Integrals of finite motion in the Schwarzschild gravitational field up to terms of order c^–2

Abstract

Up to terms containing c^–2, simple analytical dependences are derived that describe the finite motion of a test particle in the Schwarzschild geometry. Such a motion is considered as a correction to the unperturbed Keplerian motion under the condition that the ratio of the Schwarzschild radius to the radial coordinate is small. In this approximation, conserved integrals are also found that characterise the orbital motion of the particle. For this, the equations of motion are presented in the Hamiltonian form, and a number of canonical transformations of the generalised coordinates and momenta are made, which make it possible to integrate these equations. Periodic and secular contributions are derived for the osculating elements of the test particle orbit: the mean anomaly, the periapsis argument, and the semi-major axis. An algorithm for calculating the position of a particle in the c^–2 approximation is proposed, which is comparable in computational complexity to the algorithm for solving the standard Kepler problem. An estimate of the error of the obtained approximate solutions is made and the limits of their applicability are indicated.