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SamRoss
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- TL;DR Summary
- Is there a missing exponent in the authors' application of the Euler-Lagrange equation?
In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian
$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$
They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be partial derivatives there but I couldn't find the symbol for it in my Latex primer; if anyone could enlighten me, I'd appreciate it) and wrote the result...
$$\ddot r=r\dot \theta^2-\frac {GM} r$$
My question is, shouldn't the last r in the denominator be squared since it results from differentiating the GMm/r term in the Lagrangian by r?
$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$
They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be partial derivatives there but I couldn't find the symbol for it in my Latex primer; if anyone could enlighten me, I'd appreciate it) and wrote the result...
$$\ddot r=r\dot \theta^2-\frac {GM} r$$
My question is, shouldn't the last r in the denominator be squared since it results from differentiating the GMm/r term in the Lagrangian by r?