Lagrangian

In physics, a Lagrangian is a scalar function from which the time evolution, conservation laws and other important properties of a dynamical system can be obtained. In fact, in modern physics the Lagrangian is considered the most fundamental operator that describes a physical system.

The term is named after the Italian-French astronomer and mathematician Joseph Louis de Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Lagrange, known as Lagrangian mechanics, in 1788. This reformulation was necessary in order to explore mechanics in alternative Cartesian coordinate systems, such as polar coordinates, cylindrical and spherical, for which Newton's mechanics was not suitable.

The Lagrangian formalism allows to achieve both Newton's laws and Maxwell's equations, which can be deduced as the Euler-Lagrange equations of a classical Lagrangian. Likewise, the form of the Lagrangian determines the basic properties of the system in quantum field theory.

Introduction

Lagrange mechanics have its origin as a formulation of classic mechanics. It is an alternative formulation to hamiltonian mechanics. The lagrangian of a particle system is defined as the difference between its kinetic energy ${displaystyle E_{c}}$ and its potential energy ${displaystyle E_{p}}$:

${displaystyle {mathcal {L}}=E_{c}-E_{p}=T-V}$

Historically, the Lagrangian formalism arose within classical mechanics for systems with a finite number of degrees of freedom. This Lagrangian allowed to write the equations of movement of a totally general system that had restrictions of movement or was non-inertial in a very simple way.

Later the concept was generalized to systems with a non-finite number of degrees of freedom such as continuous media or physical fields. Later the concept could be generalized to quantum mechanics as well, particularly in quantum field theory.

Mathematical formalism

The Lagrangian is a scalar function defined on a certain space of possible states of the system. In a system with a finite number of degrees of freedom, the physical action is defined as a line integral over the paths of motion (1), while in a continuous system or system with a non-finite number of degrees of freedom the action is defined as a multiple integral over a 4-volume (2):

(1)${displaystyle S[mathbf {q}C]=int _{C}L dt=int _{C}L(mathbf {q} (t),{dot {mathbf {q}}}}}(t);t$,

(2)${cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFF}{cHFFFF}{cHFF}{cHFFFF}{cH00}{cHFFFFFFFFFFFF}{cHFF}{cH00}{cH00}{cHFFFFFFFFFF}{cH00}{cHFFFFFFFFFFFFFFFFFF}{cH00}{cHFFFFFF}{cH00}{cHFFFFFFFFFFFFFFFF}{cH00}{cHFFFFFFFF}{cHFFFF}{cHFF}{cH00}{cH00}{cHFF}{cHFF}{cHFFFFFFFF}{cHFF dV dt}$,

The equations of motion can be obtained from the Lagrangian form, since the trajectories of the real motion of the system are such that the previous integrals take the minimum possible value. Knowing the form of the Lagrangian in a coordinate system, the particularized Euler-Lagrange equations for the concrete Lagrangian are precisely the equations of motion.

Finite number of degrees of freedom

In the case of a system with a finite number of degrees of freedom, the state space is a finite-dimensional differential variety constructed as the tangent fiberdo TQ of a variety n- dimensional and lagrangian is a function of shape ${displaystyle {tilde {L}}:TQtimes mathbb {R} to mathbb {R} }$.

One function lagrangiana is the expression of lagrangian in a concrete coordinate system, it is related to the kinetic energy and the potential energy of the system. For example for a classic particle that moves in conventional euclid space ${displaystyle TQ=mathbb {R} ^{3}times mathbb {R} ^{3}}$ under a field of conservation forces given by function V(x,y,z), the usual lagrangian using Cartesian coordinates can be represented by the lagrangiana function:

(3)${displaystyle L(x,y,z,v_{x},v_{y},v_{z}}}={frac {m}{2}}(v_{x}{2}+v_{y}{2}}{2}+v_{z^}{2}{2})-V(x,y,z)}}}$,

The Lagrangian function is usually written in terms of any kind of generalized coordinates:

(4)${displaystyle (q_{1},...q_{n};{dot {q}_{1},...,{dot {q}}_{n};t)mapsto L(mathbf {q}{dot {mathbf {q}}}},t)in mathbb {R}}}$,

