complete set of quasi-conserved quantities for spinning particles around Kerr
• Quantum corrections to frame-dragging in scattering amplitudes
• Observables from spherically symmetric modified dispersion relations
• Effect of Particle Spin on Trajectory Deflection and Gravitational Lensing
• Gravitational analog of Faraday rotation in the magnetized Kerr and Reissner-Nordström spacetimes
• Poisson Geometry from a Dirac perspective
• Relativistic massive particle with spin-1/2, a vector bundle point of view
• Conserved non-Noether charge in general relativity: Physical definition vs. Noether's 2nd theorem
• Effective dark energy through spin-gravity coupling
• Symmetries and Covariant Poisson brackets on pre-symplectic manifolds
• Bianchi IX geometry and the Einstein-Maxwell theory
• Chiral effects in classical spinning gas
• Equatorial orbits of spinning test particle in rotating boson star
• Current Density of Majorana Bound States
• Study Zitterbewegung effect in a Quasi One-dimensional Relativistic Quantum Plasma by DHW formalization
• Spin dynamics of moving bodies in rotating black hole spacetimes
• Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field
• New symmetries, conserved quantities and gauge nature of a free Dirac field
• A variational principle, wave-particle duality, and the Schrödinger equation
• Generalized Mathisson-Papapetrou-Tulczyjew-Dixon Equations

Cited by in corpus (8)

• Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system
• Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body
• On the integrability of extended test body dynamics around black holes
• Euler-Poisson equations of a dancing spinning top, integrability and examples of analytical solutions
• Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime
• Poincaré-Chetaev equations in the Dirac's formalism of constrained systems
• Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability
• Dirac bracket and time dependent constraints