The Lagrangian, ''L'', of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics. In classical mechanics, the Lagrangian is defined as the kinetic energy, $T$, of the system minus its potential energy, $V$. In symbols,

:$L\; =\; T\; -\; V.\backslash quad$

Under conditions that are given in Lagrangian mechanics, if the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation, a particular family of partial differential equations.

The Lagrangian formulation

Importance

The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, the ''action principle'' that is used to derive the Lagrange equation was later recognized to be applicable to quantum mechanics as well.

Physical action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions.

The same principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which relates physical conserved quantities to continuous symmetries of a physical system.

Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.

Advantages over other methods

The formulation is not tied to any one coordinate system—rather, any convenient variables $\backslash phi\_i(s)$ may be used to describe the system; these variables are called "generalized coordinates" and may be any independent variable of the system (for example, strength of the magnetic field at a particular location; angle of a pulley; position of a particle in space; or degree of excitation of a particular eigenmode in a complex system). This makes it easy to incorporate constraints into a theory by defining coordinates which only describe states of the system which satisfy the constraints.

If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This is very helpful in showing that theories are consistent with either special relativity or general relativity.

Equations derived from a Lagrangian will almost automatically be unambiguous and consistent, unlike equations brought together from multiple formulations.

==="Cyclic coordinates" and conservation laws=== An important property of the Lagrangian is that conservation laws can easily be read off from it. For example, if the Lagrangian $\backslash mathcal\; L$ depends on the ''time-derivative'' $\backslash dot\; q\_i$ of a generalized coordinate, but ''not'' on $q\_i$ itself, then the ''generalized momentum'', :$p\_i:=\backslash frac\{\backslash partial\backslash mathcal\; L\}\{\backslash partial\backslash dot\; q\_i\}$, is a conserved quantity. This is a special case of Noether's theorem. Such coordinates are called "cyclic".

For example, the conservation of the generalized momentum, : $p\_2:=\backslash frac\{\backslash partial\backslash mathcal\; L\}\{\backslash partial\backslash dot\; q\_2\}$, say, can be directly seen if the Lagrangian of the system is of the form :$\backslash mathcal\; L(q\_1,q\_3,q\_4,\; \backslash dots;\; \backslash dot\; q\_1,\backslash dot\; q\_2,\backslash dot\; q\_3,\backslash dot\; q\_4,\; \backslash dots;t)\backslash ,.$

Also, if the time, ''t'', does not appear in $\backslash mathcal\; L$, then the conservation of the Hamiltonian follows. This is the energy conservation unless the potential energy depends on velocity, as in electrodynamics. More details can be found in any textbook on theoretical mechanics.

Explanation

The equations of motion are obtained by means of an action principle, written as:

:$\backslash frac\{\backslash delta\; \backslash mathcal\{S\}\}\{\backslash delta\; \backslash varphi\_i\}\; =\; 0\backslash ,.$

where the ''action'', $\backslash mathcal\{S\}$, is a functional of the dependent variables $\backslash varphi\_i(s)$ with their derivatives and ''s'' itself

:$\backslash mathcal\{S\}\backslash left[\backslash varphi\_i,\; \backslash frac\{\backslash partial\; \backslash varphi\_i\}\; \{\backslash partial\; s\}\backslash right]\; =\; \backslash int\{\; \backslash mathcal\{L\}\; \backslash left[\backslash varphi\_i\; [s],\; \backslash frac\{\backslash partial\; \backslash varphi\_i\; [s]\}\{\backslash partial\; s^\backslash alpha\},\; s^\backslash alpha\backslash right]\; \backslash ,\; \backslash mathrm\{d\}^n\; s\; \}$

and where $s\; =\; \backslash \{\; s^\backslash alpha\; \backslash \}\; \backslash !$ denotes the set of ''n'' independent variables of the system, indexed by $\backslash alpha\; =\; 1,\; 2,\; 3,\; \backslash ldots,\; n\; .$

The equations of motion obtained from this functional derivative are the Euler–Lagrange equations of this action. For example, in the classical mechanics of particles, the only independent variable is time, ''t''. So the Euler-Lagrange equations are :$\backslash frac\{d\}\{d\; t\}\backslash frac\{\backslash partial\backslash mathcal\; L\}\{\backslash partial\backslash dot\; \backslash varphi\_i\}\; =\; \backslash frac\{\backslash partial\backslash mathcal\; L\}\{\backslash partial\backslash varphi\_i\}\; .$

Dynamical systems 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.

