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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, relativistic Lagrangian mechanics is
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
applied in the context of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
.


Introduction

The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: : L = -\frac - V(\mathbf, \dot, t) \,. Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
with
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
, the relativistic
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
corresponds to total energy in a similar manner but without including
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
. The form of the Lagrangian also makes the relativistic
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
functional proportional to the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
of the path in
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. In covariant form, the Lagrangian is taken to be: : \Lambda = g_\frac \frac , where ''σ'' is an ''
affine parameter In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
'' which parametrizes the spacetime curve.


Lagrangian formulation in special relativity

Lagrangian mechanics can be formulated in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
as follows. Consider one particle (''N'' particles are considered later).


Coordinate formulation

If a system is described by a Lagrangian ''L'', the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s : \frac\frac = \frac retain their form in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
, provided the Lagrangian generates equations of motion consistent with special relativity. Here is the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
of the particle as measured in some
lab frame In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime. The term is most often used in ...
where
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
are used for simplicity, and : \mathbf = \dot = \frac = \left(\frac,\frac,\frac\right) is the coordinate velocity, the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of position r with respect to
coordinate time In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial ...
''t''. (Throughout this article, overdots are with respect to coordinate time, not proper time). It is possible to transform the position coordinates to
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...
exactly as in non-relativistic mechanics, . Taking the
total differential In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by dy = f'(x)\,dx, where f'(x) is the derivative of with resp ...
of r obtains the transformation of velocity v to the generalized coordinates, generalized velocities, and coordinate time : \mathbf = \sum_^n \frac\dot_j +\frac \,, \quad \dot_j = \frac remains the same. However, the
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
of a moving particle is different from non-relativistic mechanics. It is instructive to look at the total
relativistic energy Relativity may refer to: Physics * Galilean relativity, Galileo's conception of relativity * Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity ...
of a free test particle. An observer in the lab frame defines events by coordinates r and
coordinate time In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial ...
''t'', and measures the particle to have coordinate velocity . By contrast, an observer moving with the particle will record a different time, this is the ''
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
'', ''τ''. Expanding in a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
, the first term is the particle's
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
, plus its non-relativistic
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
, followed by higher order relativistic corrections; : E = m_0 c^2 \frac = \frac = m_0 c^2 + m_0 \dot^2 (t) + m_0 \frac + \cdots \,, where ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in vacuum. The differentials in ''t'' and ''τ'' are related by the
Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
''γ'',The
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
squared is the
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
: c^2d\tau^2 = \eta_dx^\alpha dx^\beta = c^2dt^2 - d\mathbf^2 \,, which takes the same values in all
inertial frame In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...
s of reference. Here ''ηαβ'' are the components of the
Minkowski metric In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model ...
tensor, ''dxα'' = (''cdt'', ''d''r) = (''cdt'', ''dx'', ''dy'', ''dz'') are the components of the differential position
four-vector In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
, the
summation convention In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
over the covariant and contravariant spacetime indices ''α'' and ''β'' is used, each index takes the value 0 for timelike components, and 1, 2, 3 for spacelike components, and : d\mathbf^2 \equiv d\mathbf\cdot d\mathbf \equiv dx^2 + dy^2 + dz^2 is a shorthand for the square differential of the particle's position coordinates. Dividing by ''c''2''dt''2 allows the conversion to the lab coordinate time as follows, : \frac = \frac\eta_\frac\frac= 1-\frac\frac = \frac so that : d\tau = \frac\sqrt dt = \frac \,.
: dt=\gamma(\dot)d\tau \,, \quad \gamma(\dot) = \frac \,,\quad \dot = \frac \,, \quad \dot^2 (t) = \dot(t) \cdot \dot(t)\,. where · is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
. The relativistic kinetic energy for an uncharged particle of
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
''m''0 is : T = (\gamma(\dot) - 1)m_0c^2 and we may naïvely guess the relativistic Lagrangian for a particle to be this relativistic kinetic energy minus the potential energy. However, even for a free particle for which ''V'' = 0, this is wrong. Following the non-relativistic approach, we expect the derivative of this seemingly correct Lagrangian with respect to the velocity to be the relativistic momentum, which it is not. The definition of a generalized momentum can be retained, and the advantageous connection between
cyclic coordinate In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
s and
conserved quantities A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as a function of the dependent vari ...
