# Integrals of motion (also First integrals)

1. Feb 2, 2014

### Happy

Hi all,
1. The problem statement, all variables and given/known data
I have got a system described by this lagrangian $L(\varphi ,\psi ,\vartheta ,\dot\varphi ,\dot\psi ,\dot\vartheta )=\frac{1}{2}m(\dot\varphi^2 +\dot\psi^2 +\dot\vartheta^2 )+cos(\varphi ^2+\psi ^2)$. I have to find all system's integrals of motion.

2. The attempt at a solution
From $L(\varphi ,\psi ,\vartheta ,\dot\varphi ,\dot\psi ,\dot\vartheta )=\frac{1}{2}m(\dot\varphi^2 +\dot\psi^2 +\dot\vartheta^2 )+cos(\varphi ^2+\psi ^2)$ I know that $\vartheta$ is the only cyclic coordinate. Therefore 1st integral of motion is $\frac{\partial L}{\partial \dot\vartheta }=m\dot\vartheta$.

And 2nd integral of motion is
$E=\sum_{}^{}\left(\frac{\partial L}{\partial \dot q}\dot q\right)-L=\left(\frac{\partial L}{\partial \dot \varphi }\dot \varphi +\frac{\partial L}{\partial \dot \psi }\dot \psi +\frac{\partial L}{\partial \dot \vartheta }\dot \vartheta \right)-L$

Probably there are more integrals of motion. Unfortunately, I do not know how to find them. I would be grateful if you could help me and guide me through the process of finding them all.

Help me please I really need it.

2. Feb 3, 2014

3. Feb 3, 2014

### Happy

Thank u very much, that Noether Theorem was the key.