Hamiltonian & Symmetries

Hamilton’s Equations

Hamiltonian

Recall that the canonical momentum is defined as , we define the hamiltonian as the Legendre transform of the Lagrangian, viewed as a function of :

Hamilton’s Equations

We can easily derive the following Hamilton’s equations from the Euler-Lagrange equation:

Theorem

The hamiltonian in a conservative potential is the total energy.

Proof Under the assumption that , and is a quadratic form with respect to , in a conservative system, only depends on . We invoke the lemma, and obtain .

Corollary

In particular, for a system whose Hamiltonian does not dependent explicitly on time, the Hamiltonian is conserved.

Proof We have the following equations hold:

Symmetries

Proposition

One coordinate is cyclic if and only if , which leads to a constant momentum.

Noether's Theorem

Every continuous symmetry of the action corresponds to a conserved quantity and vice versa. where is the Lagrangian, is the action and is some generalized coordinates.

Proof A proof of Noether’s theorem using symplectic geometry is given here.