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 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.

Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and integrability according to Liouville

“…According to the Liouville's theorem [1,3,4], this implies the integrability in quadratures of equations of motion of the free rigid body. Moreover, since there are no more constraints in the problem (15) with the canonical brackets (16), we can use the known formula of Hamiltonian mechanics to write the solution to the equations (18) in terms of exponential of the Hamiltonian vector field [7]…”

Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

“…Let us return to the discussion of equations of motion (23) and (24), implied by the Lagrangian (22). As we saw above, any solution of this system is of the form…”

“…Then the theory of differential equations guarantees the existence and uniqueness of a solution to the Cauchy problem. Note that to prove the existence of solutions for a mixed system of differential and algebraic equations, much more effort is required [23].…”

Poincaré–Chetaev Equations in Dirac’s Formalism of Constrained Systems

“…{M i , a j } D = − ijk a k in accordance with (35). For this unconstrained Hamiltonian system we can use the known formula of Hamiltonian mechanics to write solutions to the Equation (38) in terms of an exponential of the Hamiltonian vector field [31]. Performing the same function for the vectors b and c, we obtain the solution to Euler-Poisson Equations ( 36) and (37) as follows:…”