A resonance problem of two degrees of freedom

An analytical theory is presented for determining the motion described by a Hamiltonian of two degrees of freedom. Hamiltonians of this type are representative of the problem of an artificial Earth satellite in a near-circular orbit or a near-equatorial orbit and in resonance with a longitudinal dependent part of the geopotential. Using the classical Bohlin-von Zeipel procedure the variation of the elements is developed through a generating function expressed as a trigonometrical series. The coefficients of this series, determined in ascending powers of an auxiliary parameter, are the solutions of paired sets of ordinary differential equations and involve elliptic functions and quadrature. The first order solution accounts for the full variation of the resonance terms with the second coordinate.