An example from classical mechanics

In the rectangular coordinate system

Suppose we have a three-dimensional space and the Lagrangian

:$L(\backslash vec\{x\},\; \backslash dot\{\backslash vec\{x\}\})\; \backslash \; =\; \backslash \; \backslash frac\{1\}\{2\}\; \backslash \; m\; \backslash \; \backslash dot\{\backslash vec\{x\}\}^2\; \backslash \; -\; \backslash \; V(\backslash vec\{x\})$.

Then, the Euler–Lagrange equation is:

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash \; \backslash left(\; \backslash ,\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\_i\}\; \backslash ,\; \backslash right)\; \backslash \; -\; \backslash \; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\_i\}\; \backslash \; =\; \backslash \; 0$ where $i\; =\; 1,\; 2,\; 3$.

The derivation yields:

:$\backslash frac\{\backslash partial\; L\}\{\backslash partial\; x\_i\}\; \backslash \; =\; \backslash \; -\; \backslash \; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; x\_i\}$ :$\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\_i\}\; \backslash \; =\; \backslash \; \backslash frac\{\backslash partial\; ~\}\{\backslash partial\; \backslash dot\{x\}\_i\}\; \backslash ,\; \backslash left(\; \backslash ,\; \backslash frac\{1\}\{2\}\; \backslash \; m\; \backslash \; \backslash dot\{\backslash vec\{x\}\}^2\; \backslash ,\; \backslash right)\; \backslash \; =\; \backslash \; \backslash frac\{1\}\{2\}\; \backslash \; m\; \backslash \; \backslash frac\{\backslash partial\; ~\}\{\backslash partial\; \backslash dot\{x\}\_i\}\; \backslash ,\; \backslash left(\; \backslash ,\; \backslash dot\{x\}\_i\; \backslash ,\; \backslash dot\{x\}\_i\; \backslash ,\; \backslash right)\; =\; \backslash \; m\; \backslash ,\; \backslash dot\{x\}\_i$ :$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash \; \backslash left(\; \backslash ,\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{x\}\_i\}\; \backslash ,\; \backslash right)\; \backslash \; =\; \backslash \; m\; \backslash ,\; \backslash ddot\{x\}\_i$

The Euler–Lagrange equations can therefore be written as:

:$m\backslash ddot\{\backslash vec\{x\}\}+\backslash nabla\; V=0$

where the time derivative is written conventionally as a dot above the quantity being differentiated, and $\backslash nabla$ is the del operator.

Using this result, it can easily be shown that the Lagrangian approach is equivalent to the Newtonian one.

If the force is written in terms of the potential $\backslash vec\{F\}=-\; \backslash nabla\; V(x)$; the resulting equation is $\backslash vec\{F\}=m\backslash ddot\{\backslash vec\{x\}\}$, which is exactly the same equation as in a Newtonian approach for a constant mass object.

A very similar deduction gives us the expression $\backslash vec\{F\}=\backslash mathrm\{d\}\backslash vec\{p\}/\backslash mathrm\{d\}t$, which is Newton's Second Law in its general form.

In the spherical coordinate system

Suppose we have a three-dimensional space using spherical coordinates $r,\; \backslash theta,\; \backslash phi$ with the Lagrangian

:$L\; =\; \backslash frac\{m\}\{2\}(\backslash dot\{r\}^2+r^2\backslash dot\{\backslash theta\}^2\; +r^2\backslash sin^2\backslash theta\backslash dot\{\backslash varphi\}^2)-V(r).$

Then the Euler–Lagrange equations are:

:$m\backslash ddot\{r\}-mr(\backslash dot\{\backslash theta\}^2+\backslash sin^2\backslash theta\backslash dot\{\backslash varphi\}^2)+V\text{'}\; =0,$ :$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}(mr^2\backslash dot\{\backslash theta\})\; -mr^2\backslash sin\backslash theta\backslash cos\backslash theta\backslash dot\{\backslash varphi\}^2=0,$ :$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}(mr^2\backslash sin^2\backslash theta\backslash dot\{\backslash varphi\})=0.$

Here the set of parameters $s\_i$ is just the time $t$, and the dynamical variables $\backslash phi\_i(s)$ are the trajectories $\backslash vec\; x(t)$ of the particle.

