What I am missing here? (Partial derivative in Euler-Lagrange equation)

Question

So, I have an equation for unit sphere

$dS^2=d\theta^2+\sin^2\theta(d\phi)^2$

$L=\theta^2+\sin^2(\theta)\phi^2$

Then Euler-Lagrange equation

$\dfrac{d}{d\lambda}(\dfrac{d}{d\theta})=\dfrac{d}{d\theta}$ and using $\sin(2\theta)=2\sin(\theta)\cos(\theta)$

It results

$\theta-(\sin(\theta)\cos(\theta))\phi^2=0$ in the book but I found

$\theta+(\sin(\theta)\cos(\theta))\phi^2=0$

Also, it says for component $\phi$ it results

$\phi+2\cot(\theta)\theta\phi=0$

But I found

$4(\sin(\theta)\cos(\theta))\phi=0$

Thanks...

Answer 1

Let's first fix the notation: $L=\dot{\theta}^2+\sin^2\theta\cdot\dot{\phi}^2$ with $\dot{f}:=df/d\lambda$ and $\frac{d}{d\lambda}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial\theta}$ etc. So $\ddot{\theta}-\sin\theta\cos\theta\cdot\dot{\phi}^2=0$, as in your book (assuming they include the dots you dropped), and$$0=\frac{1}{\sin^2\theta}\frac{d}{d\lambda}(\sin^2\theta\dot{\phi})=\ddot{\phi}+2\cot\theta\cdot\dot{\theta}\dot{\phi},$$which again is what the book presumably said. (You can write $\dot{f}$ with \dot{f}.)