As for the intrinsic lagrangian, it can be written in terms of any lagrangiana function, if the general coordinates used match a local letter ${displaystyle (U,phi _{q});}$ Intrinsic lagrangian can be written as a function that satisfies:

(5)${displaystyle {tilde {L}}(p,v,t) meant_{U}=L(phi _{q}(p),(phi _{q})_{*}v,t)}}$,

Where ${displaystyle (phi _{p})_{*}=Dphi _{q};$ It's him. pushforward o differential of homeomorphism that defines the local letter. The lagrangian defined in local coordinates and defined directly over the state space are related by:

(6)${displaystyle {begin{matrix}{tilde {L}}: alienTQtimes mathbb {R} 'to 'mathbb {R} \mathbb {R}{mathbb _{mathbb}{{b}{mcHFFFF}{R}{cHFFFF}{R}{cHFF}{cHFFFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFFFF}{cHFFFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFFFF}{cHFF}{cHFFFF}{cHFFFF}{cHFF}{cHFFFF}{cHFF}{cHFFFF}{cHFF}{cHFFFF}{cHFFFF}{cHFFFF$,

The trajectories that give the time evolution of a system are differentiable curves over the configuration manifold, which can be calculated from the Euler-Lagrange equations:

(7)${displaystyle {d over dt}left({partial l over partial {dot {q}}_{i}}}right)-{partial l over partial q_{i}}}=0}$,

Infinite number of degrees of freedom

In systems with an infinite number of degrees of freedom, that is, in systems of continuous media mechanics or the classical field theory, they require a more complex description in terms of lagrangian density. In addition, the configuration space can be substantially more complicated than in the case of finite-grade systems. In fact the configuration space ${displaystyle {mathcal {F}}}$ must be a variety of infinite dimension formed by all possible variations that can have a field on a 4-varise or time-space M, and in fact in this case "trayectories" are not one-dimensional varieties but 4-varies. There is a rigorous and elegant way to build such a variety of configuration considering tangent fibers on MBut that kind of formalism will not be treated here.

La density lagrangiana is a function of the type ${displaystyle {tilde {mathcal {L}}}{mathcal {F}times Mto mathbb {R} }$ (even in the theories in which the field can take complex, there are physical reasons to continue demanding that the grenagian be a real function). Moreover, the fact that the theory is local, that is, that it meets certain requirements of physical causality, the lagrangiana density should not contain derivatives superior to the second order, otherwise certain strange violations of causality occur.

If we now consider an observer ${displaystyle O;}$ concrete we can derive, as we did for the case with a finite number of degrees of freedom, an expression of the lagrangiana density at coordinates, being able to write the action as:

(8)${displaystyle S[phi _{r}^{alpha }]equiv int _{M}{mathcal {L}}{O}{Big (}{big (}{phi _{r^}{alpha }{x),{partial phi _{r}{alpha }{big}{big},x{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{b }{$,

Given certain contour conditions on the edge of a region ${displaystyle Vsubset M}$, then the equations of the movement are given by the equations of Euler-Lagrange:

(9)${displaystyle {frac {partial }{partial x{mu }}}{left({frac {partial {mathcal {L}{O}}}{partial}{partial}{mu }{phi _{r}{alphah}}}}{right}{frac {partial {math}{$,

Incidentally, the left side is the functional derivative of the action with respect to ${displaystyle phi _{r}^{alpha }}}$.

Lagrangian in classical mechanics

In classical mechanics the Lagrangian function of a conservative system, denoted by L, is simply the difference between its kinetic energy, T, and its potential energy, V. The appropriate domain of the Lagrangian is a phase space, and must obey the Euler-Lagrange equations. The concept was originally used in a reformulation of classical mechanics known as Lagrangian mechanics. In generalized coordinates this Lagrangian usually takes the form:

(10)${displaystyle L(q_{i},{dot {q}}_{i})={frac {m}{2}{i}{i,j}g_{ij} {dot {q}}}_{i}{dot {q}}}{j}-V(q_{1},ldotsq_{n};t}}}}}}}}}}}}$,

Where ${displaystyle g_{ij}(mathbf {q})}$ is the metric tensor of the euclid space expressed in the coorrespondient generalized coordinates, which only depends on the very coordinates of the speeds ${displaystyle {dot {q}}_{i}}$.

