# Euler-Lagrange Equations for a field

• Jun 1st 2013, 01:37 AM
Kiwi_Dave
Euler-Lagrange Equations for a field
Given the functional:

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

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

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

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.
$\displaystyle \mathcal{L}=\sqrt{1+\phi_x^2+\phi_y^2+\phi_z^2}$

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

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

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

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

I have tried all kinds of things but they always become computationally complex and that seems unlikely.