# nLab Lagrangian density

Contents

### Context

#### Variational calculus

variational calculus

# Contents

## 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.

Hamiltonian$\leftarrow$ Legendre transform $\rightarrow$Lagrangian
Lagrangian correspondenceprequantizationprequantized Lagrangian correspondence

## References

Named after Joseph-Louis Lagrange.