Consider the action integral
and the corresponding Lagrange equation
We define a new variable p, called the canonical momentum, by
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
These solutions can also be represented in the yt-plane, shown below.