A note on first-order normalizations of perturbed Keplerian systems

Abstract

Techniques are developed to facilitate the transformation of a perturbed Keplerian system into Deláunay normal form at first order. The implicit dependence of the Hamiltonian on 1, the mean anomaly, through the explicit variable f, the true anomaly, or E, the eccentric anomaly, is removed through first order for terms of the form:

$$\left\{ {\begin{array}{*{20}c} {\cos } \\ {\sin } \\ \end{array} } \right\}\left( {kf + \nu } \right)and\left\{ {\begin{array}{*{20}c} {\cos } \\ {\sin } \\ \end{array} } \right\}\left( {kE + \nu } \right)$$

, where the angle ν is independent ofl andk is an integer constant. The procedure involves no expansion in the powers of the eccentricity.