Universal Power Law Scaling Near the Turning Points

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 $\left|v_\pm\right| \propto \left|x_0-x\right|^β$, where the exponent $β=1/2$ 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 $τ\propto \left|x_0-x\right|^ν$, with the exponent $ν=-1/2$. 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.

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