Lagrangian of a classical particle in rectangular coordinates

If we assume, as usual, that a classical system is formed by particles that move in a three-dimensional Euclidean space, then the metric tensor adopts the diagonal form and the Lagrangian is given by:

(11)${displaystyle l(x,y,z,{dot {x}},{dot {y},{dot {z}})={frac {m}{2}}}{dot {mathbf {r}}}}}}{^{2}-V(mathbf {r}}}}}}}}}}}$,

And so the system turns out to be inertial, and the Euler-Lagrange equations simply reduce to Newton's laws:

(12)${displaystyle m{ddot {mathbf {r}}}}}+{boldsymbol {nabla }V=0quad Rightarrow quad m{frac {d^{2}mathbf {r}}{ }{dt^{2}}}=-{fboldsymbol {b }$,

Lagrangian of a particle in spherical coordinates

In spherical coordinates (r,θ,φ) the same Lagrangian function above, particularized to the case of a potential with spherical symmetry that only depends on the radial coordinate, is expressed as:

(13)${displaystyle {frac {m}{2}}}({dot {r}}}{2}{2} +r^{2}{2}{dot {dot {theta }}}}{2}{2}sin ^{2}theta {dot {dot {phi }}}{2}{tilde {V}}}}}$,

Using the Euler-Lagrange equations, the same calculation from the previous section leads us to the equations of motion on a non-inertial frame:

${displaystyle m{frac {d^{2}r}{dt^{2}}}}-mr({dot {theta }}}{2}+sin ^{2}theta {dot {phi }}}}{^{2})=-{frac {d{tilde {V}{(r}}{d}{r}}}}{$

${displaystyle {frac {d}{dt}}(mr^{2}{dot {theta}}})-mr^{2}sin theta cos theta {dot {phi }}{{2}{2}=0}$

${displaystyle {frac {d}{dt}}(mr^{2}sin ^{2}theta {dot {dot {phi}}}})=0}$

Among the additional terms that have now appeared are the Coriolis force and the centripetal force, thus the Lagrangian formalism automatically predicts that any non-Cartesian frame of reference entails the appearance of non-inertial forces.

Lagrangian in relativistic mechanics

In relativistic mechanics the action of a particle is obtained by calculating along the world line of a particle, specifically a material particle of mass m moves along a geodesic. The action integral along an L curve is given in curvilinear coordinates by:

${displaystyle S_{m}=-int _{L}mc ds=-int _{tau _{1}{1}{tau _{2}}mc {sqrt {g_{mu}}{frac}{d {dx{mu }}{dtau}}{frac {dx{$,

If the integrand of the previous integral is introduced into the Euler-Lagrange equations, the geodesic equations are obtained:

${displaystyle {frac {d^{2}x^{mu }{dtau ^{2}}}}}}{sigmanu }Gamma _{sigma nu ^{mu}{mu}{frac {dx^{sigma }{dtau}{$,

Lagrangian in continuum mechanics

In continuum mechanics, the magnitudes that evolve over time and define the physical state of the system are related to displacement vector fields. In solid mechanics and elasticity, the Lagrangian depends on the displacement field and its derivatives, while in fluid mechanics the Lagrangian depends on the velocity field and its derivatives (ultimately related to the displacements of the particles).

Lagrangian of an elastic solid

An elastic problem is defined by the geometry of the body before being deformed, the external forces, which give rise to the term "potential" of the Lagrangian and the components of the tensor of elastic constants in fact the Lagrangian density can be written, using the Einstein summation convention, as:

${displaystyle {mathcal {L}}(u_{i},partial _{ju_{i})={begin{matrix}{frac {1}{2}}}}}}{end{matrix}}}C_{ijkl}varepsilon _{ij}varepsilon _{kl-b_{i}{i}{i}{i}{i}}}$

Where:
${displaystyle C_{ijkl};}$, are the components of the matrix or tensor of elastic constants.
${displaystyle varepsilon _{ij}={begin{matrix}{frac {1}{2}}{matrix}}(partial _{j}u_{i}+partial _{i}u_{i})}}}$, they are the components of the strainer.
${displaystyle u_{i};}$ are the components of the displacement vector, which is defined for each point of the body.
${displaystyle b_{i};}$ are forces per unit of mass, such as weight or centrifugal forces, which act on each point of the body.

