5. Canonical formalism.

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


with solutions


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