On the Ideal Resonance Problem

Abstract

The Ideal Resonance Problem is soluble in power series in $\varepsilon^ \frac{1}{2}$, where ε is a small parameter. The zeroth order approximation, in the Hori perturbation method, is given by the motion of a simple pendulum. The perturbed solution to any order in ε is expressible in elliptic functions, and is free of singularities and mixed secular terms. This solution would provide a theoretical framework for an attack on resonance problems in celestial mechanics when the latter are reducible to the ideal form. The condition of reducibility is that the Hamiltonian can be put in the form

$$F=A_{0}\left ( y_{1}\right)+A_{1}\left ( y_{1},y_{2}\right)cos2x_{1}+\phi \left ( x_{1},x_{2},y_{1},y_{2} \right )$$

, where ø is periodic in x ₁ and x ₂ with a period 2π, and the relations

$$ {A_0} = 0\left( 1 \right),{A_1} = 0\left( \varepsilon \right),\phi = o\left( \varepsilon \right) $$

are satisfied. The rate of the convergence of the solution depends on the smallness of ø.