Lagrangian

This article is about Lagrange mechanics. For other uses, see Lagrangian (disambiguation). The term Lagrangian refers to any of several mathematical concepts developed by Joseph Louis Lagrange In physics the Lagrangian is a function that characterizes

The Lagrangian, $L$ , of a dynamical system is a function that summarizes the dynamics of the system. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. In classical mechanics, the Lagrangian is defined as the kinetic energy, $T$ , of the system minus its potential energy, $V$ . The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system ^[1]^:270 In symbols,

$L = T - V.\quad$

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. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

The Lagrange formulation

Importance

The Lagrange formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation is now recognized to be applicable to quantum mechanics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

Physical action and quantum-mechanical phase (waves) are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. This article discusses the history of the principle of least action In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system

The same principle, and the Lagrange formalism, are tied closely to Noether's theorem, which relates physical conserved quantities to continuous symmetries of a physical system. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

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. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has A first quantization of a physical system is a semi- classical treatment of Quantum mechanics, in which particles or physical objects are treated using quantum In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

Advantages over other methods

The formulation is not tied to any one coordinate system -- rather, any convenient variables $\varphi_i(s)$ 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). By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency 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 just thrown together from various sources.

Explanation

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

$\frac{\delta \mathcal{S}}{\delta \varphi_i} = 0$

where the action, S, is a functional

$\mathcal{S}[\varphi_i] = \int{\mathcal{L}[\varphi_i(s)]{}\,\mathrm{d}^ns},$

and where ${}{}{}{}\ s_\alpha$ denotes the set of parameters of the system. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics

The equations of motion obtained by means of the functional derivative are identical to the usual Euler–Lagrange equations. In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions 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. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematics, Plateau's problem is to show the existence of a Minimal surface with a given boundary a problem raised by Joseph-Louis Lagrange in

An example from classical mechanics

In the rectangular coordinate system

Suppose we have a three-dimensional space and the Lagrangian

$L(\vec{x}, \dot{\vec{x}}) \ = \ \frac{1}{2} \ m \ \dot{\vec{x}}^2 \ - \ V(\vec{x})$ . Three-dimensional space is a geometric model of the physical Universe in which we live

Then, the Euler–Lagrange equation is:

$\frac{d~}{dt} \ \left( \, \frac{\partial L}{\partial \dot{x}_i} \, \right) \ - \ \frac{\partial L}{\partial x_i} \ = \ 0$

where $i = 1,2,3$ .

The derivation yields:

$\frac{\partial L}{\partial x_i} \ = \ - \ \frac{\partial V}{\partial x_i}$

$\frac{\partial L}{\partial \dot{x}_i} \ = \ \frac{\partial ~}{\partial \dot{x}_i} \, \left( \, \frac{1}{2} \ m \ \dot{\vec{x}}^2 \, \right) \ = \ \frac{1}{2} \ m \ \frac{\partial ~}{\partial \dot{x}_i} \, \left( \, \dot{x}_i \, \dot{x}_i \, \right) = \ m \, \dot{x}_i$

$\frac{d~}{dt} \ \left( \, \frac{\partial L}{\partial \dot{x}_i} \, \right) \ = \ m \, \ddot{x}_i$

The Euler–Lagrange equations can therefore be written as:

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

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

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

A very similar deduction gives us the expression $\vec{F}=\mathrm{d}\vec{p}/\mathrm{d}t$ , 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 $r,θ,φ$ with the Lagrangian

$\frac{m}{2}(\dot{r}^2+r^2\dot{\theta}^2 +r^2\sin^2\theta\dot{\varphi}^2)-V(r).$

Then the Euler–Lagrange equations are:

$m\ddot{r}-mr(\dot{\theta}^2+\sin^2\theta\dot{\varphi}^2)+V' =0,$

$\frac{\mathrm{d}}{\mathrm{d}t}(mr^2\dot{\theta}) -mr^2\sin\theta\cos\theta\dot{\varphi}^2=0,$

$\frac{\mathrm{d}}{\mathrm{d}t}(mr^2\sin^2\theta\dot{\varphi})=0.$

Here the set of parameters $s i$ is just the time $t$ , and the dynamical variables $φ i (s)$ are the trajectories $\vec x(t)$ of the particle. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial

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". By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately

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 ofen 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

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

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

and the particle's gravitational potential energy is:

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

Thus the Lagrangian is:

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

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

$0 = \delta\int{L[t] \, \mathrm{d}t} = \int{\delta L[t] \, \mathrm{d}t}$

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

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

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

and thus

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

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:

$- m c^2 \frac{d \tau[t]}{d t} = - m c^2 \sqrt {1 - \frac{v^2 [t]}{c^2}}$

$= -m c^2 + {1 \over 2} m v^2 [t] + {1 \over 8} m \frac{v^4 [t]}{c^2} + \dots$

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

where $c \!$ meters per second is the speed of light in vacuum, $\tau \!$ seconds is the proper time (i. e. time measured by a clock moving with the particle) and $v^2 [t] = \dot{\vec{x}}[t] \cdot \dot{\vec{x}}[t].$ Notice that the second term in the series is just the classical kinetic energy. Suppose the particle has electrical charge $q\!$ coulombs and is in an electromagnetic field with scalar potential $\phi \!$ volts (a volt is a joule per coulomb) and vector potential $\vec{A}$ volt seconds per meter. A scalar Potential is a fundamental concept in Vector analysis and Physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion In Vector calculus, a vector potential is a Vector field whose curl is a given vector field The Lagrangian of a special relativistic test particle in an electromagnetic field is:

