Question

What is the hamiltonian of a system? (and please don’t say its teh total energy because its only the total energy when it’s time independent.

What is the difference between the momonetum P and the canonical momentum in the hamiltonian ? from a physical and mathimatical point of view ?

when is the canonical momentum the same as the mechanical momentum ?

please explain the connection between the canonical parameter q and its conanical momemtum p in the hamiltonian.

Answer 1

Hamiltonian and Lagrangian dynamics are superior to Newtonian dynamics.

If you know the lagrangian of a system, you can calculate the Hamiltonian of the same system. Even though the dimension of Hamiltonian of a system is equal to that of energy it need not be total energy. It can be some linear combination of kinetic and potential energy usually. The Hamiltonian of a system can be calculated by the following equation

momentum in usual since is mass times velocity, but Hamiltonian dynamics, the dimension of a canonical momentum need not be that of momentum,. Since this canonical momentum Say P is associated to a generalised coordinate Q which may not have the dimensions of length (it may be an angle or something else). Canonical momentum just means that it is associated to a the generalised coordinate while momentum in usual sense is a vector which is defined as already I did above.

If the generalised coordinate is some position say x component of the body, then the canonical momentum is P_x

The partial derivative of Hamiltonian with p will give you time derivative of q