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From Hamiltonian to Lagrangian 1

  1. Sep 6, 2008 #1

    malawi_glenn

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    1. The problem statement, all variables and given/known data

    [tex]H = p_1p_2 + q_1q_2[/tex]

    Find the corresponding Lagrangian, [itex]q_i[/itex] are generelized coordinates and
    [itex]p_i[/itex] are canonical momenta.

    2. Relevant equations

    [tex]H = \dot{q}_ip_i - L[/tex]

    [tex] p_i = \frac{\partial L}{\partial \dot{q}_i}[/tex]

    [tex] \dot{q}_i = \frac{\partial H}{\partial p_i}[/tex]


    3. The attempt at a solution

    Using these relations, I found:


    [tex]L = \dot{q}_ip_i - H[/tex]

    [tex]L = p_1p_2 + p_2p_1 - p_1p_2 + q_1q_2 = p_1p_2 - q_1q_2 = [/tex]

    [tex]\frac{\partial L}{\partial \dot{q}_1}\frac{\partial L}{\partial \dot{q}_2}-q_1q_2 [/tex]

    Am I supposed to solve this nasty PDE? Or have I forgot something really fundamental?
     
    Last edited: Sep 6, 2008
  2. jcsd
  3. Sep 6, 2008 #2

    malawi_glenn

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    I had missed something fundamental, its solved now
     
  4. Sep 6, 2008 #3
    Hi,

    it is only a bagatelle, but if you write the Hamilton function in generel, not for a concret case, then you schould write it like that:

    [tex] \mathcal{H}(q_{1} \ldots q_{s}, p_{1} \ldots p_{s}, t) = \sum\limits_{i=1}^{s} p_{i} \dot{q}_{i} - \mathcal{L}(q_{1} \ldots q_{s}, \dot{q}_{1} \ldots \dot{q}_{s},t)[/tex]

    [tex] & s = 3N-m \text{ with N dimensions and m constraints}[/tex]

    all the best
     
  5. Sep 6, 2008 #4

    malawi_glenn

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    I know, I already listed that eq. under "relevant eq's".

    Aslo I have solved the problem, no need to post.

    Also, it seems I can't marked this thread as solved in the "old way", why is that?
     
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