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

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

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

which is

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

which is the equation for the Lorentz force where

$\vec{E}[\vec{x},t] = - \nabla\phi [\vec{x},t] - \partial_t{\vec{A}} [\vec{x},t]$

$\vec{B}[\vec{x},t] = \nabla \times \vec{A} [\vec{x},t]$

General relativistic test particle

In general relativity, the first term generalizes (includes) both the classical kinetic energy and interaction with the Newtonian gravitational potential. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 It becomes:

$- m c^2 \frac{d \tau[t]}{d t}$

$= - m c \sqrt {- g_{\alpha\beta}[x[t]] \frac{d x^{\alpha}[t]}{d t} \frac{d x^{\beta}[t]}{d t}} .$

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

$L[t] = - m c \sqrt {- g_{\alpha\beta}[x[t]] \frac{d x^{\alpha}[t]}{d t} \frac{d x^{\beta}[t]}{d t}} + q \frac{d x^{\gamma}[t]}{d t} A_{\gamma}[x[t]] .$

If the four space-time coordinates $x^{\alpha}\!$ are given in arbitrary units (i. e. unit-less), then $g_{\alpha\beta}\!$ meters squared is the rank 2 symmetric metric tensor which is also the gravitational potential. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space Also, $A_{\gamma}\!$ 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

$c\, \sqrt {1 - \frac{v^2 [t]}{c^2}} = \sqrt {- ( - c^2 + v^2 [t])} .$

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:

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

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

$\mathcal{S} [\varphi_i] = \int{\mathcal{L} [\varphi_i (x)]\, \mathrm{d}^4x}$

The Lagrangian is then the spatial integral of the Lagrangian density. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS However, $\mathcal{L}$ 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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, the principle of locality is that distant objects cannot have direct influence on one another an object is influenced directly only by its immediate surroundings In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally In Physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable $\vec x$ is incorporated into the index $i$ or the parameters $s$ in $\varphi_i(s)$ . Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of $\mathcal{L}$ , and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Feynman diagrams. In quantum field theory (QFT the forces between particles are mediated by other particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described

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 $[\vec{x},t]$ . For example, in the case of Newtonian gravity, the Lagrangian density integrated over space-time gives you an equation which, if solved, would yield $\zeta [\vec{x},t]$ . This $\zeta [\vec{x},t]$ , 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 $\mathcal{L}$ joules per cubic meter. The interaction term $m \zeta \!$ is replaced by a term involving a continuous mass density $\mu \!$ 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:

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

where $G \!$ meters cubed per kilogram second squared is the gravitational constant. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass Variation of the integral with respect to $\zeta \!$ gives:

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

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

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

and thus

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

which yields Gauss's law for gravity. In Physics, Gauss' law for gravity, also known as Gauss' flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal

Electromagnetism in special relativity

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

$\mathcal{L}[\vec{x},t] = - \rho [\vec{x},t] \phi [\vec{x},t] + \vec{j} [\vec{x},t] \cdot \vec{A} [\vec{x},t] + {\epsilon_0 \over 2} {E}^2 [\vec{x},t] - {1 \over {2 \mu_0}} {B}^2 [\vec{x},t] .$

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

$0 = - \rho [\vec{x},t] + \epsilon_0 \nabla \cdot \vec{E} [\vec{x},t]$

which yields Gauss' law.

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

$0 = \vec{j} [\vec{x},t] + \epsilon_0 \partial_t \vec{E} [\vec{x},t] - {1 \over \mu_0} \nabla \times \vec{B} [\vec{x},t]$

which yields Ampère's law.

Electromagnetism in general relativity

For the Lagrangian of gravity in general relativity, see Einstein-Hilbert action. The Einstein-Hilbert action in General relativity is the action that yields the Einstein's field equations when varied to obtain Equations The Lagrangian of the electromagnetic field is:

$\mathcal{L}[x] = + J^{\gamma}[x] A_{\gamma}[x] - {1 \over 4\mu_0} F_{\mu \nu}[x] F_{\alpha \beta}[x] g^{\mu\alpha}[x] g^{\nu\beta}[x] \sqrt{\frac{-1}{c^2} \mathrm{det} [g[x]]}.$

If the four space-time coordinates $x^{\alpha}\!$ are given in arbitrary units, then: $\mathcal{L}$ joule seconds is the Lagrangian, a scalar density; $J^{\gamma}\!$ coulombs is the current, a vector density; and $F_{\mu \nu}\!$ volt seconds is the electromagnetic tensor, a covariant antisymmetric tensor of rank two. The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is Notice that the determinant under the square root sign is applied to the matrix of components of the covariant metric tensor $g_{\alpha\beta}\!$ , and $g^{\alpha\beta}\!$ is its inverse. Notice that the units of the Lagrangian changed because we are integrating over $x^0, x^1, x^2, x^3\!$ which are unit-less rather than over $t, x, y, z \!$ 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 $\frac{-1}{c^2}$ 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 $- c^2 \!$ in special relativity).

