# Lagrangian

A **Lagrangian** **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}[\varphi_i] }**
of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \varphi_i(s)}**
which concisely describes the equations of motion of the system. The equations of motion are obtained by means of an action principle, written as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta \mathcal{S}}{\delta \varphi_i} = 0 }**

where the action **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}[\varphi_i] = \int{\mathcal{L}[\varphi_i(s)]{}\,d^ns}, }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {}{}{}{}\ s_\alpha }**
denoting the set of parameters of the system.

The equations of motion obtained by means of the functional derivative are identical to the usual Euler-Lagrange equations. Dynamical system whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as *Lagrangian dynamical systems*. Examples of Lagrangian dynamical systems range from the (classical version of the) Standard Model, to Newton's equations, to purely mathematical problems such as geodesic equations and Plateau's problem.

## Contents

## An example from classical mechanics

The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In this context, the Lagrangian is usually taken to be the kinetic energy of a mechanical system minus its potential energy.

Suppose we have a three dimensional space and the Lagrangian

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix}\frac{1}{2}\end{matrix} m\dot{\vec{x}}^2-V(\vec{x}).}**

Then, the Euler-Lagrange equation is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot\vec{x}+\nabla V=0}**
where the time derivative is written conventionally as a dot above the quantity being differentiated, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla}**
is the del operator.

Using this result we can easily show that the Lagrangian approach is equivalent to the Newtonian one. We write the force in terms of the potential **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}=- \nabla V(x)}**
; then the resulting equation is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}=m\ddot{\vec{x}}}**
, which is exactly the same equation as in a Newtonian approach for a constant mass object. A very similar deduction gives us the expression **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F}=d\vec{p}/dt}**
, which is Newton's Second Law in its general form.

Suppose we have a three-dimensional space in spherical coordinates, r, θ, φ with the Lagrangian

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{m}{2}(\dot{r}^2+r^2\dot{\theta}^2 +r^2\sin^2\theta\dot{\varphi}^2)-V(r).}**

Then the Euler-Lagrange equations are:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot{r}-mr(\dot{\theta}^2+\sin^2\theta\dot{\varphi}^2)+V' =0,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}(mr^2\dot{\theta}) -mr^2\sin\theta\cos\theta\dot{\varphi}^2=0,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}(mr^2\sin^2\theta\dot{\varphi})=0.}**

Here the set of parameters **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ s_i}**
is just the time **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ t}**
, and the dynamical variables **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \phi_i(s)}**
are the trajectories **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x(t)}**
of the particle.

## Lagrangians and Lagrangian densities in field theory

In field theory, occasionally a distinction is made between the Lagrangian **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}**
, of which the action is the time integral

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \int{L \, dt}}**

and the *Lagrangian density* **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}}**
, which one integrates over all space-time to get the action:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S [\varphi_i] = \int{\mathcal{L} [\varphi_i (x)]\, d^4x}}**

The Lagrangian is then the spatial integral of the Lagrangian density. However, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}}**
is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in relativistic theories since it is a locally defined, Lorentz scalar field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec x}**
is incorporated into the index **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i}**
or the parameters **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s}**
in **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_i(s)}**
. Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}}**
, and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.

## Electromagnetic Lagrangian

Generally, in Lagrangian mechanics, the Lagrangian is equal to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = T - V }**

where *T* is kinetic energy and *V* is potential energy. Given an electrically charged particle with mass *m* and charge *q*, with velocity **v** in an electromagnetic field with scalar potential φ and vector potential **A**, the particle's kinetic energy is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = {1 \over 2} m \mathbf{v} \cdot \mathbf{v} }**

and the particle's potential energy is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = q\phi - {q \over c} \mathbf{v} \cdot \mathbf{A} }**

where *c* is the speed of light. Then the electromagnetic Lagrangian is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = {1 \over 2} m \mathbf{v} \cdot \mathbf{v} - q\phi + {q \over c} \mathbf{v} \cdot \mathbf{A} . }**

## Lagrangians in Quantum Field Theory

### Quantum Electrodynamic Lagrangian

The Lagrangian density for QED is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} = \bar \psi (i \not \!\, D - m) \psi - {1 \over 4} F_{\mu \nu} F^{\mu \nu}}**

where ψ is a spinor, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar \psi := \psi^\dagger \gamma^0 }**
(**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma^0}**
being a gamma matrix), **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{\mu\nu}}**
is the electromagnetic tensor, *D* is the gauge covariant derivative, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \not \!\, D }**
is Feynman notation for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma^\sigma D_\sigma }**
.

### Dirac Lagrangian

The Lagrangian density for a Dirac field is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} = \bar \psi (i \not \! \; \partial - m) \psi = 0 }**.

### Quantum Chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is [1] [2] [3]

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} = -{1\over 4} F^\alpha {}_{\mu\nu} F_\alpha {}^{\mu\nu} - \sum_n \bar \psi_n (\not\!\, D_\mu + m_n) \psi_n }**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_\mu}**
is the QCD gauge covariant derivative, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^\alpha {}_{\mu\nu} }**
is the gluon field strength tensor.

## Mathematical formalism

Suppose we have an *n*-dimensional manifold, M and a target manifold T. Let **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{C}}**
be the configuration space of smooth functions from M to T.

Before we go on, let's give some examples:

- In classical mechanics, in the Hamiltonian formalism, M is the one dimensional manifold
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}}**, representing time and the target space is the cotangent bundle of space of generalized positions. - In field theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m real-valued scalar fields, φ
_{1},...,φ_{m}, then the target manifold is**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^m}**. If the field is a real vector field, then the target manifold is isomorphic to**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n}**. There is actually a much more elegant way using tangent bundles over M, but we will just stick to this version.

Now suppose there is a functional, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S:\mathcal{C}\rightarrow \mathbb{R}}**
, called the action. Note that it is a mapping to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}}**
, not **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}}**
; this has to do with physical reasons.

In order for the action to be local, we need additional restrictions on the action. If **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi\in\mathcal{C}}**
, we assume S[φ] is the integral over M of a function of φ, its derivatives and the position called the **Lagrangian**, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}(\varphi,\partial\varphi,\partial\partial\varphi, ...,x)}**
. In other words,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall\varphi\in\mathcal{C}\, S[\varphi]\equiv\int_M d^nx \mathcal{L}(\varphi(x),\partial\varphi(x),\partial\partial\varphi(x), ...,x).}**

Most of the time, we will also assume in addition that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives; this is only a matter of convenience, though, and is not true in general! We will make this assumption for the rest of this article.

Given boundary conditions, basically a specification of the value of φ at the boundary if M is compact or some limit on φ as x approaches **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}**
(this will help in doing integration by parts), the subspace of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{C}}**
consisting of functions, φ such that all functional derivatives of S at φ are zero and φ satisfies the given boundary conditions is the subspace of on shell solutions.

The solution is given by the **Euler-Lagrange equations** (thanks to the boundary conditions),

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta}{\delta\varphi}S=-\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}\right)+ \frac{\partial\mathcal{L}}{\partial\varphi}=0.}**

Incidentally, the left hand side is the functional derivative of the action with respect to φ.

## See also

- Lagrangian mechanics
- Noether's theorem
- Hamiltonian mechanics
- functional derivative
- functional integral
- action principle
- coherence condition

## External links

- Christoph Schiller (2005),
*Global descriptions of motion: the simplicity of complexity*, Motion Mountain

ca:Lagrangià de:Lagrangefunktion es:Lagrangiano it:Lagrangiana nl:Lagrangiaan pl:Lagranżjan pt:Lagrangiana sl:Lagrangeeva funkcija