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$$\mathcal L \left( t, x, y, u(x,y,t),\frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) = −\frac{1}{2} \left( \frac{\partial u}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial x} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} \right)^2 + \frac{1}{2}\left( u \right)^2$$

Not really sure where to begin for computing the Euler-Lagrange equation for a Lagrangian in one variable, let alone the required $2$ variables.

I also can't seem to find similar examples online to grasp the concept. Could you give me any help on how to begin, and key ideas to keep in mind? How would you begin to compute the Euler Lagrange for one variable?

Thank you

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Any good book on variational calculus or optimal control should supply you with explanations and plenty examples.

Also, your notation is a bit hard to understand. Try stating which variables are functions of which arguments. My understanding of your equation is:

$$ L \left( t, x, y, u(x,y,t),\frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) = −\frac{1}{2} \left( \frac{\partial u}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial x} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} \right)^2 + \frac{1}{2}\left( \frac{d u}{d t} \right)^2 $$

See the code for the equation above:

L \left( t, x, y, u(x,y,t),\frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) = −\frac{1}{2} \left( \frac{\partial u}{\partial t} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial x} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} \right)^2 + \frac{1}{2}\left( \frac{d u}{d t} \right)^2
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${\cal L}$ is (minus) the massive Klein-Gordon (KG) Lagrangian density in 2+1D. The Euler-Lagrange (EL) equation $$ \frac{d}{dt} \frac{\partial {\cal L}}{\partial u_t}+\frac{d}{dx} \frac{\partial {\cal L}}{\partial u_x} + \frac{d}{dy}\frac{\partial {\cal L}}{\partial u_y} ~=~ \frac{\partial {\cal L}}{\partial u} $$ becomes the massive KG equation in 2+1D $$ -u_{tt}+u_{xx}+u_{yy}~=~u.$$

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