Motion of the planets: the calculation and visualization in Mathcad

This article describes use of Mathcad mathematical package to solve problem of the motion of two, three and four material points under the influence of gravitational forces on the planar motion and in three-dimensional space. The limits of accuracy of numerical methods for solving ordinary differential equations are discussed. Usual concept of Kepler hours with uneven movement arrows illustrates Kepler’s second law. Mathcad animation tools are used to illustrate solutions and links with animations are provided.

💡 Research Summary

The paper presents a complete workflow for modeling, solving, and visualizing the gravitational motion of two, three, and four point masses using the commercial mathematical software Mathcad. Beginning with a brief historical context, the authors note that while dedicated N‑body simulators exist, Mathcad’s combination of symbolic computation, built‑in ordinary differential equation (ODE) solvers, and rich graphics capabilities makes it an attractive platform for both teaching and exploratory research.

The theoretical foundation follows directly from Newton’s law of universal gravitation. For each body i the acceleration is expressed as

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