will continue to apply. The momenta can be used to "reverse-engineer" the Lagrangian. For the case of the free massive particle, in Cartesian coordinates, the ''x'' component of relativistic momentum is : p_x = \frac = \gamma(\dot)m_0 \dot\,,\quad and similarly for the ''y'' and ''z'' components. Integrating this equation with respect to ''dx''/''dt'' gives : L = -\frac + X(\dot,\dot) \,, where ''X'' is an arbitrary function of ''dy''/''dt'' and ''dz''/''dt'' from the integration. Integrating ''p''''y'' and ''p''''z'' obtains similarly : L = -\frac + Y(\dot,\dot) \,,\quad L = -\frac + Z(\dot,\dot) \,, where ''Y'' and ''Z'' are arbitrary functions of their indicated variables. Since the functions ''X'', ''Y'', ''Z'' are arbitrary, without loss of generality we can conclude the common solution to these integrals, a possible Lagrangian that will correctly generate all the components of relativistic momentum, is : L = -\frac\,, where . Alternatively, since we wish to build a Lagrangian out of relativistically invariant quantities, take the action as proportional to the integral of the
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
in
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
, the length of the particle's
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
between proper times ''τ''1 and ''τ''2, : S = \varepsilon \int_^ d\tau = \varepsilon \int_^ \frac \,,\quad L = \frac = \varepsilon\sqrt\,, where ''ε'' is a constant to be found, and after converting the proper time of the particle to the coordinate time as measured in the lab frame, the integrand is the Lagrangian by definition. The momentum must be the relativistic momentum, : \mathbf = \frac = \left(\frac\right)\gamma(\dot)\dot = m_0 \gamma(\dot)\dot \,, which requires ''ε'' = −''m''0''c''2, in agreement with the previously obtained Lagrangian. Either way, the position vector r is absent from the Lagrangian and therefore cyclic, so the Euler–Lagrange equations are consistent with the constancy of relativistic momentum, : \frac\frac = \frac \quad \Rightarrow \quad \frac (m_0 \gamma(\dot)\dot ) = 0 \,, which must be the case for a free particle. Also, expanding the relativistic free particle Lagrangian in a power series to first order in , : L = -m_0 c^2 \left 1 + \frac\left(- \frac\right) + \cdots \right\approx -m_0 c^2 + \frac\dot^2 \,, in the non-relativistic limit when v is small, the higher order terms not shown are negligible, and the Lagrangian is the non-relativistic kinetic energy as it should be. The remaining term is the negative of the particle's rest energy, a constant term which can be ignored in the Lagrangian. For the case of an interacting particle subject to a potential ''V'', which may be non-conservative, it is possible for a number of interesting cases to simply subtract this potential from the free particle Lagrangian, : L = -\frac - V(\mathbf, \dot, t) \,. and the Euler–Lagrange equations lead to the relativistic version of
Newton's second law Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
. The derivative of relativistic momentum with respect to the time coordinate is equal to the force acting on the particle: : \mathbf = \frac\frac - \frac = \frac(m_0 \gamma(\dot)\dot)\,, assuming the potential ''V'' can generate the corresponding force F in this way. If the potential cannot obtain the force as shown, then the Lagrangian would need modification to obtain the correct equations of motion. Although this has been shown by taking Cartesian coordinates, it follows due to invariance of Euler Lagrange equations, that it is also satisfied in any arbitrary co-ordinate system as it physically corresponds to action minimization being independent of the co-ordinate system used to describe it. In a similar manner, several properties in Lagrangian mechanics are preserved whenever they are also independent of the specific form of the Lagrangian or the laws of motion governing the particles. For example, it is also true that if the Lagrangian is explicitly independent of time and the potential ''V''(r) independent of velocities, then the total relativistic energy : E = \frac\cdot\dot - L = \gamma(\dot)m_0c^2 + V(\mathbf) is conserved, although the identification is less obvious since the first term is the relativistic energy of the particle which includes the rest mass of the particle, not merely the relativistic kinetic energy. Also, the argument for homogeneous functions does not apply to relativistic Lagrangians. The extension to ''N'' particles is straightforward, the relativistic Lagrangian is just a sum of the "free particle" terms, minus the potential energy of their interaction; : L = - c^2 \sum_^N \frac - V(\mathbf_1, \mathbf_2, \ldots, \dot_1,\dot_2,\ldots, t) \,, where all the positions and velocities are measured in the same lab frame, including the time. The advantage of this coordinate formulation is that it can be applied to a variety of systems, including multiparticle systems. The disadvantage is that some lab frame has been singled out as a preferred frame, and none of the equations are '' manifestly covariant'' (in other words, they do not take the same form in all frames of reference). For an observer moving relative to the lab frame, everything must be recalculated; the position r, the momentum p, total energy ''E'', potential energy, etc. In particular, if this other observer moves with constant relative velocity then
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s must be used. However, the action will remain the same since it is Lorentz invariant by construction. A seemingly different but completely equivalent form of the Lagrangian for a free massive particle, which will readily extend to general relativity as shown below, can be obtained by inserting : d\tau = \frac\sqrt dt \,, into the Lorentz invariant action so that : S = \varepsilon \int_^ \frac\sqrt dt \quad\Rightarrow\quad L = \frac\sqrt where is retained for simplicity. Although the line element and action are Lorentz invariant, the Lagrangian is ''not'', because it has explicit dependence on the lab coordinate time. Still, the equations of motion follow from
Hamilton's principle In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single funct ...
: \delta S = 0\,. Since the action is proportional to the length of the particle's worldline (in other words its trajectory in spacetime), this route illustrates that finding the stationary action is asking to find the trajectory of shortest or largest length in spacetime. Correspondingly, the equations of motion of the particle are akin to the equations describing the trajectories of shortest or largest length in spacetime, ''
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s''. For the case of an interacting particle in a potential ''V'', the Lagrangian is still : L = \frac\sqrt - V , which can also extend to many particles as shown above, each particle has its own set of position coordinates to define its position.


