(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind.

Suppose I have two functions [tex]\phi(t)[/tex] and [tex]\chi(t)[/tex] and the potential V which is a function of these two. Suppose I introduce new variables [tex]\sigma[/tex] and [tex]s[/tex] such that

[tex]d\sigma = \cos\theta d\phi + \sin\theta e^{b(\phi)} d\chi[/tex] (1)

[tex]ds = \cos\theta e^{b(\phi)} d\chi - \sin\theta d\phi[/tex] (2)

where

[tex]\cos\theta = \frac{\dot{\phi}}{\sqrt{\dot{\phi}^2 + e^{2b(\phi)}\dot{\chi}^2}}[/tex] (3)

[tex]\sin\theta = \frac{e^{b(\phi)}\dot{\chi}}{\sqrt{\dot{\phi}^2 + e^{2b(\phi)}\dot{\chi}^2}}[/tex] (4)

where the overdot represents the derivative wrt t.

Denote partial derivatives as follows: [tex]A_x \equiv \frac{\partial A}{\partial x}[/tex].

I need to find second partial derivatives of V wrt to new variables in terms of second partial derivatives wrt old variables (i.e. [tex]V_{\sigma\sigma}, V_{\sigma s}, V_{ss}[/tex] in terms of [tex]V_{\phi\phi},V_{\phi\chi}[/tex] and [tex]V_{\chi\chi}[/tex]).

2. Relevant equations

3. The attempt at a solution

The way I went about this is as follows (using as an example [tex]V_{\sigma\sigma}[/tex]):

[tex]V = V_{\phi}d\phi + V_{\chi}d\chi[/tex] solving [tex]d\phi[/tex] and [tex]d\chi[/tex] from (1) and (2)

[tex]\Rightarrow V = V_{\phi}(\cos\theta d\sigma - \sin\theta ds) + V_{\chi}e^{-b(\phi)}(\sin\theta d\sigma + \cos\theta ds)[/tex]

[tex]\Rightarrow V = (V_{\phi}\cos\theta + V_{\chi}e^{-b(\phi)}\sin\theta) d\sigma + (-V_{\phi}\sin\theta + V_{\chi}e^{-b(\phi)}\cos\theta) ds[/tex]

[tex]\Rightarrow V_{\sigma} = V_{\phi}\cos\theta + V_{\chi}e^{-b(\phi)}\sin\theta[/tex] (5)

This seems right so far. Now taking [tex]V_{\sigma}[/tex] as the new function and repeating the exact same steps I get (just replacing [tex]V[/tex] with [tex]V_{\sigma}[/tex] in the above result):

[tex]V_{\sigma\sigma} = V_{\sigma\phi}\cos\theta + V_{\sigma\chi}e^{-b(\phi)}\sin\theta[/tex] and taking the apropriate derivatives of (5)

[tex]\Rightarrow V_{\sigma\sigma} = V_{\phi\phi}\cos^2\theta + 2 V_{\phi\chi}e^{-b(\phi)}\sin\theta\cos\theta + V_{\chi\chi}e^{-2b(\phi)}\sin^2\theta - b_{\phi}V_{\chi}e^{-b(\phi)}\sin\theta\cos\theta[/tex]

What makes me suspicious is the last term which arises because of the [tex]\phi[/tex] dependence of b. For example it makes mixed derivatives non-symmetric i.e. [tex]V_{\sigma s} \neq V_{s \sigma}[/tex]. Could that be right? I'm not 100 % sure of my method of arriving at the result so it would be great if someone with a firmer understanding of the mathematics involved could check this. Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Relating 2nd order partial derivatives in a coordinate transformation.

**Physics Forums | Science Articles, Homework Help, Discussion**