nLab Lagrangian density



Variational calculus


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For a classical mechanical system, the laws of motion can be expressed in terms of an action principle: the actual paths must be the (locally) extremal paths of the action functional.

In one of the formulations of the classical mechanics, called Lagrangean formalism, every mechanical system is characterized by its configuration space and a single function called Lagrangian which determines the laws of motion (the initial configuration should be given independently).

The Lagrangian, Lagrangian function or Lagrangean L=L(q,q,t)L = L(q,\stackrel{\cdot}q,t) is a real valued function of the points in configuration space and their time derivatives (for some sytems also depending on time), such that the corresponding action principle can be expressed as Euler-Lagrange equations: for all ii,

ddt(Lq i)Lq i=0 \frac{d}{dt} \left( \frac{\partial L}{\partial \stackrel{\cdot}{q}_i} \right) - \frac{\partial L}{\partial {q}_i} = 0

Here q=(q 1,,q n)q = (q_1,\ldots, q_n) is the coordinate in the configuration space.

For continuum systems satisfying reasonable locality, Lagrangians can be expressed in terms of integrating a local quantity, so-called Lagrangian density.


For XX a (spacetime/worldvolume) smooth manifold of dimension nn, let EXE \to X be a vector bundle, to serve as the field bundle for the nn-dimensional field theory Lagrangian to be defined.

Denote the jet bundle by j EXj_\infty E \to X and write Ω ,(j E)\Omega^{\bullet, \bullet}(j_\infty E) be the corresponding variational bicomplex.


A local Lagrangian on fields given by the field bundle EXE \to X is given by an element

LΩ n,0(j E), L \in \Omega^{n,0}(j_\infty E) \,,

hence a horizontal differential form of degree nn on the jet bundle of EE.

The local Lagrangian itself is the pullback of this along the jet prolongation map j :Γ X(E)Γ(j E)j_\infty \colon \Gamma_X(E) \longrightarrow \Gamma(j_\infty E), hence the differential form-valued functional on the space of sections of EE given by

L:(ϕΓ(E))L(j ϕ)Ω n(X). L : (\phi \in \Gamma(E)) \mapsto L(j_\infty \phi) \in \Omega^n(X) \,.

The integral (for compact XX)

XL(j ()):Γ(E) \int_X L(j_\infty(-)) \;\colon\; \Gamma(E) \longrightarrow \mathbb{R}

is the corresponding local action functional.

Hamiltonian\leftarrow Legendre transform \rightarrowLagrangian
Lagrangian correspondenceprequantizationprequantized Lagrangian correspondence


Named after Joseph-Louis Lagrange.

Textbook account in the context of gauge theory:

Discussion in the convenient context of smooth sets:

See also

Last revised on December 29, 2023 at 13:15:35. See the history of this page for a list of all contributions to it.