Covariant formulation

In the covariant formulation, time is placed on equal footing with space, so the coordinate time as measured in some frame is part of the configuration space alongside the spatial coordinates (and other generalized coordinates). For a particle, either
massless In particle physics, a massless particle is an elementary particle whose invariant mass is zero. At present the only confirmed massless particle is the photon. Other particles and quasiparticles Standard Model gauge bosons The photon (carrier of ...
or massive, the Lorentz invariant action is (abusing notation) : S = \int_^ \Lambda(x^\nu(\sigma),u^\nu(\sigma),\sigma) d\sigma , where lower and upper indices are used according to
covariance and contravariance of vectors In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Briefly, a contravariant vecto ...
, ''σ'' is an ''
affine parameter In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conne ...
'', and is the
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three ...
of the particle. For massive particles, ''σ'' can be the arc length ''s'', or proper time ''τ'', along the particle's world line, : ds^2 = c^2d\tau^2 = g_ d x^\alpha d x^\beta . For massless particles, it cannot because the proper time of a massless particle is always zero; : g_ d x^\alpha d x^\beta = 0\,. For a free particle, the Lagrangian has the form : \Lambda = g_\frac \frac where the irrelevant factor of 1/2 is allowed to be scaled away by the scaling property of Lagrangians. No inclusion of mass is necessary since this also applies to massless particles. The Euler–Lagrange equations in the spacetime coordinates are : \frac\frac - \frac = \frac + \Gamma^\alpha_ \frac\frac = 0\,, which is the geodesic equation for affinely parameterized geodesics in spacetime. In other words, the free particle follows geodesics. Geodesics for massless particles are called "null geodesics", since they lie in a "
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
" or "null cone" of spacetime (the null comes about because their inner product via the metric is equal to 0), massive particles follow "timelike geodesics", and hypothetical particles that travel faster than light known as
tachyons A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists posit that faster-than-light particles cannot exist because they are inconsistent with the known laws of physics. If such particles ...
follow "spacelike geodesics". This manifestly covariant formulation does not extend to an ''N''-particle system, since then the affine parameter of any one particle cannot be defined as a common parameter for all the other particles.


