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

Karlo Lelas, Dario Jukić

TL;DR

The paper addresses solving classical mechanics problems with positive-definite potentials by factorizing the total energy using complex numbers.The method introduces a time-dependent phase $phi(t)$ via $v = \sqrt{2E/m}\cos\phi(t)$ and $\sqrt{U}=\sqrt{E}\sin\phi(t)$, enabling exact solutions for several systems.The authors derive exact solutions for the simple harmonic oscillator, homogeneous gravitational potential, inverse-cube repulsive, and the damped harmonic oscillator, plus an accurate energy-decay approximation in the weak-damping regime.This approach provides an accessible undergraduate teaching tool, clarifies method limitations, and points to extensions to other dissipative and central-force problems.

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.

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

TL;DR

The paper addresses solving classical mechanics problems with positive-definite potentials by factorizing the total energy using complex numbers.The method introduces a time-dependent phase $phi(t)$ via $v = \sqrt{2E/m}\cos\phi(t)$ and $\sqrt{U}=\sqrt{E}\sin\phi(t)$, enabling exact solutions for several systems.The authors derive exact solutions for the simple harmonic oscillator, homogeneous gravitational potential, inverse-cube repulsive, and the damped harmonic oscillator, plus an accurate energy-decay approximation in the weak-damping regime.This approach provides an accessible undergraduate teaching tool, clarifies method limitations, and points to extensions to other dissipative and central-force problems.

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.
Paper Structure (13 sections, 78 equations, 1 figure)

This paper contains 13 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simple harmonic oscillator
  3. Homogeneous gravitational potential energy
  4. Repulsive potential energy with the inverse-cube force law
  5. Central force field: effective potential energy
  6. Dissipative system: damped harmonic oscillator
  7. Exact solution and exact energy decay
  8. Approximation of the energy decay and approximate analytical solution in the weak damping regime
  9. Limitations of our approach
  10. Conclusion and outlook
  11. Acknowledgments
  12. Author contributions
  13. Solution to the integral \ref{['fazaDHOintegral']} and phase $\phi(t)$ for underdamped case

Figures (1)

  • Figure 1: Solid red curve corresponds to the approximate solution $x_{approx}(t)$, i.e. \ref{['Xapprox']}, solid blue curve to $x_1(t)=x_0e^{-\gamma t}\cos(\omega_0t)$, and black dotted curve to exact analytical solution \ref{['XDHOsolution']}. All curves are obtained with $\gamma=0.1\omega_0$, and $T_0=2\pi/\omega_0$ is the period of undamped oscillator. (a) We show a shorter time span in order to see more clearly the better match between $x_{approx}(t)$ and the exact solution in vicinity of the turning points, compared to $x_1(t)$. (b) Comparison of $x_{approx}(t)$ and the exact solution over a larger time span.