Resonance

Abstract

Delaunay was the first astronomer to use the mechanics of Hamilton and Jacobi to obtain the approximated solution of the equations of motion of a celestial body. His lunar theory [22] is a pioneer work in many respects. We credit Delaunay with the introduction of the set of angle-action variables ℓ, g, h, L, G, H in which the Lagrange equations for the variation of the orbital elements under a perturbation are canonical. His theory of the motion of the Moon is not a collection of clever tricks, as other theories in the old Celestial Mechanics. Having obtained the variation equations in canonical form, his problem was to find the solutions of the differential equations defined by the Hamiltonian

$$ H = H_0 (J) + \varepsilon \sum\limits_{h \in D} {A_h } (J)cos(h|\theta ), $$

(4.1)

where the canonical variables are J ≡ (J₁, ..., J_N) and θ ≡ (θ₁, ..., θ_N), ε is a small parameter and D ⊂ Z^N. The technique adopted by Delaunay is methodologically very clear. He defined an operation and performed it, successively, almost 500 times. This operation starts with the choice of one argument (h₁|θ) in (4.1) and the consideration of the dynamical system defined by the abridged Hamiltonian

$$ \mathcal{F}_1 = H_0 (J) + \varepsilon A_{h_1 } (J)cos(h_1 |\theta ). $$

(4.2)

This system is integrable, since the angles θ_i are present only through the linear combination (h₁|θ).