Examples in special relativity


Special relativistic 1d free particle

For a 1d relativistic
free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
, the Lagrangian is : L = - m_0 c^2 \sqrt \,. This results in the following equation of motion: : m_0\ddot \, \frac = 0 \,. :


Special relativistic 1d harmonic oscillator

For a 1d relativistic
simple harmonic oscillator In mechanics and physics, simple harmonic motion (sometimes abbreviated as ) is a special type of periodic function, periodic motion an object experiences by means of a restoring force whose magnitude is directly proportionality (mathematics), ...
, the Lagrangian is : L = - m c^2 \sqrt - \fracx^2 \,. where ''k'' is the spring constant.


Special relativistic constant force

For a particle under a constant force, the Lagrangian is : L = - m c^2 \sqrt - mgx \,, where ''g'' is the force per unit mass. This results in the following equation of motion: : \frac\ddot = -g \,. Which, given initial conditions of : \begin x(t=0) &= x_0 \\ \dot(t=0) &= v_0 \end results in the position of the particle as a function of time being : x(t) = x_0 + \frac\left frac - \sqrt\right\,. : :


Special relativistic test particle in an electromagnetic field

In special relativity, the Lagrangian of a massive charged test particle in an electromagnetic field modifies to : L = - m c^2 \sqrt - q \phi + q \dot \cdot \mathbf \,. The Lagrangian equations in r lead to the
Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
law, in terms of the
relativistic momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. I ...
: \frac\left(\frac \right) = q \mathbf + q \dot \times \mathbf \,. In the language of
four-vector In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
s and
tensor index notation In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, the Lagrangian takes the form : L(\tau) = \fracm u^\mu(\tau)u_\mu(\tau) + qu^\mu(\tau)A_\mu(x) , where ''uμ'' = ''dxμ''/''dτ'' is the
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three ...
of the test particle, and ''Aμ'' the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
. The Euler–Lagrange equations are (notice the total derivative with respect to proper time instead of
coordinate time In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial ...
) : \frac - \frac\frac = 0 obtains : qu^\mu\frac = \frac (m u_\nu + q A_\nu) \,. Under the
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
with respect to proper time, the first term is the relativistic momentum, the second term is : \frac = \frac \frac = \frac u^\mu \,, then rearranging, and using the definition of the antisymmetric
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, gives the covariant form of the Lorentz force law in the more familiar form, : \frac (m u_\nu) = qu^\mu F_ \,,\quad F_ = \frac - \frac \,.


Lagrangian formulation in general relativity

The Lagrangian is that of a single particle plus an interaction term ''L''I : L = - m c^2 \frac + L_\text \,. Varying this with respect to the position of the particle ''x''α as a function of time ''t'' gives : \begin \delta L & = m \frac \delta \left( g_ \frac \frac \right) + \delta L_\text \\ & = m \frac \left( g_ \delta x^ \frac \frac + 2 g_ \frac \frac \right) + \frac \delta x^ + \frac \frac \\ & = \frac12 m g_ \delta x^ \frac \frac - \frac \left( m g_ \frac \right) \delta x^ + \frac \delta x^ - \frac \left( \frac \right) \delta x^ + \frac \,. \end This gives the equation of motion : 0 = \frac12 m g_ \frac \frac - \frac \left( m g_ \frac \right) + f_ where : f_ = \frac - \frac \left( \frac \right) is the non-gravitational force on the particle. (For ''m'' to be independent of time, we must have .) Rearranging gets the force equation : \frac \left( m \frac \right) = - m \Gamma^_ \frac \frac + g^ f_ where Γ is the
Christoffel symbol In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
, which describes the gravitational field. If we let : p^ = m \frac be the (kinetic) linear momentum for a particle with mass, then : \frac = - \Gamma^_ p^ \frac + g^ f_ and : \frac = \frac hold even for a massless particle.


