Ignorable coordinates in the ideal resonance problem

Abstract

If a dynamical system ofN degrees of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form

$$F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 , \mu<< 1.$$

Herey is the momentum-vectory _k withk=1, 2,...,N, andx ₁ is thecritical argument.

A first-orderglobal solution,x ₁(t) andy ₁(t), for theactive variables of the problem, has been given in Garfinkelet al. (1971). Sincex _k fork>1 are ignorable coordinates, it follows that

$$y_\kappa = const., k > 1.$$

The solution is completed here by the construction of the functionsx _k(t) fork>1, derivable from the new HamiltonianF′(y′) and the generatorS(x, y′) of the von Zeipel canonical transformation used in the cited paper.

The solution is subject to thenormality condition, derived in a previous paper fork=1, and extended here to 2≤k≤N. It is shown that the condition is satisfied in the problem of the critical inclination provided it is satisfied fork=1.