Substituting the previous Lagrangian density in the Euler-Lagrange equations and applying the symmetry conditions of the elastic constants tensor below, we arrive at:

${bHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFF}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cHFFFFFFFFFFFFFFFF}{cHFFFF}{cHFFFF}{cHFFFF}{cHFFFFFFFF}{cHFFFFFF}{cHFFFF$

Finally, the Euler-Lagrange equations result in the equilibrium equations of an elastic solid:

${displaystyle {frac {partial }{partial x_{j}}}}}{left({frac {partial {mathcal {L}}{partial}{j}{j}{i}{i}}}{right)}{frac {partial {mathcal} {L}{partial}{$

For dynamic problems, it is enough to extend the previous Lagrangian with the derivatives of the displacement:

${displaystyle {tilde {mathcal {L}}}(u_{i},partial _{j}u_{i},partial _{t}u_{i})={mathcal {L}(u}{i}{i},partial _{j}u_{i}$

Lagrangian of a fluid

Lagrangian in classical field theory

A physical field is any type of magnitude that presents both spatial and temporal variation. This type of physical entities requires treatment by means of Lagrangian densities, since they are not representable as systems with a finite number of degrees of freedom. In addition, its rigorous treatment generally requires the use of relativistic mechanics to explain its propagation. The fields that classical field theory usually deals with:

• Electromagnetic field, which is the field associated with the interaction of charged particles, and which ultimately explains the properties of conventional matter, such as the properties of solids, liquids and gases, phenomena such as color, light, etc.
• Gravitational field, it is a relatively weak type of field compared to the electromagnetic field, but as its effect is cumulative, it is the only relevant to cosmic scale to explain the evolution of the universe.
• Quantum fields treated classically, which allow to formulate first approaches for free fields that are useful when it comes to the evolution of quantum fields with interaction.

Lagrangian of the electromagnetic field

The Lagrangian of the electromagnetic field is given by a scalar built from the electromagnetic field tensor:

${displaystyle S_{c,em}[F_{mu nu },Omega ]=-{frac {1}{16pi c}}int _{Omega }F_{mu nu }F^{mu nu }dOmega }$

In fact this Lagrangian can be rewritten in terms of the electric and magnetic fields to give (in cgs units):

${displaystyle S_{c,em}[mathbf {E}mathbf {Bomega]=-{frac {1}{8pi }{int _{mathbb} {R}}{x1}{x1}{Big}{bc(E}{2}{bf {B}{2}{c}{2}{bc}{c}{c}}{b$

Introducing this Lagrangian into the Euler-Lagrange equations, the result is Maxwell's equations and applying a generalized Legrendre transformation, the expression for electromagnetic energy is obtained:

${displaystyle E_{em}={frac {1}{8pi }}}int _{mathbb {R} ^{3}}left(mathbf {E} ^{2}+mathbf {B} ^{2}right)dV}$

Lagrangian of the gravitational field

In general relativity the gravitational field is seen as a manifestation of the curved geometry of the time space, therefore the lagrangiana formulation of the gravitational field relativly treated must involve some climbing related to the metric tensor and its first derivatives (equivalently the symbols of Christoffel ${displaystyle Gamma _{ij}^{k}}}$or with the curvature tensor. It can be proved that it is not possible to find any climbing that involves only the components of the metric tensor and the symbols of Christoffel, since by some transformation of coordinates the latter can be cancelled (which is precisely the content of the so-called equivalence principle).

It is interesting that the scalar curvature R, gives us an adequate form of action: although it contains second derivatives of the metric tensor, the variation of its action integral on a region can end up being expressed in terms of only first derivatives. In fact the common form of the action integral for the gravitational field most commonly in the theory of general relativity is:

${displaystyle S_{c,g}[g_{mu nu },omega ]=-{frac {c^{3}{16pi G}}{int _{omega }R {sqrt {-g}}domega }$

Some metric theories of gravitation such as the relativistic theory of gravitation use a slightly more complicated Lagrangian that includes terms associated with the mass of the graviton:

${cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFFFF}{cHFFFFFF}{cHFFFF}{cHFFFF}{cHFFFFFF}{cHFF}{cHFFFFFFFFFF}{cH}{cHFFFF}{cHFFFFFFFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFF}{cHFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH}{cHFF$

Where:

${displaystyle R;}$It's the scalar curvature of space-time.

