SCOPE OF PARTIAL DERIVATIVES

[JMMM]

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:

\(\displaystyle \frac{{{\partial }^{2}}U}{\partial {{r}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left[ \dot{r}\cdot \frac{\partial U}{\partial \dot{r}} \right]\)

 

[Steven G]

Have you tried to prove the inequality? If yes, then show us your work. No one on this forum will do your work for you as that will not be helpful.

 

[Dr.Peterson]

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:

\(\displaystyle \frac{{{\partial }^{2}}U}{\partial {{r}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}}{\partial {{t}^{2}}}\left[ \dot{r}\cdot \frac{\partial U}{\partial \dot{r}} \right]\)

Context will also be helpful: Why do you think this should be true? How do you intend to use it? Where did it come from?

 

[JMMM]

Have you tried to prove the inequality? If yes, then show us your work. No one on this forum will do your work for you as that will not be helpful.

Hello, Steven G

I read the forum conditions and I understand perfectly what you are telling me. Naturally, I tried many ways before posting the question on the forum.
A simple way is to obtain (with Mathematica or Derive, in my case) the partial derivatives given considering as an independent variable \(\displaystyle \dot{r}\) (the only true ones are \(\displaystyle r\) and \(\displaystyle t\)). But I have doubts about that method, because I think I am thus ignoring relevant information.
I can't think how to proceed otherwise, that's why I didn't want to condition the help by exposing my fruitless efforts.
Although this demonstration is independent, it could help in the demonstration of another question that I have uploaded to the forum 'Could a Legendre transform...'.

 

[JMMM]

Context will also be helpful: Why do you think this should be true? How do you intend to use it? Where did it come from?

Thank you, Dr.Peterson, for your advice.

I can tell you about the context, but I am afraid it will be of little help, being little known. The problem arises in Weber Electrodynamics, considering Phipps' modernization. I am trying to demonstrate an old idea of Professor J.P. Wesley. The equality which I ask if satisfied, would be related to the so-called one-dimensional wave equation. This problem, although independent, could help in the demonstration of another one I have uploaded to the forum 'Could a Legendre transform...'.