We have the following function
\(\displaystyle U=U(r(t),\,\,\dot{r}(t))=\frac{\alpha }{r}\sqrt{1-\frac{{{{\dot{r}}}^{2}}}{{{c}^{2}}}}\)
being \(\displaystyle \alpha \) and \(\displaystyle c\) are positive constants, and \(\displaystyle \dot{r}={dr}/{dt}\;\).
I would like to know if it is possible to prove the following equality:
\(\displaystyle \frac{{{\partial }^{2}}U}{\partial {{r}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left[ \dot{r}\cdot \frac{\partial U}{\partial \dot{r}} \right]\)
\(\displaystyle U=U(r(t),\,\,\dot{r}(t))=\frac{\alpha }{r}\sqrt{1-\frac{{{{\dot{r}}}^{2}}}{{{c}^{2}}}}\)
being \(\displaystyle \alpha \) and \(\displaystyle c\) are positive constants, and \(\displaystyle \dot{r}={dr}/{dt}\;\).
I would like to know if it is possible to prove the following equality:
\(\displaystyle \frac{{{\partial }^{2}}U}{\partial {{r}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left[ \dot{r}\cdot \frac{\partial U}{\partial \dot{r}} \right]\)