Lagrangian field theory ed

Lagrangian mechanics for fields.

Theory ed

In field theory we go from a particle with coordinates or path \( q^i ( t ) \) to fields \( \phi^i ( \vec r , t ) = \phi^i ( x^\mu ) = \phi^i ( x ) \).

Also we have to turn the Lagrangian \( L ( q^i , \dot q^i ) \) into the Lagrangian density \( \mathcal L ( \phi^i , \partial_\mu \phi^i ) \). The action functional we need to minimize will turn into the integral over a space-time volume

\[ S = \int_{M} \mathcal L ( \phi^i , \partial_\mu \phi^i ) \, d^nx \]

That leads to the Euler-Lagrange-Equations for fields:

\[ \partial_\mu \frac{ \partial \mathcal L }{ \partial ( \partial_\mu \phi^i ) } = \frac{ \partial \mathcal L }{ \partial \phi^i } \]

Examples ed

Klein-Gordon ed

Electrodynamics ed