Lagrangians in quantum field theory

Dirac Lagrangian

The Lagrangian density for a Dirac field is:

$\mathcal{L} = \bar \psi (i \hbar c \not\!D - mc^2) \psi$

where $\psi\!$ is a Dirac spinor, $\bar \psi = \psi^\dagger \gamma^0$ is its Dirac adjoint, $D\!$ is the gauge covariant derivative, and $\not\!D$ is Feynman notation for $\gamma^\sigma D_\sigma\!$ . In Quantum field theory, a fermionic field is a Quantum field whose quanta are Fermions that is they obey Fermi-Dirac statistics. In Quantum field theory, Dirac spinor is the Bispinor in the plane-wave solution \psi = \omega_\vec{p}\e^{-ipx} \ of the In Quantum field theory, the Dirac adjoint \bar\psi of a Dirac spinor \ \psi is defined to be the dual Spinor \ The gauge covariant derivative (ˌgeɪdʒ koʊˌvɛəriənt dɪˈrɪvətɪv is like a generalization of the Covariant derivative used in General relativity. In the study of Dirac fields in Quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the '''Dirac'''

Quantum electrodynamic Lagrangian

The Lagrangian density for QED is:

$\mathcal{L}_{\mathrm{QED}} = \bar \psi (i \hbar c\not\!D - mc^2) \psi - {1 \over 4\mu_0} F_{\mu \nu} F^{\mu \nu}$

where $F^{\mu \nu}\!$ is the electromagnetic tensor

Quantum chromodynamic Lagrangian

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

$\mathcal{L}_{\mathrm{QCD}} = \sum_n \bar \psi_n (i \hbar c\not\!D - m_n c^2) \psi_n - {1\over 4} G^\alpha {}_{\mu\nu} G_\alpha {}^{\mu\nu}$

where $D\!$ is the QCD gauge covariant derivative, and $G^\alpha {}_{\mu\nu}\!$ is the gluon field strength tensor. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the The gauge covariant derivative (ˌgeɪdʒ koʊˌvɛəriənt dɪˈrɪvətɪv is like a generalization of the Covariant derivative used in General relativity. In Physics, the field strength of a field is the magnitude of its vector value

Mathematical formalism

Suppose we have an n-dimensional manifold, $M$ , and a target manifold, $T$ . A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Let $\mathcal{C}$ be the configuration space of smooth functions from $M$ to $T$ . In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

Examples

In classical mechanics, in the Hamiltonian formalism, $M$ is the one-dimensional manifold $\mathbb{R}$ , representing time and the target space is the cotangent bundle of space of generalized positions. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another
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. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS For example, if there are m real-valued scalar fields, $φ 1,. In Mathematics, the real numbers may be described informally in several different ways In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point . . ,φ m$ , then the target manifold is $\mathbb{R}^m$ . If the field is a real vector field, then the target manifold is isomorphic to $\mathbb{R}^n$ . In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective There is actually a much more elegant way using tangent bundles over $M$ , but we will just stick to this version. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the

Mathematical development

Consider a functional, $\mathcal{S}:\mathcal{C}\rightarrow \mathbb{R}$ , called the action. For functional analysis as used in psychology see the Functional analysis (psychology article In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation Physical reasons determine that it is a mapping to $\mathbb{R}$ , not $\mathbb{C}$ .

In order for the action to be local, we need additional restrictions on the action. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation If $\varphi\in\mathcal{C}$ , we assume $\mathcal{S}[\varphi]$ is the integral over $M$ of a function of $φ$ , its derivatives and the position called the Lagrangian, $\mathcal{L}(\varphi,\partial\varphi,\partial\partial\varphi, ...,x)$ . The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In other words,

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

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 $φ$ at the boundary if $M$ is compact or some limit on $φ$ as x approaches $\infty$ (this will help in doing integration by parts), the subspace of $\mathcal{C}$ 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. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints For a different notion of boundary related to Manifolds see that article In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. In Physics, particularly in Quantum field theory, configurations of a physical system that satisfy classical Equations of motion are called on shell

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

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

The left hand side is the functional derivative of the action with respect to $φ$ . In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation

References

^ Torby, Bruce (1984). In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. You may also be looking for Functional integration (neurobiology or Functional integration (sociology. This article discusses the history of the principle of least action Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In recent years there has been renewed interest in covariant classical field theory. For the pseudoscientific "scalar field theory" see " Scalar field theory (pseudoscience " In Theoretical physics, In Fluid dynamics and finite-deformation plasticity the Lagrangian reference frame is a way of looking at fluid motion where the observer follows individual fluid "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4.

Christoph Schiller (2005), Global descriptions of motion: the simplicity of complexity, Motion Mountain
David Tong Classical Dynamics (Cambridge lecture notes)

Dictionary

-adjective

of or relating to Joseph Louis Lagrange

-noun

(mathematics) the Lagrangian function

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