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Integrals of motion (also First integrals)

  1. Feb 2, 2014 #1
    Hi all,
    1. The problem statement, all variables and given/known data
    I have got a system described by this lagrangian [itex]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)[/itex]. I have to find all system's integrals of motion.

    2. The attempt at a solution
    From [itex]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)[/itex] I know that [itex]\vartheta[/itex] is the only cyclic coordinate. Therefore 1st integral of motion is [itex]\frac{\partial L}{\partial \dot\vartheta }=m\dot\vartheta [/itex].

    And 2nd integral of motion is
    [itex]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[/itex]

    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. jcsd
  3. Feb 3, 2014 #2
  4. Feb 3, 2014 #3
    Thank u very much, that Noether Theorem was the key.
     
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