Classical Perturbation Theories

Abstract

Many general perturbation theories devised since the nineteenth century were founded on the powerful tools of Hamiltonian mechanics. They aimed at solving the specific problem of finding the solutions of the canonical system of 2N differential equations

$$ \frac{{dq_i }} {{dt}} = \frac{{\partial H}} {{\partial p_i }}, \frac{{dp_i }} {{dt}} = - \frac{{\partial H}} {{\partial q_i }}, $$

(3.1)

(i = 1, 2, ..., N), where the Hamiltonian H is the energy of the system and may be written as

$$ H = \mathcal{H}_0 (q,p) + \mathcal{R}(q,p,\varepsilon ) $$

(3.2)

with q ≡ (q₁, ..., q_N), p ≡ (p₁, ..., p_N). $\mathcal{H}_0$ is the Hamiltonian of a separable system and $\mathcal{R}$ is a disturbing potential, analytical in some small parameter ε, and vanishing for ε = 0.