Symmetry of Hamiltonian Systems

摘要

In the present study we use the formalism of Hamiltonian system onsymplectic manifold due to Reeb, given in Abraham and Marsden and Arnold toderive the equation of motion for a particle on a line in a plane with a spring forceand for a free particle in n-space. The time flows for both the problems mentionedabove are also determined and proved that the determined flow is a Hamiltonianflow i.e., the symmetry of a Hamiltonian system. A non-Hamiltonian flow is alsoconsidered and it is shown that by changing the symplectic form and the phasespace of the system we can convert it into a Hamiltonian flow. The translation androtational symmetry related to linear and angular momentum respectively for themotion of a free particle in n-space is also considered, which is useful in reducingthe phase space of a mechanical system.