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Math Help - Euler-Lagrange Equations for a field

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    Euler-Lagrange Equations for a field

    Given the functional:

    J= \int_R  \sqrt{1+\phi_x^2+\phi_y^2+\phi_z^2}  dxdydz

    a. find the canonical mementa and construct the Hamiltonian tensor components, and
    b. write the Euler-Lagrange equations

    Notation:
    \frac{\partial \phi}{\partial x^{\mu}} \equiv \partial _{\mu} \phi \equiv \phi _ {\mu}

    Now I think can do part a (soln below) but I am strugling with part b. Obviously if I have part a wrong then that would make part b "difficult"

    Part a.
    \mathcal{L}=\sqrt{1+\phi_x^2+\phi_y^2+\phi_z^2}

    The canonical momentum is
    p^{\mu}=\frac{\partial L}{\partial \phi_{\mu}}=\frac{ \phi_{\mu}}{\mathcal{L}}

    The Hamiltonian tensor is
    \mathcal{H}_\rho ^{.\mu}=\frac {-1} {\mathcal{L}} \left( \begin{array}{ccc}1+\phi^2_y+\phi^2_z & -\phi_x\phi_y & -\phi_x\phi_z \\ -\phi_x\phi_y             & 1+\phi^2_x+\phi^2_z & -\phi_y\phi_z \\ -\phi_x\phi_z             & -\phi_y\phi_z             & 1+\phi^2_x+\phi^2_y \end{array} \right)

    Part b.
    The functional does not depend explicitly on the coordinates or on phi. So the Euler-Lagrange equations are:

    \partial_\mu p^\mu=0 and \partial _\mu \mathcal{H}_\rho ^{.\mu}=0

    I have tried all kinds of things but they always become computationally complex and that seems unlikely.
    Last edited by Kiwi_Dave; June 1st 2013 at 01:52 AM.
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