Lagrangian density

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

This article uses ${\displaystyle {\mathcal {L}}}$ for the Lagrangian density, and L for the Lagrangian.

The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a manifold. The dependent variables (q) are replaced by the value of a field at that point in spacetime φ(x, y, z, t) so that the equations of motion are obtained by means of an action principle, written as:

where the action, ${\displaystyle {\mathcal {S}}}$, is a functional of the dependent variables φ_i(s) with their derivatives and s itself

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