Despite the use of standard variables such as $x$, the Lagrangian allows the use of any coordinates, which do not need to be orthogonal. These are "generalized coordinates".

Lagrangian of a test particle

A test particle is a particle whose mass and charge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like electrons and up-quarks are more complex and have additional terms in their Lagrangians.

Classical test particle with Newtonian gravity

Suppose we are given a particle with mass $m\; \backslash !$ kilograms, and position $\backslash vec\{x\}$ meters in a Newtonian gravitation field with potential $\backslash zeta\; \backslash !$ joules per kilogram. The particle's world line is parameterized by time $t\backslash !$ seconds. The particle's kinetic energy is:

:$T[t]\; =\; \{1\; \backslash over\; 2\}\; m\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash dot\{\backslash vec\{x\}\}[t]$

and the particle's gravitational potential energy is:

:$V[t]\; =\; m\; \backslash zeta\; [\backslash vec\{x\}\; [t],t]\; .$

Then its Lagrangian is $L\; \backslash !$ joules where

:$L[t]\; =\; T[t]\; -\; V[t]\; =\; \{1\; \backslash over\; 2\}\; m\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash dot\{\backslash vec\{x\}\}[t]\; -\; m\; \backslash zeta\; [\backslash vec\{x\}\; [t],t]\; .$

Varying $\backslash vec\{x\}\backslash !$ in the integral (equivalent to the Euler–Lagrange differential equation), we get

:$0\; =\; \backslash delta\backslash int\{L[t]\; \backslash ,\; \backslash mathrm\{d\}t\}\; =\; \backslash int\{\backslash delta\; L[t]\; \backslash ,\; \backslash mathrm\{d\}t\}$ :$=\; \backslash int\{m\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash dot\{\backslash delta\; \backslash vec\{x\}\}[t]\; -\; m\; \backslash nabla\; \backslash zeta\; [\backslash vec\{x\}\; [t],t]\; \backslash cdot\; \backslash delta\; \backslash vec\{x\}[t]\; \backslash ,\; \backslash mathrm\{d\}t\}.$

Integrate the first term by parts and discard the total integral. Then divide out the variation to get

:$0\; =\; -\; m\; \backslash ddot\{\backslash vec\{x\}\}[t]\; -\; m\; \backslash nabla\; \backslash zeta\; [\backslash vec\{x\}\; [t],t]$

and thus

:$m\; \backslash ddot\{\backslash vec\{x\}\}[t]\; =\; -\; m\; \backslash nabla\; \backslash zeta\; [\backslash vec\{x\}\; [t],t]\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,(1)$

is the equation of motion — two different expressions for the force.

Special relativistic test particle with electromagnetism

In special relativity, the form of the term which gives rise to the derivative of the momentum must be changed; it is no longer the kinetic energy. It becomes:

:$-\; m\; c^2\; \backslash frac\{d\; \backslash tau[t]\}\{d\; t\}\; =\; -\; m\; c^2\; \backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}$ :$=\; -m\; c^2\; +\; \{1\; \backslash over\; 2\}\; m\; v^2\; [t]\; +\; \{1\; \backslash over\; 8\}\; m\; \backslash frac\{v^4\; [t]\}\{c^2\}\; +\; \backslash dots$

(In special relativity, the energy of a free test particle is $m\; c^2\; \backslash frac\{dt\}\{d\; \backslash tau\; [t]\}\; =\; \backslash frac\{m\; c^2\}\{\backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}\}\; =\; +m\; c^2\; +\; \{1\; \backslash over\; 2\}\; m\; v^2\; [t]\; +\; \{3\; \backslash over\; 8\}\; m\; \backslash frac\{v^4\; [t]\}\{c^2\}\; +\; \backslash dots$)

where $c\; \backslash !$ meters per second is the speed of light in vacuum, $\backslash tau\; \backslash !$ seconds is the proper time (i.e. time measured by a clock moving with the particle) and $v^2\; [t]\; =\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash dot\{\backslash vec\{x\}\}[t].$ The second term in the series is just the classical kinetic energy. Suppose the particle has electrical charge $q\backslash !$ coulombs and is in an electromagnetic field with scalar potential $\backslash phi\; \backslash !$ volts (a volt is a joule per coulomb) and vector potential $\backslash vec\{A\}$ volt seconds per meter. The Lagrangian of a special relativistic test particle in an electromagnetic field is:

