Physicists Develop Alternative Method for Solving Classical Mechanics Problems Using Energy Factorization
Researchers have developed a new approach to solving classical mechanics problems by factorizing total mechanical energy using complex numbers, yielding exact analytical solutions for systems including the harmonic oscillator, projectile motion, and damped oscillators. The method offers an alternative to the standard technique of solving Newton's equations of motion directly. The work, accepted for publication in the European Journal of Physics, is aimed at undergraduate physics education and opens avenues for further research.
A paper by Karlo Lelas and colleagues, accepted in the European Journal of Physics, introduces an 'energy factorization' approach to classical mechanics in which the total mechanical energy of a system is expressed as a product of complex-number factors. Using this technique, the authors derive exact analytical solutions for the simple harmonic oscillator, vertical projectile motion, motion under a repulsive inverse cube force, and the linearly damped harmonic oscillator. For the weakly damped harmonic oscillator, the method yields an excellent approximation of energy decay and a new approximate analytical solution. The authors argue the approach is pedagogically suitable as an alternative to Newton's equations for undergraduate courses. The paper also discusses the limitations of the method and highlights insights it provides, as well as opportunities for further research. A second version of the preprint, submitted in June 2026, more precisely defines the energy factorization approach compared to the original January 2026 submission.
What's missing
The paper discusses limitations of the energy factorization approach but does not specify which classes of problems or potential energy functions fall outside its scope. It is also unclear whether the method generalizes to systems without positive definite potential energy or to higher-dimensional problems.
What different sources said
- arXiv physicsCenter
An alternative approach to several important systems in classical mechanics: energy factorization
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