Examples in general relativity


General relativistic test particle in an electromagnetic field

In
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the first term generalizes (includes) both the classical kinetic energy and the interaction with the gravitational field. For a charged particle in an electromagnetic field, the Lagrangian is given by : L(x,\dot) = - m c^2 \sqrt + q \frac A_(x(\tau))\,. If the four spacetime coordinates ''x''''μ'' are given in arbitrary units (i.e. unitless), then ''g''''μν'' is the rank 2 symmetric
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, which is also the gravitational potential. Also, ''A''''μ'' is the electromagnetic 4-vector potential. There exists an equivalent formulation of the relativistic Lagrangian, which has two advantages: * it allows for a generalization to massless particles and tachyons; * it is based on an energy functional instead of a length functional, such that it does not contain a square root. In this alternative formulation, the Lagrangian is given by : L(x,\dot,e) = \frac g_( x(\lambda) ) \, \frac \frac - \frac + q \, A_\mu(x(\lambda)) \frac, where \lambda is an arbitrary affine parameter and e is an auxiliary parameter that can be viewed as an einbein field along the worldline. In the original Lagrangian with the square root the energy-momentum relation appears as a
primary constraint In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of ...
that is also a
first class constraint In physics, a first-class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultan ...
. In this reformulation this is no longer the case. Instead, the energy-momentum relation appears as the equation of motion for the auxiliary field e. Therefore, the constraint is now a
secondary constraint In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion. A secondary constraint is one that is not primary—in other words it holds when the equations of ...
that is still a
first class constraint In physics, a first-class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultan ...
, reflecting the invariance of the action under reparameterization of the affine parameter \lambda. After the equation of motion has been derived, one must gauge fix the auxiliary field e. The standard gauge choice is as follows: * If m^2 > 0, one fixes e = , m, ^. This choice automatically fixes \lambda=\tau, i.e. the affine parameter is fixed to be the proper time. * If m^2 < 0, one fixes e = , m \, c, ^. This choice automatically fixes \lambda=s, i.e. the affine parameter is fixed to be the proper length. * If m=0, there is no choice that fixes the affine parameter \lambda to a physical parameter. Consequently, there is some freedom in fixing the auxiliary field. The two common choices are: ** Fix e = 1. In this case, e does not carry a dependence on the affine parameter \lambda, but the affine parameter is measured in units of time per unit of mass, i.e.
lambda Lambda (; uppercase , lowercase ; , ''lám(b)da'') is the eleventh letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoen ...
T/M. ** Fix e = c^2 E(\lambda), where E is the energy of the particle. In this case, the affine parameter is measured in units of time, i.e.
lambda Lambda (; uppercase , lowercase ; , ''lám(b)da'') is the eleventh letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoen ...
T, but e retains a dependence on the affine parameter \lambda.


See also

*
Relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non- quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities o ...
*
Fundamental lemma of the calculus of variations In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero ...
*
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
*
Functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
*
Generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...
*
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
*
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
*
Lagrangian analysis Lagrangian analysis is the use of Lagrangian coordinates to analyze various problems in continuum mechanics. Lagrangian analysis may be used to analyze currents and flows of various materials by analyzing data collected from gauges/sensors embed ...
(applications of Lagrangian mechanics) *
Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium (mechanics), equilibrium for small-mass objects under the gravity, gravitational influence of two massive orbit, orbiting b ...
*
Lagrangian system In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of . In classical mechanics, many dynamical systems are L ...
*
Non-autonomous mechanics Non-autonomous mechanics describe non- relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space o ...
*
Restricted three-body problem In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocity, velocities (or momentum, momenta) of three point masses orbiting each other in space and then calculate their subsequent trajector ...
*
Plateau's problem 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 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...


Footnotes


Citations


References

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Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
General relativity