Lagrangian density Definition and 54 Threads

Hello, I took an Electrodynamics course this semester, where we derived Maxwell's equations from the field's Lagrangian density. As a motivation, we "looked" for a scalar (in the relativistic sense) having something to do with EM fields - and had we found one we would have declared it a...

we know the lagrangian l=ke-pe right in case of fields is called "lagrangian density" let particle with mass "m" and position "x" it kientic energy= 1/2(mv^2) so lagrangian =1/2(mv^2)-v(x) , v(x)=potential energy in case the field lagrangian density how can i determine the lagragian...

Why does the Lagrangian density for the EM field in a dielectric medium take the form d^3 \bf x \left[ \epsilon \bf E^2 - \bf B^2 \right]? I can see that the expression for Lagrangian density has units of energy per unit volume as you would expect but that's about it. Much appreciated.

Here's the problem. For a neutral vector field V_{\mu} we have the Lagrangian density \mathcal{L} = -\frac{1}{2}(\partial_{\mu}V_{\nu})(\partial^{\mu}V^{\nu})+\frac{1}{2}(\partial_{\mu}V^{\mu})(\partial_{\nu}V^{\nu})+\frac{1}{2}m^2V_{\mu}V^{\mu} We are then going to use the Euler-Lagrange...