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

TL;DR

This paper introduces an energy factorization method using complex numbers to analytically solve various classical mechanics problems, offering an alternative to traditional Newtonian approaches and suitable for undergraduate teaching.

Contribution

The paper presents a novel energy factorization approach that simplifies solving classical mechanics problems and provides new analytical solutions and approximations.

Findings

Exact solutions for harmonic oscillator, projectile motion, and inverse cube force.

Approximate energy decay and solutions for weakly damped oscillators.

Method is suitable for undergraduate physics education.

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.