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

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.

💡 Research Summary

The paper introduces a novel pedagogical method for solving a variety of one‑dimensional classical mechanics problems whose potential energy is positive‑definite. Starting from the energy conservation law
( \frac{1}{2}mv^{2}+U(x)=E )
the authors factor the left‑hand side as a product of two complex conjugates:
((\sqrt{E},e^{i\phi})(\sqrt{E},e^{-i\phi})).
By equating the real and imaginary parts they obtain two fundamental relations that hold for any such system:
( v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t) ) and ( U(x(t))=\sqrt{E}\sin\phi(t) ).
Thus the dynamics is reduced to determining a single phase function (\phi(t)).

The method is applied to several canonical problems.

1. Simple harmonic oscillator (SHO) – With (U=\frac{1}{2}kx^{2}) the phase satisfies (\dot\phi=\omega_{0}) where (\omega_{0}=\sqrt{k/m}). Integration yields (\phi(t)=\omega_{0}t+\phi_{0}) and the familiar solution (x(t)=A\sin(\omega_{0}t+\phi_{0})) with amplitude (A=\sqrt{2E/k}).

2. Uniform gravitational field – For (U=mgx) the constant acceleration (a=-g) gives (v(t)=v_{0}-gt). Substituting into the velocity relation produces (\cos\phi(t)=\sqrt{m/(2E)},(v_{0}-gt)) and, after using (\sin^{2}\phi=1-\cos^{2}\phi), the standard kinematic result (x(t)=h+v_{0}t-\frac{1}{2}gt^{2}).

3. Repulsive inverse‑cube force – With (U=K/(2x^{2})) the phase obeys (\sin^{2}\phi,\dot\phi=-2E\sqrt{K/m}). Integration gives (\phi(t)=\arctan(2E\sqrt{K/m},t+\cot\phi_{0})). For the common initial condition (x_{0}>0,;v_{0}=0) (so (\phi_{0}=\pi/2)) the trajectory simplifies to (x(t)=\sqrt{x_{0}^{2}+ (K/m) t^{2}}).

4. Central‑force problems – By introducing the effective potential (U_{\text{eff}}=U+L^{2}/(2mr^{2})) the same factorization applies. For potentials proportional to (1/r^{2}) (e.g., a dipole field) the effective potential retains the inverse‑square form, and the inverse‑cube solution carries over. For the Kepler potential (U\propto 1/r) the phase integral can be performed, but the resulting expression for (r(t)) is not closed‑form, mirroring the well‑known limitation of the standard approach.

5. Damped harmonic oscillator (linear damping) – Energy now decays: (E(t)=\frac{1}{2}m v^{2}+ \frac{1}{2}k x^{2}). The same relations hold with (E) replaced by (E(t)). Using the power‑loss law (\dot E=-b v^{2}) leads to a differential equation for the phase:
   (\dot\phi + \gamma\omega_{0}\sin 2\phi = \omega_{0}) with (\gamma=b/(2m)).
   Integrating yields (\phi(t)=\arctan!\big