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Euler–Lagrange equation

The Euler–Lagrange equation is a fundamental second-order ordinary differential equation in the calculus of variations, providing the necessary condition for a function to extremize (minimize or maximize) a given functional, typically expressed as an integral over a path or curve.^[1] In its standard form for a functional J = \int_a^b L(x, y(x), y'(x)) \, dx, where L is the Lagrangian integrand depending on the independent variable x, the function y(x), and its derivative y'(x), the equation is \frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0.^[1] This equation arises from setting the first variation of the functional to zero, analogous to setting the derivative to zero in ordinary calculus for extrema.^[2] The equation is named after the mathematicians Leonhard Euler and Joseph-Louis Lagrange, who independently developed its core ideas in the mid-18th century while addressing problems in optimization and mechanics.^[1] Euler first formulated the equation in a letter dated April 15, 1743, and elaborated it geometrically and intuitively in his 1744 treatise Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti, where he applied it to problems like the brachistochrone and isoperimetric curves using elementary methods without modern variational notation.^[3]^[4] Lagrange extended and formalized the approach starting in 1755 through correspondence with Euler, introducing a more analytical framework, and fully presented it in his 1788 work Mécanique Analytique, where he derived the equation for systems with multiple variables and constraints, emphasizing its role in deriving equations of motion from a principle of least action.^[5]^[6] In physics and engineering, the Euler–Lagrange equation is central to Lagrangian mechanics, where the Lagrangian L = T - V (with T as kinetic energy and V as potential energy) yields the equations of motion for conservative systems, generalizing Newton's laws to complex configurations involving constraints or generalized coordinates.^[7] It also finds applications in field theories, such as electromagnetism and general relativity, where it determines the field equations from action principles, and in optimization problems across mathematics, including geodesics on surfaces and minimal surfaces.^[2] Extensions to higher dimensions or time-dependent systems lead to more general forms, such as those for multiple functions or with non-holonomic constraints, underscoring its versatility in modern theoretical frameworks.^[1]

Background

Historical development

The roots of the Euler–Lagrange equation trace back to the late 17th century, when the Bernoulli brothers laid foundational work in variational problems. Jakob Bernoulli explored isoperimetric problems, which involve finding curves that maximize enclosed area for a fixed perimeter, providing early insights into optimization of functionals.^[8] These investigations influenced his brother Johann Bernoulli, who in 1696 posed the brachistochrone problem—a challenge to determine the curve allowing the fastest descent between two points under gravity—in the journal Acta Eruditorum, sparking widespread interest in methods for extremizing integrals.^[9] Leonhard Euler advanced this field significantly in the mid-18th century, beginning with early treatises around 1736 on variational principles and culminating in his comprehensive 1744 publication Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, where he introduced systematic variations of functions and derived a general differential equation for extremal curves.^[10] Euler's approach transformed the ad hoc solutions of earlier problems into a unified framework for the calculus of variations.^[11] Joseph-Louis Lagrange reformulated Euler's methods during 1760–1762 in his Essai d'une nouvelle méthode pour déterminer les maxima et minima des formules intégrales définies, published in the Mémoires de l'Académie des Sciences de Turin, by employing the delta (δ) notation for infinitesimal variations and integrating it into analytical mechanics for broader applicability.^[12] Lagrange expanded these ideas in his seminal Mécanique analytique (1788), emphasizing algebraic manipulation over geometric intuition. The equation, initially termed Euler's differential equation, became known as the Euler–Lagrange equation in the 19th century to recognize both pioneers' contributions.^[5]

Calculus of variations

The calculus of variations is a branch of mathematics dedicated to finding functions that extremize functionals, which are mappings from a set of admissible functions to the real numbers.^[13] Unlike ordinary calculus, where extrema are sought for functions of finitely many variables, the calculus of variations addresses infinite-dimensional optimization problems by treating functions themselves as the variables.^[14] A fundamental example of a functional is the integral form

