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. $v_{\pm }\to 0$ as $x\to x_0$, according to the power law scaling $\left|v_\pm\right| \propto \left|x_0-x\right|^{\beta}$,wheretheexponent$\beta=1/2$ 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 $\tau \propto {\mid x_0-x\mid }^{\nu }$, with the exponent $\nu =-1\mathrm{/}2$. 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.