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.