J = \int_a^b L\bigl(x, y(x), y'(x)\bigr) \, dx,

where y: [a, b] \to \mathbb{R} is a differentiable function, y'(x) = dy/dx, and L is a given smooth function called the integrand or Lagrangian.^[15] This setup generalizes problems in physics, geometry, and optimization, where the goal is to determine y(x) that minimizes or maximizes J while respecting boundary conditions, such as fixed endpoints y(a) = y_a and y(b) = y_b.^[14] The variational problem thus involves identifying the extremal function y(x) among all possible curves connecting the endpoints that yields the optimal value of the functional.^[15] To solve such problems, one introduces the concept of a variation of the function, defined as a small perturbation \delta y(x) = \epsilon \eta(x), where \epsilon is an infinitesimal scalar and \eta(x) is a smooth test function vanishing at the boundaries, \eta(a) = \eta(b) = 0, to preserve the endpoint conditions.^[14] Substituting the varied function \tilde{y}(x) = y(x) + \epsilon \eta(x) into the functional yields J[\tilde{y}] = J[y + \epsilon \eta], which can be expanded in a Taylor series with respect to \epsilon.^[15] The first variation \delta J is the linear term in this expansion, corresponding to the Gateaux derivative of J at y in the direction of \eta, and is given by \delta J = \left. \frac{d}{d\epsilon} J[y + \epsilon \eta] \right|_{\epsilon=0}.^[14] For y(x) to be a stationary point of the functional—analogous to a critical point in finite dimensions—the first variation must vanish for every admissible \eta(x), so \delta J = 0.^[15] This condition of zero first variation leads directly to the Euler–Lagrange equation, which emerges as the necessary differential equation governing the extremal functions in the calculus of variations.^[13]

Core Formulation

Statement

The Euler–Lagrange equation arises as the necessary condition for a function y(x) to extremize the functional J = \int_a^b L(x, y(x), y'(x)) \, dx in the calculus of variations, where L(x, y, y') denotes the Lagrangian and y' = dy/dx.^[1] Assuming L is continuously differentiable with respect to x, y, and y', and considering one independent variable x with dependent variable y(x) involving only the first-order derivative y', the extremizing function y(x) satisfies the second-order ordinary differential equation

\frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0.

^[1] If the endpoints x = a and x = b are fixed such that y(a) and y(b) are prescribed, the equation holds with these boundary values; otherwise, if an endpoint is free, y(x) must additionally satisfy the corresponding natural boundary condition \frac{\partial L}{\partial y'} = 0 at that point.^[1] This equation expresses a balance between the direct variation term \frac{\partial L}{\partial y}, analogous to a "force," and the total derivative \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right), representing an "inertial" contribution from the dependence on the slope y'.^[16] In the special case where L does not explicitly depend on y (so \frac{\partial L}{\partial y} = 0), the equation simplifies to \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0, implying \frac{\partial L}{\partial y'} = constant, which serves as a precursor to the Beltrami identity for conserved quantities in variational problems.^[1]

Derivation

The derivation of the Euler–Lagrange equation proceeds from the principle of stationarity applied to the variational functional in the calculus of variations. Consider the functional

J = \int_a^b L(x, y(x), y'(x)) \, dx,

where L is the Lagrangian (or integrand) depending on the independent variable x, the dependent function y(x), and its first derivative y'(x) = dy/dx, with fixed endpoints y(a) and y(b). This functional represents the quantity to be extremized, such as arc length, action, or energy, subject to the boundary conditions.^[15] To determine the function y(x) that makes J stationary, introduce a small perturbation to the path: let y_\epsilon(x) = y(x) + \epsilon \eta(x), where \epsilon is a small scalar parameter and \eta(x) is an arbitrary smooth variation satisfying the boundary conditions \eta(a) = \eta(b) = 0. The perturbed functional is then

J[y_\epsilon] = \int_a^b L\big(x, y + \epsilon \eta, y' + \epsilon \eta'\big) \, dx.

For stationarity at \epsilon = 0, the first-order change in J, known as the first variation \delta J, must vanish. Expanding to first order in \epsilon and differentiating with respect to \epsilon at \epsilon = 0 yields

\delta J = \left. \frac{d}{d\epsilon} J[y_\epsilon] \right|_{\epsilon=0} = \int_a^b \left[ \frac{\partial L}{\partial y} \eta + \frac{\partial L}{\partial y'} \eta' \right] dx = 0,

which must hold for all admissible variations \eta(x).^[15] To simplify this condition, integrate the second term by parts. The term involving \eta' is

\int_a^b \frac{\partial L}{\partial y'} \eta' \, dx = \left[ \frac{\partial L}{\partial y'} \eta \right]_a^b - \int_a^b \eta \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) dx.

The boundary term vanishes because \eta(a) = \eta(b) = 0, leaving

\delta J = \int_a^b \left[ \frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) \right] \eta \, dx = 0.

Since this integral is zero for every arbitrary smooth \eta(x) with the given boundary conditions, the integrand must identically vanish, yielding the Euler–Lagrange equation

\frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0.