:$L[t]\; =\; -\; m\; c^2\; \backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}\; -\; q\; \backslash phi\; [\backslash vec\{x\}[t],t]\; +\; q\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash vec\{A\}\; [\backslash vec\{x\}[t],t]$

Varying this with respect to $\backslash vec\{x\}$, we get

:$0\; =\; -\; \backslash frac\{d\}\{d\; t\}\backslash left(\backslash frac\{m\; \backslash dot\{\backslash vec\{x\}\}[t]\}\; \{\backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}\}\backslash right)\; -\; q\; \backslash nabla\backslash phi\; [\backslash vec\{x\}[t],t]\; -\; q\; \backslash partial\_t\{\backslash vec\{A\}\}\; [\backslash vec\{x\}[t],t]\; -\; q\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash nabla\backslash vec\{A\}\; [\backslash vec\{x\}[t],t]\; +\; q\; \backslash nabla\{\backslash vec\{A\}\}\; [\backslash vec\{x\}[t],t]\; \backslash cdot\; \backslash dot\{\backslash vec\{x\}\}[t]$

which is

:$\backslash frac\{d\}\{d\; t\}\backslash left(\backslash frac\{m\; \backslash dot\{\backslash vec\{x\}\}[t]\}\; \{\backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}\}\backslash right)\; =\; q\; \backslash vec\{E\}[\backslash vec\{x\}[t],t]\; +\; q\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash times\; \backslash vec\{B\}\; [\backslash vec\{x\}[t],t]$

which is the equation for the Lorentz force where

:$\backslash vec\{E\}[\backslash vec\{x\},t]\; =\; -\; \backslash nabla\backslash phi\; [\backslash vec\{x\},t]\; -\; \backslash partial\_t\{\backslash vec\{A\}\}\; [\backslash vec\{x\},t]$ :$\backslash vec\{B\}[\backslash vec\{x\},t]\; =\; \backslash nabla\; \backslash times\; \backslash vec\{A\}\; [\backslash vec\{x\},t]$

General relativistic test particle

In general relativity, the first term generalizes (includes) both the classical kinetic energy and interaction with the Newtonian gravitational potential. It becomes:

:$-\; m\; c^2\; \backslash frac\{d\; \backslash tau[t]\}\{d\; t\}$ :$=\; -\; m\; c\; \backslash sqrt\; \{-\; g\_\{\backslash alpha\backslash beta\}[x[t]]\; \backslash frac\{d\; x^\{\backslash alpha\}[t]\}\{d\; t\}\; \backslash frac\{d\; x^\{\backslash beta\}[t]\}\{d\; t\}\}\; .$

The Lagrangian of a general relativistic test particle in an electromagnetic field is:

:$L[t]\; =\; -\; m\; c\; \backslash sqrt\; \{-\; g\_\{\backslash alpha\backslash beta\}[x[t]]\; \backslash frac\{d\; x^\{\backslash alpha\}[t]\}\{d\; t\}\; \backslash frac\{d\; x^\{\backslash beta\}[t]\}\{d\; t\}\}\; +\; q\; \backslash frac\{d\; x^\{\backslash gamma\}[t]\}\{d\; t\}\; A\_\{\backslash gamma\}[x[t]]\; .$

If the four space-time coordinates $x^\{\backslash alpha\}\backslash !$ are given in arbitrary units (i.e. unit-less), then $g\_\{\backslash alpha\backslash beta\}\backslash !$ meters squared is the rank 2 symmetric metric tensor which is also the gravitational potential. Also, $A\_\{\backslash gamma\}\backslash !$ volt seconds is the electromagnetic 4-vector potential. Notice that a factor of ''c'' has been absorbed into the square root because it is the equivalent of

