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Encyclopedia > Lagrangian

A Lagrangian mathcal{L}[varphi_i] of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables  varphi_i(s) and concisely describes the equations of motion of the system. The equations of motion are obtained by means of an action principle, written as A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... Joseph-Louis Lagrange Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 – April 10, 1813; b. ... Partial plot of a function f. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... In physics, equations of motion are equations that describe the behavior of a system (e. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

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

where the action is a functional mathcal{S}[varphi_i] = int{mathcal{L}[varphi_i(s)]{},mathrm{d}^ns}, In mathematics, the term functional is applied to certain functions. ...


{}{}{}{} s_alpha denoting the set of parameters of the system. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... The factual accuracy of this article is disputed. ...


The equations of motion obtained by means of the functional derivative are identical to the usual Euler-Lagrange equations. 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. In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... Plateaus problem is to show the existence of a minimal surface with a given boundary. ...


The Lagrange formulation of mechanics is important not just for its broad applications (see below) but also for its role in advancing deep understanding of physics. Although Lagrange sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation is now recognized to be deeply tied to quantum mechanics: 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. 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; and 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. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Fig. ... Phase is an overloaded word used for: instantaneous phase: the current position in the cycle of something that changes cyclically phase shift: a constant difference/offset between two instantaneous phases, particularly when one is a standard reference Waves are amplitudes that change cyclically, often modeled as sinusoidal functions of time... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between continuous symmetries and conservation laws. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between continuous symmetries and conservation laws. ... 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 wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if...

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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. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Kinetic energy is the energy that a body possesses as a result of its motion. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


Suppose we have a three dimensional space and the Lagrangian :For other senses of this word, see dimension (disambiguation). ...

begin{matrix}frac{1}{2}end{matrix} mdot{vec{x}}^2-V(vec{x}).

Then, the Euler-Lagrange equation is mddot{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. In vector calculus, del is a vector differential operator represented by the nabla symbol, ∇. In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del can be defined as or alternatively, where is the standard basis in R3. ...


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 vec{F}=- nabla V(x); then the resulting equation is vec{F}=mddot{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.


Suppose we have a three-dimensional space in spherical coordinates, r, θ, φ with the Lagrangian This article describes some of the common coordinate systems that appear in elementary mathematics. ...

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

Then the Euler-Lagrange equations are:

mddot{r}-mr(dot{theta}^2+sin^2thetadot{varphi}^2)+V' =0,
frac{mathrm{d}}{mathrm{d}t}(mr^2dot{theta}) -mr^2sinthetacosthetadot{varphi}^2=0,
frac{mathrm{d}}{mathrm{d}t}(mr^2sin^2thetadot{varphi})=0.

Here the set of parameters  s_i is just the time  t, and the dynamical variables  phi_i(s) are the trajectories vec x(t) of the particle.

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Lagrangians and Lagrangian densities in field theory

In field theory, occasionally a distinction is made between the Lagrangian L, of which the action is the time integral There are two types of field theory in physics: Classical field theory, the theory and dynamics of classical fields. ...

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

and the Lagrangian density mathcal{L}, which one integrates over all space-time to get the action: In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ...

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

The Lagrangian is then the spatial integral of the Lagrangian density. 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. 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. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... 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. ... ‹The template below has been proposed for deletion. ... In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...

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Electromagnetic Lagrangian

Generally, in Lagrangian mechanics, the Lagrangian is equal to

L = TV

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 It has been suggested that this article or section be merged with Potential. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ...

T = {1 over 2} m mathbf{v} cdot mathbf{v}

and the particle's potential energy is

V = qphi - {q over c} mathbf{v} cdot mathbf{A}

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

L = {1 over 2} m mathbf{v} cdot mathbf{v} - qphi + {q over c} mathbf{v} cdot mathbf{A} .
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Lagrangians in Quantum Field Theory

Note that in the following, ħ = c = 1.

