Lagrangian field theory specifies an action by integrating a Lagrangian density over spacetime. Requiring its first variation to vanish for compactly supported field variations yields the field Euler-Lagrange equations. The variational Lagrangian is the integrand, whereas this field-theoretic formalism specifies the continuum fields and their dynamics.
A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.
A gauge-covariant kinetic term replaces ordinary derivatives by covariant derivatives. Expanding it generates interactions between the matter field and one or more gauge fields, including contact vertices with two gauge bosons.
A mass term is a quadratic term without derivatives whose coefficient sets the mass scale of a free field.
A field interaction term is nonlinear in the fields and produces the interaction vertices of perturbation theory.
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In the context of field theory and theoretical physics, the Lagrangian is a mathematical function that encapsulates the dynamics of a system. It is a central concept in the Lagrangian formulation of mechanics, which has been extended to fields in the context of quantum field theory and classical field theory.
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