nLab Lagrangian density

Idea

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 $L = L(q,\stackrel{\cdot}q,t)$ 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 $i$,

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

Here $q = (q_1,\ldots, q_n)$ 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.

Definition

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

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

Definition

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

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

hence a horizontal differential form of degree $n$ on the jet bundle of $E$.

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

$$L : (\phi \in \Gamma(E)) \mapsto L(j_\infty \phi) \in \Omega^n(X) \,.$$

The integral (for compact $X$)

$$\int_X L(j_\infty(-)) \;\colon\; \Gamma(E) \longrightarrow \mathbb{R}$$

is the corresponding local action functional.

Related concepts

• action functional

• extended Lagrangian,

• BV-Lagrangian density, gauge fixing Lagrangian density

• traditional Lagrangian mechanics and Hamiltonian mechanics are naturally embedding into local prequantum field theory by the notion of prequantized Lagrangian correspondences

• Lagrangian quantum field theory

• non-Lagrangian field theory

Hamiltonian	$\leftarrow$ Legendre transform $\rightarrow$	Lagrangian
Lagrangian correspondence	prequantization	prequantized Lagrangian correspondence

References

Named after Joseph-Louis Lagrange.

Textbooks:

• Garth Warner: Lagrangian Mechanics, EPrint Collection, University of Washington (2009) [hdl:1773/4606, pdf, pdf]

Textbook account in the context of gauge theory:

• Marc Henneaux, Claudio Teitelboim, §1.1.1 in: Quantization of Gauge Systems, Princeton University Press (1992) [ISBN:9780691037691, jstor:j.ctv10crg0r]

Discussion in the convenient context of smooth sets:

• Grigorios Giotopoulos, Hisham Sati, §3 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]

See also

• Wikipedia, Lagrangian mechanics

• Daniel Freed, Lecture 2 of Five lectures on supersymmetry