Integrals of motion (also First integrals)

[Happy]

Hi all,

Homework Statement

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.

 

[Sunfire]

More on "integrals of motion" is found here: http://en.wikipedia.org/wiki/Constant_of_motion#Integral_of_motion

Are you familiar with Noether Theorem? It can be used to find all conservation laws for your system. Then you can examine them individually and see which ones do not depend on time.

 

[Happy]

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