Numerical simulation of the Kepler problem in Wolfram Mathematica

Abstract

In this article, we consider the classical Kepler problem, emphasizing its fundamental features, implications, and broad applications in celestial mechanics and astrophysics. The Kepler problem, which describes the motion of bodies under the influence of Newton’s law of universal gravitation, remains one of the cornerstones of classical mechanics and continues to play a crucial role in modern space science. For pedagogical and illustrative purposes, we employ Wolfram Mathematica to visualize a variety of examples of planetary motion within the Solar System. This computational approach allows us to highlight the interplay between theory and numerical simulations, offering students and researchers an intuitive way to explore orbital dynamics. The Kepler problem is formulated and solved numerically, and we demonstrate how variations in initial conditions - such as position and velocity - lead to distinct orbital trajectories. As representative cases, we analyze the motion of the Earth and selected artificial satellites under the gravitational field of the Sun, stressing the relevance of these simulations for practical applications, such as mission design and orbital prediction. In addition, we examine elliptical orbits in detail and numerically confirm the conservation of orbital angular momentum and total mechanical energy, thereby reinforcing key principles of mechanics. We also validate all three of Kepler’s laws within our numerical framework, demonstrating their consistency with the trajectories derived from Newtonian gravity. The results obtained in this work are versatile and can be effectively applied at multiple educational levels. They are suitable for advanced high school programs in physics and astronomy, as well as for undergraduate laboratory courses in physics, astronomy, mathematics, and engineering. Beyond their pedagogical value, the presented methodology offers a simple yet powerful framework for initiating research-oriented activities. Overall, this study provides both educational and scientific insights, serving as a bridge between classical theory, computational methods, and modern applications in space science.