nLab
Lagrangian

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 $i$,

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

Here $q=(q_1,\dots ,q_n)$ 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) smooth manifold of dimension $n$ and $E\to X$ a field bundle with jet bundle $j_{\mathrm{\infty }}E\to X$ let ${\Omega }^{•,•}(j_{\mathrm{\infty }}E)$ be the corresponding variational bicomplex.

A Lagrangian on $E$ is an element

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

regarded as a differential form-valued functional on the space of sections:

$$L:(\varphi \in \Gamma (E))\mapsto L(j_{\mathrm{\infty }}\varphi ).$$L : (\phi \in \Gamma(E)) \mapsto L(j_\infty \phi) \,.

The integral

$${\int }_{X}L(j_{\mathrm{\infty }}(-)):\Gamma (\varphi )\to ℝ$$\int_X L(j_\infty(-)) : \Gamma(\phi) \to \mathbb{R}

is the corresponding local action functional.

Related concepts

• action functional

• extended Lagrangian