:$c\backslash ,\; \backslash sqrt\; \{1\; -\; \backslash frac\{v^2\; [t]\}\{c^2\}\}\; =\; \backslash sqrt\; \{-\; (\; -\; c^2\; +\; v^2\; [t])\}\; .$ Note that this notion has been directly generalized from special relativity

Lagrangians and Lagrangian densities in field theory

The time integral of the Lagrangian is called the action denoted by $S$. In field theory, a distinction is occasionally made between the Lagrangian $L$, of which the action is the time integral:

:$\backslash mathcal\{S\}\; =\; \backslash int\{L\; \backslash ,\; \backslash mathrm\{d\}t\}$

and the ''Lagrangian density'' $\backslash mathcal\{L\}$, which one integrates over all space-time to get the action:

:$\backslash mathcal\{S\}\; [\backslash varphi\_i]\; =\; \backslash int\{\backslash mathcal\{L\}\; [\backslash varphi\_i\; (x)]\backslash ,\; \backslash mathrm\{d\}^4x\}$

The Lagrangian is then the spatial integral of the Lagrangian density. However, $\backslash mathcal\{L\}$ 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 $\backslash vec\; x$ is incorporated into the index $i$ or the parameters $s$ in $\backslash varphi\_i(s)$. Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of $\backslash mathcal\{L\}$, and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams.

Selected fields

To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move and allow the calculation of those fields. The equations below will not give you the equations of motion of a test particle in the field but will instead give you the potential (field) induced by quantities such as mass or charge density at any point $[\backslash vec\{x\},t]$. For example, in the case of Newtonian gravity, the Lagrangian density integrated over space-time gives you an equation which, if solved, would yield $\backslash zeta\; [\backslash vec\{x\},t]$. This $\backslash zeta\; [\backslash vec\{x\},t]$, when substituted back in equation (1), the Lagrangian equation for the test particle in a Newtonian gravitational field, provides the information needed to calculate the acceleration of the particle.

Newtonian gravity

The Lagrangian (density) is $\backslash mathcal\{L\}$ joules per cubic meter. The interaction term $m\; \backslash zeta\; \backslash !$ is replaced by a term involving a continuous mass density $\backslash mu\; \backslash !$ kilograms per cubic meter. This is necessary because using a point source for a field would result in mathematical difficulties. The resulting Lagrangian for the classical gravitational field is:

:$\backslash mathcal\{L\}[\backslash vec\{x\},t]\; =\; -\; \backslash mu\; [\backslash vec\{x\},t]\; \backslash zeta\; [\backslash vec\{x\},t]\; -\; \{1\; \backslash over\; 8\; \backslash pi\; G\}\; (\backslash nabla\; \backslash zeta\; [\backslash vec\{x\},t])^2$

where $G\; \backslash !$ meters cubed per kilogram second squared is the gravitational constant. Variation of the integral with respect to $\backslash zeta\; \backslash !$ gives:

:$0\; =\; -\; \backslash mu\; [\backslash vec\{x\},t]\; \backslash delta\backslash zeta\; [\backslash vec\{x\},t]\; -\; \{2\; \backslash over\; 8\; \backslash pi\; G\}\; (\backslash nabla\; \backslash zeta\; [\backslash vec\{x\},t])\; \backslash cdot\; (\backslash nabla\; \backslash delta\backslash zeta\; [\backslash vec\{x\},t])\; .$

Integrate by parts and discard the total integral. Then divide out by $\backslash delta\backslash zeta\; \backslash !$ to get:

:$0\; =\; -\; \backslash mu\; [\backslash vec\{x\},t]\; +\; \{1\; \backslash over\; 4\; \backslash pi\; G\}\; \backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash zeta\; [\backslash vec\{x\},t]$

and thus

:$4\; \backslash pi\; G\; \backslash mu\; [\backslash vec\{x\},t]\; =\; \backslash nabla^2\; \backslash zeta\; [\backslash vec\{x\},t]$

which yields Gauss's law for gravity.