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Quantum Electrodynamic Lagrangian

The Lagrangian density for QED is Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ...

mathcal{L}_{QED} = bar psi (i not !, D - m) psi - {1 over 4} F_{mu nu} F^{mu nu}

where ψ is a spinor, bar psi = psi^dagger gamma^0 is its Dirac adjoint, Fμν is the electromagnetic tensor, D is the gauge covariant derivative, and not !, D is Feynman notation for γσDσ. To meet Wikipedias quality standards, this article or section may require cleanup. ... The Dirac adjoint of a Dirac spinor is defined to be the dual spinor , where denotes the time-like Dirac matrix. ... In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ... The gauge covariant derivative (pronounced: [geɪdÊ’ koÊŠvÉ›riÉ™nt dɪrɪvÉ™tɪv]) is like a generalization of the covariant derivative used in general relativity. ... In the study of Dirac fields in quantum field theory, Feynman invented the convenient Feynman slash notation. ...

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Dirac Lagrangian

The Lagrangian density for a Dirac field is In physics, a Dirac field is a fermionic field (usually a quantized field, as usual in quantum field theory) associated with spin 1/2 fermions such as the electron or muon. ...

mathcal{L} = bar psi (i not ! ; partial - m) psi.
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Quantum Chromodynamic Lagrangian

The Lagrangian density for quantum chromodynamics is [1] [2] [3] Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ...

mathcal{L}_{QCD} = -{1over 4} F^alpha {}_{munu} F_alpha {}^{munu} + sum_n bar psi_n (inot!, D - m_n) psi_n

where D is the QCD gauge covariant derivative, and Fαμν is the gluon field strength tensor. The gauge covariant derivative (pronounced: [geɪdÊ’ koÊŠvÉ›riÉ™nt dɪrɪvÉ™tɪv]) is like a generalization of the covariant derivative used in general relativity. ... ʐIn physics, the field strength of a field is the magnitude of its vector (spatial) value. ...

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Mathematical formalism

Suppose we have an n-dimensional manifold, M and a target manifold T. Let mathcal{C} be the configuration space of smooth functions from M to T. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...


Before we go on, let's give some 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.
  • 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, φ1,...,φm, 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. 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, mathcal{S}:mathcal{C}rightarrow mathbb{R}, called the action. Note that it is a mapping to mathbb{R}, not mathbb{C}; this has to do with physical reasons. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... Space has been an interest for philosophers and scientists for much of human history. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ...


In order for the action to be local, we need additional restrictions on the action. If varphiinmathcal{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,partialvarphi,partialpartialvarphi, ...,x). In other words, In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

forallvarphiinmathcal{C},   mathcal{S}[varphi]equivint_M mathrm{d}^nx mathcal{L} big( varphi(x),partialvarphi(x),partialpartialvarphi(x), ...,x big).

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 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, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... The word Boundary has a variety of meanings. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... Screenshot (from SSCX Star Warzone). ... In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...


The solution is given by the Euler-Lagrange equations (thanks to the boundary conditions), In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

frac{deltamathcal{S}}{deltavarphi}=-partial_mu left(frac{partialmathcal{L}}{partial(partial_muvarphi)}right)+ frac{partialmathcal{L}}{partialvarphi}=0.

Incidentally, the left hand side is the functional derivative of the action with respect to φ. In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

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See also

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In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... In physics, functional integration is integration over certain infinite-dimensional spaces. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... A contour plot of the effective potential of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between continuous symmetries and conservation laws. ... In recent years, there has been renewed interest in the covariant formalism of classical field theory. ... For the quantum mechanical scalar field theory which is a field theory of spinless particles, see Scalar field (physics) Scalar field theory (SWT) is a set of fringe theories in a model which posits that there is a basic mechanism that produces the electric field and the magnetic field. ...

References


  Results from FactBites:
 
Lagrangian point - Wikipedia, the free encyclopedia (2070 words)
The Lagrangian points (IPA: [lə.'grɒn.dʒi.ən] or [la.'grã.ʒi.ən]; also Lagrange point, L-point, or libration point), are the five positions in interplanetary space where a small object affected only by gravity can theoretically be stationary relative to two larger objects (such as a satellite with respect to the Earth and Moon).
The Lagrangian points constructed at each point in time as in the circular case form stationary elliptical orbits which are similar to the orbits of the massive bodies.
This fact is independent of the circularity of the orbits, and it implies that the elliptical orbits traced by the Lagrangian points are solutions of the equation of motion of the third body.
Lagrangian - definition of Lagrangian in Encyclopedia (755 words)
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 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.
  More results at FactBites »

 
 

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