[LECS/CM-04] Hamiltonian Form of Dynamics

The Hamiltonian of a charged particle in electromagnetic field is derived starting from the Lagrangian.

The Hamiltonian  for a charged particle in electromagnetic field is invariant under the gauge transformation \[\vec A \longrightarrow \vec A^\prime = \vec A -\nabla \Lambda\]if the canonical momentum is taken to transform as \[\vec p \longrightarrow \vec p^\prime -(e/c) \nabla \Lambda.\]

We define Poisson bracket and the Hamiltonian equations of motion are written in terms of Poisson brackets. The equation for time evolution of a dynamical variable is written in terms of Poisson brackets. It is proved that a dynamical variable \(F\), not having explicit time dependence, is constant of motion if its Poisson bracket with the Hamiltonian is zero.

We list important properties of Poisson brackets. Poisson bracket theoerem statement that the Poisson bracket of two integrals of motion is again an integral of motion, and its proof is given. A generalization of the theorem is given without proof.

In the canonical formulation of mechanics, the  state of a system is represented by a point in phase space. As time evolves, the system moves along a path in phase space. Principle of least action is  formulated in phase space. The Hamilton's equations motion follow if we demand  that, for infinitesimal variations, with coordinates at the end point fixed, the action be extremum. No restrictions on variations in momentum are imposed.

A Summary of Lagrangian and Hamiltonian formalisms is given in tabular form.