Integrable Systems

2Hamilton–Jacobi Theory

In this chapter we study some notions concerning the variational principle and will establish the Euler–Lagrange differential equation. We then introduce the Legendre transformation that allows to associate a system of differential equations of the first order, Hamilton’s canonical equations, with this second-order equation. In the study of a Hamiltonian system and in order to simplify the system in question, it will sometimes be necessary to perform transformations. Not all transformations transform a Hamiltonian system into a Hamiltonian system, only the canonical transformations have this effect. Such transformations will be studied. We will then introduce the Hamilton–Jacobi equation; it is a nonlinear partial differential equation, and its resolution requires the knowledge of a canonical transformation, which is generally difficult to determine. As applications, we will study the geodesics, the harmonic oscillator and the Kepler problem as well as the simple pendulum.

2.1. Euler–Lagrange equation

Let γ = {(t, q) : q = q(t), t₁ ≤ t ≤ t₂} be a curve defined on a differentiable manifold and connecting two parameter points (t₁, q₁) and (t₂, q₂). We consider the functional

[2.1]

where , is a function of class 𝒞². The variational problem is to find the function ...