This differential equation governs the extremal paths. In a modern perspective, the condition \delta J = 0 corresponds to the Gateaux derivative of the functional J at y being zero in every direction \eta, providing a rigorous foundation in functional analysis for the variational principle.^[15]^[17]

Examples

Brachistochrone problem

The brachistochrone problem involves finding the curve y = y(x) between two points, typically from (0, 0) to (a, b) with y measured downward, that minimizes the time for a particle to slide from rest under constant gravity without friction.^[18] The speed v at position y follows from conservation of potential energy, giving v = \sqrt{2 g y}, where g is the acceleration due to gravity.^[19] The infinitesimal time element is dt = ds / v, with arc length ds = \sqrt{dx^2 + dy^2} = \sqrt{1 + (y')^2} \, dx, so the total time is

T = \int_0^a \frac{\sqrt{1 + (y')^2}}{\sqrt{2 g y}} \, dx = \frac{1}{\sqrt{2 g}} \int_0^a \sqrt{\frac{1 + (y')^2}{y}} \, dx.

^[19] Minimizing T is equivalent to minimizing the functional J = \int_0^a L(y, y') \, dx, where the Lagrangian is L = \sqrt{\frac{1 + (y')^2}{y}}, as the constant factor $1/\sqrt{2 g} does not affect the minimizing path.^[19] Since L has no explicit dependence on x, the Beltrami identity from the Euler–Lagrange framework simplifies the equation to L - y' \frac{\partial L}{\partial y'} = C, where C is a constant.^[20] To apply this, compute \frac{\partial L}{\partial y'} = \frac{y'}{\sqrt{y(1 + (y')^2)}}. Then,

L - y' \frac{\partial L}{\partial y'} = \sqrt{\frac{1 + (y')^2}{y}} - y' \cdot \frac{y'}{\sqrt{y(1 + (y')^2)}} = \frac{1}{\sqrt{y(1 + (y')^2)}} = C.

^[19] Rearranging yields y(1 + (y')^2) = \frac{1}{C^2}, or (y')^2 = \frac{1/C^2 - y}{y} = \frac{k^2 - y}{y}, where k = 1/C. Thus, y' = \sqrt{\frac{k^2 - y}{y}} (taking the positive root for the descending path).^[19] Separating variables gives dx = \sqrt{\frac{y}{k^2 - y}} \, dy. Integrating this differential equation leads to the parametric solution of a cycloid generated by a circle of radius r = k^2 / 2:

x(\theta) = r (\theta - \sin \theta), \quad y(\theta) = r (1 - \cos \theta),

with \theta ranging from 0 to some \theta_0 chosen to match the endpoint (a, b).^[18] This curve is independent of g, as the gravitational constant factors out of the functional and does not influence the geometry of the minimizing path.^[19] The cycloid provides a true minimum, as confirmed by comparing travel times to other curves like the straight line, which takes longer.^[18]

Minimal surface of revolution

The problem of finding the surface of revolution that minimizes the area between two parallel coaxial rings is a classic application of the Euler–Lagrange equation to geometric optimization. Consider two rings of radii r_1 and r_2 located at x = a and x = b along the x-axis. The generating curve y = y(x) is rotated about the x-axis to form an axisymmetric surface, and the area functional to minimize is

A = 2\pi \int_a^b y \sqrt{1 + (y')^2} \, dx,

with boundary conditions y(a) = r_1 and y(b) = r_2.^[21] The integrand of this functional serves as the Lagrangian L(y, y') = y \sqrt{1 + (y')^2}, which is independent of the explicit appearance of the independent variable x. For such Lagrangians, the full Euler–Lagrange equation \frac{\partial L}{\partial y} - \frac{d}{dx} \left( \frac{\partial L}{\partial y'} \right) = 0 simplifies via the Beltrami identity to L - y' \frac{\partial L}{\partial y'} = c, where c is a constant of integration. Here, \frac{\partial L}{\partial y'} = \frac{y y'}{\sqrt{1 + (y')^2}}, so substituting yields

y \sqrt{1 + (y')^2} - y' \cdot \frac{y y'}{\sqrt{1 + (y')^2}} = c.

This simplifies to

\frac{y}{\sqrt{1 + (y')^2}} = c.

