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Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics . Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom .
File:Lagrangian vs Eulerian [further explanation needed] Eulerian perspective of fluid velocity versus Lagrangian depiction of strain. In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
Associated with the field is a Lagrangian density (,, ˙,,) defined in terms of the field and its space and time derivatives at a location r and time t. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in ...
For example, renormalization in QED modifies the mass of the free field electron to match that of a physical electron (with an electromagnetic field), and will in doing so add a term to the free field Lagrangian which must be cancelled by a counterterm in the interaction Lagrangian, that then shows up as a two-line vertex in the Feynman diagrams.
Lagrangian (field theory), a formalism in classical field theory; Lagrangian point, a position in an orbital configuration of two large bodies; Lagrangian coordinates, a way of describing the motions of particles of a solid or fluid in continuum mechanics; Lagrangian coherent structure, distinguished surfaces of trajectories in a dynamical system
In physics, the Euler–Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It was first obtained by Werner Heisenberg and Hans Heinrich Euler [1] in 1936. By treating the vacuum as a medium, it predicts rates of quantum electrodynamics (QED) light interaction processes. [clarification needed]
For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons.
The Lagrangian for a complex scalar field can be motivated as follows. For two scalar fields and the Lagrangian has the form (,) = [] + [] (+),which can be written more concisely introducing a complex scalar field defined as