Electromagnetism in special relativity

The interaction terms $-\; q\; \backslash phi\; [\backslash vec\{x\}[t],t]\; +\; q\; \backslash dot\{\backslash vec\{x\}\}[t]\; \backslash cdot\; \backslash vec\{A\}\; [\backslash vec\{x\}[t],t]$ are replaced by terms involving a continuous charge density $\backslash rho\; \backslash !$ coulombs per cubic meter and current density $\backslash vec\{j\}\; \backslash !$ amperes per square meter. The resulting Lagrangian for the electromagnetic field is:

:$\backslash mathcal\{L\}[\backslash vec\{x\},t]\; =\; -\; \backslash rho\; [\backslash vec\{x\},t]\; \backslash phi\; [\backslash vec\{x\},t]\; +\; \backslash vec\{j\}\; [\backslash vec\{x\},t]\; \backslash cdot\; \backslash vec\{A\}\; [\backslash vec\{x\},t]\; +\; \{\backslash epsilon\_0\; \backslash over\; 2\}\; \{E\}^2\; [\backslash vec\{x\},t]\; -\; \{1\; \backslash over\; \{2\; \backslash mu\_0\}\}\; \{B\}^2\; [\backslash vec\{x\},t]\; .$

Varying this with respect to $\backslash phi\; \backslash !$, we get

:$0\; =\; -\; \backslash rho\; [\backslash vec\{x\},t]\; +\; \backslash epsilon\_0\; \backslash nabla\; \backslash cdot\; \backslash vec\{E\}\; [\backslash vec\{x\},t]$

which yields Gauss' law.

Varying instead with respect to $\backslash vec\{A\}$, we get

:$0\; =\; \backslash vec\{j\}\; [\backslash vec\{x\},t]\; +\; \backslash epsilon\_0\; \backslash partial\_t\; \backslash vec\{E\}\; [\backslash vec\{x\},t]\; -\; \{1\; \backslash over\; \backslash mu\_0\}\; \backslash nabla\; \backslash times\; \backslash vec\{B\}\; [\backslash vec\{x\},t]$

which yields Ampère's law.

Electromagnetism in general relativity

For the Lagrangian of gravity in general relativity, see Einstein-Hilbert action. The Lagrangian of the electromagnetic field is:

:$\backslash mathcal\{L\}[x]\; =\; +\; J^\{\backslash gamma\}[x]\; A\_\{\backslash gamma\}[x]\; -\; \{1\; \backslash over\; 4\backslash mu\_0\}\; F\_\{\backslash mu\; \backslash nu\}[x]\; F\_\{\backslash alpha\; \backslash beta\}[x]\; g^\{\backslash mu\backslash alpha\}[x]\; g^\{\backslash nu\backslash beta\}[x]\; \backslash sqrt\{\backslash frac\{-1\}\{c^2\}\; \backslash mathrm\{det\}\; [g[x]]\}.$

If the four space-time coordinates $x^\{\backslash alpha\}\backslash !$ are given in arbitrary units, then: $\backslash mathcal\{L\}$ joule seconds is the Lagrangian, a scalar density; $J^\{\backslash gamma\}\backslash !$ coulombs is the current, a vector density; and $F\_\{\backslash mu\; \backslash nu\}\backslash !$ volt seconds is the electromagnetic tensor, a covariant antisymmetric tensor of rank two. Notice that the determinant under the square root sign is applied to the matrix of components of the covariant metric tensor $g\_\{\backslash alpha\backslash beta\}\backslash !$, and $g^\{\backslash alpha\backslash beta\}\backslash !$ is its inverse. Notice that the units of the Lagrangian changed because we are integrating over $x^0,\; x^1,\; x^2,\; x^3\backslash !$ which are unit-less rather than over $t,\; x,\; y,\; z\; \backslash !$ which have units of seconds meters cubed. The electromagnetic field tensor is formed by anti-symmetrizing the partial derivative of the electromagnetic vector potential; so it is not an independent variable. The square root is needed to convert that term into a scalar density instead of just a scalar, and also to compensate for the change in the units of the variables of integration. The factor of $\backslash frac\{-1\}\{c^2\}$ inside the square root is needed to normalize it so that the square root will reduce to one in special relativity (since the determinant is $-\; c^2\; \backslash !$ in special relativity).

