(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.

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

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