# 5. Canonical formalism.

Consider the action integral

and the corresponding Lagrange equation

We define a new variable p, called the canonical momentum, by

If

we might according to the implicit function theorem solve for   as

In particular we may write the Hamiltonian as

Example 4: Consider a particle of mass m and potential energy V(y), moving in one dimension. Then

This implies that

the regular momentum. If we solve for   we get

The Hamiltonian thus becomes

the total energi expressed in the position y and the momentum p.

We also note that

Lagrange’s equations can thus be written as a system of equations in the variables y and p,

Definition: The equations above are called Hamilton’s equations.

Example 5: Consider the harmonic oscillator in Example 1. Then

Hamilton’s equations are in this case

We solve these equations in the yp-plane (the so called  phase plane). Division yields

This is a separable equation. Integration gives

where C is a constant. Geometrically this is a family of ellipses in the phase plane.

This are the curves the system develops along in the phase space.
Remark: Euler-Lagrange’s equation for the harmonic oscillator is

with solutions

These solutions can also be represented in the yt-plane, shown below.