Solving for the derivative gives y' = \pm \sqrt{\frac{y^2}{c^2} - 1}, and separating variables and integrating produces the explicit solution y(x) = c \cosh \left( \frac{x - x_0}{c} \right), where x_0 is determined by boundary conditions; the resulting surface is the catenoid.^[21]^[22] Leonhard Euler first derived this catenoid in 1744 as the surface of revolution with minimal area, proving it satisfies the necessary conditions from the calculus of variations.^[23]^[24] The catenoid provides a local minimum for the area functional only when the ring separation |b - a| is sufficiently small, specifically below a critical value approximately $1.325 \times \max(r_1, r_2); for larger separations, the global minimizer consists of two separate disks spanning the rings, as the catenoid becomes unstable.^[21]^[25]

Generalizations

Higher-order derivatives

In the calculus of variations, the Euler–Lagrange equation can be extended to functionals that depend on higher-order derivatives of the extremizing function. Consider a functional of the form

J = \int_a^b L\left(x, y, y', y'', \dots, y^{(n)}\right) \, dx,

where y^{(k)} = \frac{d^k y}{dx^k} denotes the k-th derivative of y with respect to x, and L is a smooth integrand depending on up to the n-th order derivative. The necessary condition for y to extremize J is the generalized Euler–Lagrange equation,

\sum_{k=0}^n (-1)^k \frac{d^k}{dx^k} \left( \frac{\partial L}{\partial y^{(k)}} \right) = 0.

This equation reduces to the standard first-order form when n=1. For n=0, it simplifies to \frac{\partial L}{\partial y} = 0. The equation is a (2n)-th order ordinary differential equation in general, reflecting the increased complexity introduced by higher derivatives. The derivation proceeds from the first variation of the functional. For a varied path y + \epsilon \eta, where \eta is an admissible variation vanishing at the endpoints along with its first n-1 derivatives (under fixed boundary conditions), the condition \delta J = 0 for arbitrary \eta yields

\int_a^b \left[ \sum_{k=0}^n \frac{\partial L}{\partial y^{(k)}} \eta^{(k)} \right] dx = 0.

Applying integration by parts n times to each term involving \eta^{(k)} transfers the derivatives to the coefficients \frac{\partial L}{\partial y^{(k)}}, resulting in boundary terms at x = a and x = b plus the integral \int_a^b \eta \sum_{k=0}^n (-1)^k \frac{d^k}{dx^k} \left( \frac{\partial L}{\partial y^{(k)}} \right) dx = 0. Since \eta is arbitrary, the integrand must vanish, giving the generalized equation, provided the boundary terms vanish. If some derivatives of y are not specified at the boundaries, natural boundary conditions arise from setting the coefficients of the non-vanishing boundary terms to zero. For a second-order case (n=2), these include conditions such as \frac{\partial L}{\partial y'} - \frac{d}{dx} \frac{\partial L}{\partial y''} = 0 and \frac{\partial L}{\partial y''} = 0 at each endpoint, ensuring the variation is stationary without prescribed values. Higher-order cases involve analogous higher-derivative expressions for the remaining boundary terms.^[26] A representative application occurs in elasticity theory for the deflection of a slender beam under transverse loading, modeled by the Euler–Bernoulli theory. The total potential energy functional is

J = \int_a^b \left[ \frac{1}{2} EI (y'')^2 - q(x) y \right] dx,

where EI is the flexural rigidity, q(x) is the distributed load, and the Lagrangian L = \frac{1}{2} EI (y'')^2 - q y depends on y and y'' (n=2). Applying the generalized Euler–Lagrange equation yields \frac{d^2}{dx^2} (EI y'') - q = 0, or equivalently (EI y'')'' = q, the fourth-order beam equation governing static deflection. Natural boundary conditions, such as those for a free end, include EI y'' = 0 (zero moment) and \frac{d}{dx} (EI y'') = 0 (zero shear).

Multiple functions

In variational problems involving multiple dependent functions y_1(x), \dots, y_m(x) that vary with a single independent variable x, the objective is to extremize a functional of the form

J[y_1, \dots, y_m] = \int_a^b L(x, y_1, \dots, y_m, y_1', \dots, y_m') \, dx,

where L denotes the Lagrangian, assumed sufficiently smooth, and primes indicate derivatives with respect to x.^[27] The stationarity condition yields a system of m first-order Euler–Lagrange equations, one for each i = 1, \dots, m:

\frac{\partial L}{\partial y_i} - \frac{d}{dx} \left( \frac{\partial L}{\partial y_i'} \right) = 0.

