Universal Power Law Scaling Near the Turning Points

TL;DR

This paper demonstrates that particle velocity near turning points follows a universal power law with specific exponents, revealing a phase transition-like behavior in classical dynamics.

Contribution

It analytically and numerically establishes universal power law scaling of velocity and time near turning points, independent of particle mass and force.

Findings

Velocity scales as |x₀ - x|^{1/2} near turning points.

Time spent diverges as |x₀ - x|^{-1/2} near turning points.

Behavior resembles critical phenomena in phase transitions.

Abstract

We show analytically and numerically that, the velocity $v_\pm$ of a particle near the turning points $x_0$ vanishes, i. e. $v_\pm\rightarrow 0$ as $x\rightarrow x_0$, according to the power law scaling $\left|v_\pm\right| \propto \left|x_0-x\right|^{\beta}$, where the exponent $\beta=1/2$ is independent of the particle mass and the force acting on it. We also show that, the time spends it any particle at each small interval $dx$ near the turning points diverges as $\tau\propto \left|x_0-x\right|^{\nu}$, with the exponent $\nu=-1/2$. Behavior we find here is very similar to power law scaling that had been found near the critical points for systems which undergo a phase transition.