Electromagnetism using differential forms

Using differential forms, the electromagnetic action in vacuum on a (pseudo-) Riemannian manifold $\backslash mathcal\; M$ can be written as (using natural units, $c=\backslash varepsilon\_0=1$) :$S[A]\; =\; \backslash int\_\{\backslash mathcal\; M\}\; \backslash left(-\backslash frac12\backslash mathrm\; d\; A\backslash wedge*\backslash mathrm\; dA+A\backslash wedge\backslash mathcal\; S\backslash right)$ Here, $A$ stands for the electromagnetic potential 1-form, and $\backslash mathcal\; S$ is the current 3-form. Note that Lagrangian is exactly the same thing as in the paragraph above, only that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Expanding the action integral into a basis yields the lengthy Lagrangian expression. Variation of the expression leads to :$\backslash mathrm\; d*\backslash mathrm\; dA=\backslash mathcal\; S$ These are Maxwell's equations for the electromagnetic potential. Substituting $F=\backslash mathrm\; dA$ immediately yields the equations for the fields, :$\backslash mathrm\; dF=0$ :$\backslash mathrm\; d*F=\backslash mathcal\; S$

Lagrangians in quantum field theory

Dirac Lagrangian

The Lagrangian density for a Dirac field is:

:$\backslash mathcal\{L\}\; =\; \backslash frac\{i\; \backslash hbar\; c\}\{2\}\; \backslash left(\backslash bar\; \backslash psi\; \backslash not\backslash !\{\backslash partial\}\backslash psi\; -\; \backslash psi\; \backslash overline\; \{\backslash not\backslash !\{\backslash partial\}\; \backslash psi\}\; \backslash right)\; -\; mc^2\; \backslash bar\backslash psi\; \backslash psi$

where $\backslash psi\backslash !$ is a Dirac spinor (annihilation operator), $\backslash bar\; \backslash psi\; =\; \backslash psi^\backslash dagger\; \backslash gamma^0$ is its Dirac adjoint (creation operator) and $\backslash not\backslash !\{\backslash partial\}$ is Feynman notation for $\backslash gamma^\backslash sigma\; \backslash partial\_\backslash sigma\backslash !$.

Quantum electrodynamic Lagrangian

The Lagrangian density for QED is:

:$\backslash mathcal\{L\}\_\{\backslash mathrm\{QED\}\}\; =\; \backslash frac\{i\backslash hbar\; c\}\{2\}\; \backslash left(\backslash bar\; \backslash psi\; \backslash not\backslash !\backslash !\; D\; \backslash psi\; -\; \backslash psi\; \backslash overline\; \{\backslash not\backslash !\backslash !\; D\; \backslash psi\}\; \backslash right)\; -\; mc^2\; \backslash bar\backslash psi\; \backslash psi\; -\; \{1\; \backslash over\; 4\backslash mu\_0\}\; F\_\{\backslash mu\; \backslash nu\}\; F^\{\backslash mu\; \backslash nu\}$

where $F^\{\backslash mu\; \backslash nu\}\backslash !$ is the electromagnetic tensor, $D\backslash !$ is the gauge covariant derivative, and $\backslash not\backslash !\backslash !D$ is Feynman notation for $\backslash gamma^\backslash sigma\; D\_\backslash sigma\backslash !$.

Quantum chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is :

:$\backslash mathcal\{L\}\_\{\backslash mathrm\{QCD\}\}\; =\; \backslash sum\_n\; \backslash left(\; \backslash frac\{i\backslash hbar\; c\}\{2\}\; (\backslash bar\backslash psi\_n\backslash not\backslash !\backslash !\; D\; \backslash psi\_n\; -\; \backslash psi\_n\; \backslash overline\; \{\backslash not\backslash !\backslash !\; D\; \backslash psi\_n\})\; -\; m\_n\; c^2\; \backslash bar\backslash psi\_n\; \backslash psi\_n\; \backslash right)\; -\; \{1\backslash over\; 4\}\; G^\backslash alpha\; \{\}\_\{\backslash mu\backslash nu\}\; G\_\backslash alpha\; \{\}^\{\backslash mu\backslash nu\}$

where $D\backslash !$ is the QCD gauge covariant derivative, $n=1...6$ counts the quark types, and $G^\backslash alpha\; \{\}\_\{\backslash mu\backslash nu\}\backslash !$ is the gluon field strength tensor.

