Let the function:
\(\displaystyle U\left( r(t),\ v(t) \right)=\frac{\alpha }{r}\sqrt{\ 1-\frac{{{v}^{2}}}{{{c}^{2}}}}\)
where \(\displaystyle \alpha \) and \(\displaystyle c\) are positive constants and \(\displaystyle v(t)={dr}/{dt}\;\).
Consider the value \(\displaystyle t'\) of the variable \(\displaystyle t\), such that
\(\displaystyle {t}'=t-{r(t)}/{c}\;\)
If we evaluate \(\displaystyle U\) at \(\displaystyle t=t'\), we obtain:
[math]U\left( r({t}'),v({t}') \right)=U\left( r(t-{r(t)}/{c}\;),v(t-{r(t)}/{c}\;) \right)[/math] (A)
The question is whether it can be shown that (A) is of the following form:
\(\displaystyle U\left( r(t),v(t) \right)-v(t)\cdot \frac{\partial U}{\partial v}\left( r(t),v(t) \right)\) (B)
\(\displaystyle U\left( r(t),\ v(t) \right)=\frac{\alpha }{r}\sqrt{\ 1-\frac{{{v}^{2}}}{{{c}^{2}}}}\)
where \(\displaystyle \alpha \) and \(\displaystyle c\) are positive constants and \(\displaystyle v(t)={dr}/{dt}\;\).
Consider the value \(\displaystyle t'\) of the variable \(\displaystyle t\), such that
\(\displaystyle {t}'=t-{r(t)}/{c}\;\)
If we evaluate \(\displaystyle U\) at \(\displaystyle t=t'\), we obtain:
[math]U\left( r({t}'),v({t}') \right)=U\left( r(t-{r(t)}/{c}\;),v(t-{r(t)}/{c}\;) \right)[/math] (A)
The question is whether it can be shown that (A) is of the following form:
\(\displaystyle U\left( r(t),v(t) \right)-v(t)\cdot \frac{\partial U}{\partial v}\left( r(t),v(t) \right)\) (B)