Applied Physics and Mathematics Annotation << Back ON APSIDAL PRECESSION OF NON-ELLIPTICAL OVAL ORBITS AS AN EXISTING SOLUTION TO THE NEWTONIAN TWO-BODY PROBLEM. PART 1 A.V. KUKUSHKIN Based on the review of the recent astronomical data on the motion of eclipsing binary stars’ components it has been demonstrated that the revisiting of the classical two-body problem can be topical and practical if new classical solutions to this problem for finite orbits with apsidal precession can be found. This new perspective can be obtained when the test mass point’s motion is described as ‘complex’ motion where the vector of the body’s velocity is considered a sum of the relative and translational angular velocities; thanks to this step, the orbits with apsidal precession receive a mathematical description. The notions of uniform and non-uniform orbital precession are introduced. For the latter case, the fluctuation of the shape of the reference relative elliptical orbit is possible to be described. In reality the relative orbit can be not the Keplerian ellipse, as it is generally accepted in astronomy, but a smooth oval defined by the Newtonian equation of motion. It is demonstrated that this problem can be formulated and solved if the complex orbital plane is substituted for the real one from the beginning. Keywords: eclipsing binary stars, binary pulsars, exoplanets of the ‘very hot Jupiter’ type, orbital function, rotating complex plane, uniform orbital precession, non-uniform orbital precession. DOI: 10.25791/pfim.01.2022.1220 Pp. 10-30.