Mathematical formalism

Suppose we have an ''n''-dimensional manifold, $M$, and a target manifold, $T$. Let $\backslash mathcal\{C\}$ be the configuration space of smooth functions from $M$ to $T$.

Examples

In classical mechanics, in the Hamiltonian formalism, $M$ is the one-dimensional manifold $\backslash mathbb\{R\}$, 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, $\backslash phi\_\{1\},\; ...,\; \backslash phi\_\{m\}$, then the target manifold is $\backslash mathbb\{R\}^m$. If the field is a real vector field, then the target manifold is isomorphic to $\backslash mathbb\{R\}^n$. There is actually a much more elegant way using tangent bundles over $M$, but we will just stick to this version.

Mathematical development

Consider a functional, $\backslash mathcal\{S\}:\backslash mathcal\{C\}\backslash rightarrow\; \backslash mathbb\{R\}$, called the action. Physical reasons determine that it is a mapping to $\backslash mathbb\{R\}$, not $\backslash mathbb\{C\}$.

In order for the action to be local, we need additional restrictions on the action. If $\backslash varphi\backslash in\backslash mathcal\{C\}$, we assume $\backslash mathcal\{S\}[\backslash varphi]$ is the integral over $M$ of a function of $\backslash phi$, its derivatives and the position called the Lagrangian, $\backslash mathcal\{L\}(\backslash varphi,\backslash partial\backslash varphi,\backslash partial\backslash partial\backslash varphi,\; ...,x)$. In other words,

:$\backslash forall\backslash varphi\backslash in\backslash mathcal\{C\},\; \backslash \; \backslash \; \backslash mathcal\{S\}[\backslash varphi]\backslash equiv\backslash int\_M\; \backslash mathrm\{d\}^nx\; \backslash mathcal\{L\}\; \backslash big(\; \backslash varphi(x),\backslash partial\backslash varphi(x),\backslash partial\backslash partial\backslash varphi(x),\; ...,x\; \backslash big).$

It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.

Given boundary conditions, basically a specification of the value of $\backslash phi$ at the boundary if $M$ is compact or some limit on $\backslash phi$ as x approaches $\backslash infty$ (this will help in doing integration by parts), the subspace of $\backslash mathcal\{C\}$ consisting of functions, $\backslash phi$ such that all functional derivatives of $S$ at $\backslash phi$ are zero and $\backslash phi$ 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),

:$\backslash frac\{\backslash delta\backslash mathcal\{S\}\}\{\backslash delta\backslash varphi\}=-\backslash partial\_\backslash mu\; \backslash left(\backslash frac\{\backslash partial\backslash mathcal\{L\}\}\{\backslash partial(\backslash partial\_\backslash mu\backslash varphi)\}\backslash right)+\; \backslash frac\{\backslash partial\backslash mathcal\{L\}\}\{\backslash partial\backslash varphi\}=0.$

The left hand side is the functional derivative of the action with respect to $\backslash phi$.

See also

Calculus of variations

Covariant classical field theory

Functional derivative

Functional integral

Generalized coordinates

Hamiltonian mechanics

Lagrangian and Eulerian coordinates

Euler–Lagrange equation

Lagrangian mechanics

Lagrangian point

Lagrangian system

Noether's theorem

Principle of least action

Scalar field theory

Einstein-Maxwell-Dirac equations

Notes

References

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

David Tong Classical Dynamics (Cambridge lecture notes)

Category:Fundamental physics concepts Category:Dynamical systems Category:Mathematical and quantitative methods (economics)

ar:لاغرانجيان bg:Оператор на Лагранж ca:Lagrangià cs:Lagrangeova funkce de:Lagrange-Dichte et:Lagrange'i funktsioon el:Λαγκρανζιανή συνάρτηση es:Lagrangiano fr:Lagrangien gl:Lagranxiana ko:라그랑지안 it:Lagrangiana he:לגראנז'יאן hu:Lagrange-függvény nl:Lagrangiaan ja:ラグランジアン pl:Lagranżjan pt:Função de Lagrange ru:Лагранжиан sq:Formalizmi i Lagranzhit sk:Lagrangeova funkcia sl:Lagrangeeva funkcija sv:Lagrangefunktion uk:Лагранжиан zh:拉格朗日量