An alternative approach to several important systems in classical mechanics: energy factorization

2026/01/26 by Karlo Lelas, Dario Jukić · 1 voice
Physics and Astronomy · #physics.class-ph

paper · pdf · doi:10.1088/1361-6404/ae7857

arxiv published 2026/01/26 · arxiv updated 2026/06/07

Abstract

We show how several important classical problems, with positive definite potential energy, can be solved by starting from the factorization of the total mechanical energy using complex numbers. In particular, we derive in a new way exact analytical solutions for: simple harmonic oscillator, vertical projectile motion, motion under a repulsive inverse cube force, and damped harmonic oscillator (with linear damping). We also show how this approach easily yields an excellent approximation of the energy decay and a new approximate analytical solution in the case of a weakly damped harmonic oscillator. Our derivations are suitable for undergraduate physics teaching as an alternative to solving Newton's equations of motion. In addition, we comment on the limitations of our approach, but also on the insights it provides and opportunities for further research.

Discussions

• Karlo Lelas, Dario Juki\'c: An alternative approach to several important systems in classical mechanics: energy factorization https://arxiv.org/abs/2601.18957 https://arxiv.org/pdf/2601.18957 https:// [bsky, 0 points, 0 comments]

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