${displaystyle G,c;}$, are the constant of gravitation and the speed of light.

${displaystyle g_{mu nu }, gamma _{mu nu };}$ are the components of the actual metric (pseudo)riemannian and the underlying Minkowski space.

${displaystyle {sqrt {-g}},{sqrt {-gamma}}}}}$, are calculated from the determinants of the effective metric and minkowskiana, calculated at the same coordinates.

${displaystyle m_{g};}$It's the mass of the graviton.

Lagrangian in quantum field theory

In quantum mechanics the Lagrangian is a functional defined on the Hilbert space of the physical system under consideration. In quantum field theory, fields are generally overdefined distributions over space-time whose values are operators.

In quantum field theory the Lagrangian of interaction determines the form of the exponent of the exponential of the propagator. As usual, said exponential is computed as a power series in which each term is associated with a Feynman diagram.

Lagrangian for Dirac's equation

Dirac's equation describes fermionic particles with spin 1/2, in fact the equation describes said particles as a fermionic field. That fermionic field equation that represents the particles can be derived from a Lagrangian density. Specifically, for a free fermionic field without interaction, the Lagrangian density from which the Dirac equation can be derived is given by:

${displaystyle {mathcal {L}}={bar {psi }}(ihbar cgamma ^{mu }partial _{mu }-mc^{2})psi }$

Where:

${displaystyle psi ,}$ is a Dirac thorn representing the fermionic field of particles.

${displaystyle {bar {psi}}=psi ^{dagger }gamma ^{0}}$ is Dirac's attachment to the previous thorn.

${displaystyle partial _{mu },}$ is the partial derivative regarding the coordinates.

Lagrangian for QED

The Lagrangian of quantum electrodynamics or QED includes a commutative gauge field that represents the quantum analogue of electromagnetic potential in interaction with charged particles of the fermionic type (electrons, quarks,...). The usual starting Lagrangian for QED is usually taken as:

${cHFFFFFF}{cHFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFF}{cHFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFF}{cHFF}{cHFFFF}{cHFF}{cHFF}{cHFFFF}{cHFFFFFFFF}{cHFF}{cHFF}{cHFFFF}{cHFF}{cHFF}{cHFF}{cHFFFFFFFFFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFF}{cHFFFF}{cHFF}{cHFF}{cHFF}{cHFFFF}{cH}{cHFF}{cHFF}{cH}{cH$

Where:

${displaystyle psi ,}$ is the ferminonic field that represents the particles with electric charge.

${displaystyle {bar {psi}}}$ It's Dirac's attached field.

${displaystyle gamma ^{mu },}$, they are the matrices of Dirac who covariately intervene in the equation of Dirac for the feminiums.

${displaystyle e,}$It's the electrical load of the particle.

${displaystyle F_{mu nu },}$It's the electromagnetic field tensor.

${displaystyle D_{mu }=partial _{mu }+ieA_{mu }}}$, is the covariant derivative associated with the field.

Lagrangian of QCD

Quantum chromodynamics or QCD that describes the interaction between quarks and the gluon field can be described by the following Euclidean action, with Lagrangian given by:

$## #################################################################################################################################################$

Where:

${displaystyle psi({bar {psi }}})}$, Dirac's thorn depicting the fermionic fields describing the quarks (and his deputy of Dirac).

${displaystyle gamma _{mu },}$ represents Dirac's matrices.

${displaystyle D_{mu }=partial _{mu }-iglambda _{a}A^{a}$, is the covariant derivative associated with the gluonic gauge field.

${displaystyle G_{mu nu }^{a}}$, it is the gluonic field tensor, analogous to the electromagnetic field tensor.

${displaystyle lambda _{a},}$, are the matrices of Gell-Mann for their(3) that satisfy the switching rules ${displaystyle [lambda _{a},lambda _{b}]=if_{abc}lambda _{c},}$

${displaystyle u^{a},}$ is the thorn of Faddeev-Popov's "ghost" field.