# 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. 