The equation of Binet in classical and relativistic orbital mechanics

Abstract

Abstract Binet’s equation provides a direct way to obtain the geometric shape of orbits in a central force field. It is well known that in Newtonian gravitation Binet’s equation leads to all the conic curves as solutions for an inverse-square force. Its standard derivation in higher-education textbooks relies on the use of polar coordinates and the conservation of angular momentum and energy. Here, an alternative derivation is presented that emphasizes the physical interpretation of the problem by considering the motion as a superposition of a vertical fall toward the attracting centre plus an inertial motion in the horizontal and vertical directions. That is, we show how Binet’s equation arises from the horizontal and vertical infinitesimal displacements of a body in free fall and in inertial motion. This deduction uses elementary concepts of infinitesimal calculus and connects closely with the original geometric and kinematical viewpoint of Newton and his predecessors and contemporaries. It is a view that is fully consistent with the maths taught in undergraduate courses. Second, we derive the relativistic version of Binet’s equation for the Schwarzschild-(anti-)de Sitter metric. This derivation, which is novel, directly relates the coordinates involved in Binet’s equation without the need to introduce potentials or the use of Killing vectors. It is also an adequate step forward in the use of mathematical physics that can be introduced in advanced undergraduate or graduate courses of general relativity. Finally, we tackle some controversies related to the role of the cosmological constant in the trajectory of photons in a Schwarzschild-(anti-)de Sitter or even in Reissner–Nordström–(anti-)de Sitter spacetimes. In this last step of the work, it is shown to students how a thorough yet concise mathematical analysis can solve disputes that arise even in high-level research articles.

The authors' abstract, as published at the source. European Journal of Physics, 2026 · DOI ↗