This system generalizes the single-function Euler–Lagrange equation, which corresponds to the scalar case m=1.^[27] The equations are typically coupled when L contains cross terms mixing distinct y_i and y_j' (for i \neq j), resulting in interdependent ordinary differential equations that must be solved simultaneously for all functions.^[27] A classical example occurs in finding geodesics on a two-dimensional surface, where the arc-length functional for a curve parametrized by coordinates y_1(x) and y_2(x) leads to coupled Euler–Lagrange equations whose solutions describe the shortest paths between points.^[27] Another instance arises in the variational formulation of the double pendulum, where the action integral based on kinetic and potential energies of the two angles \theta_1(t) and \theta_2(t) (with time t as the independent variable) produces a coupled system governing the dynamics.^[28] Conservation laws for such systems follow from Noether's theorem: if the Lagrangian remains invariant under a continuous symmetry transformation of the multiple functions and their derivatives, a corresponding conserved quantity exists, generalizing the single-function result to multi-component settings.^[29]

Multiple variables

In the calculus of variations, the Euler–Lagrange equation generalizes to problems involving a single scalar function u(\mathbf{x}) of multiple independent variables \mathbf{x} = (x_1, \dots, x_d) in a domain \Omega \subset \mathbb{R}^d. The objective is to extremize the functional

J = \int_\Omega L(x_1, \dots, x_d, u, \frac{\partial u}{\partial x_1}, \dots, \frac{\partial u}{\partial x_d}) \, dV,

where L is the Lagrangian density depending on the position variables, the function value, and its first partial derivatives, and dV = dx_1 \dots dx_d is the volume element.^[15] This formulation arises naturally in problems like finding minimal surfaces or steady-state fields, extending the one-dimensional case where d=1.^[30] The corresponding Euler–Lagrange equation is

\frac{\partial L}{\partial u} - \sum_{i=1}^d \frac{\partial}{\partial x_i} \left( \frac{\partial L}{\partial (\partial u / \partial x_i)} \right) = 0.

This partial differential equation must hold at every point in \Omega.^[30] To derive it, consider an admissible variation \delta u such that \delta u = 0 on the boundary \partial \Omega. The first variation of the functional is

\delta J = \int_\Omega \left( \frac{\partial L}{\partial u} \delta u + \sum_{i=1}^d \frac{\partial L}{\partial (\partial u / \partial x_i)} \frac{\partial (\delta u)}{\partial x_i} \right) dV = 0

for a stationary point. Applying integration by parts in multiple dimensions to each term in the sum yields

\int_\Omega \sum_{i=1}^d \frac{\partial}{\partial x_i} \left( \frac{\partial L}{\partial (\partial u / \partial x_i)} \right) \delta u \, dV = \int_{\partial \Omega} \sum_{i=1}^d \frac{\partial L}{\partial (\partial u / \partial x_i)} n_i \delta u \, dS,

where the divergence theorem has been used, with \mathbf{n} = (n_1, \dots, n_d) the outward unit normal and dS the surface element. Since \delta u = 0 on \partial \Omega, the boundary integral vanishes, leaving the Euler–Lagrange equation upon invoking the fundamental lemma of the calculus of variations.^[30]^[15] A classic example is the variational principle for harmonic functions, where the functional is

J = \int_\Omega \frac{1}{2} |\nabla u|^2 \, dV = \int_\Omega \frac{1}{2} \sum_{i=1}^d \left( \frac{\partial u}{\partial x_i} \right)^2 dV,

corresponding to L = \frac{1}{2} \sum_{i=1}^d p_i^2 with p_i = \partial u / \partial x_i. Substituting into the Euler–Lagrange equation gives \partial L / \partial u = 0 and \partial L / \partial p_i = p_i, so

- \sum_{i=1}^d \frac{\partial}{\partial x_i} \left( \frac{\partial u}{\partial x_i} \right) = -\Delta u = 0,

the Laplace equation \Delta u = 0, whose solutions minimize the Dirichlet energy functional subject to fixed boundary values.^[13] If essential boundary conditions are not specified (i.e., \delta u \neq 0 on \partial \Omega), natural boundary conditions emerge from setting the surface integral to zero:

\sum_{i=1}^d \frac{\partial L}{\partial (\partial u / \partial x_i)} n_i = 0 \quad \text{on} \quad \partial \Omega.

For the harmonic function example, this simplifies to \partial u / \partial n = 0, the Neumann condition of zero normal derivative.^[30]^[13]

Field theories

In field theories, particularly those describing physical systems in relativistic spacetime, the Euler–Lagrange framework is extended to functionals that depend on fields varying continuously over four-dimensional Minkowski space. The action functional for a scalar field \phi(x^\mu) is given by

S[\phi] = \int L(\phi, \partial_\mu \phi, x^\mu) \, d^4 x,

where L is the Lagrangian density, \partial_\mu denotes the partial derivative with respect to spacetime coordinates x^\mu, and the integral is over all spacetime volume.^[31] This formulation arises naturally in classical field theory as the continuous analog of the particle action, extremizing which yields the equations of motion.^[32] The Euler–Lagrange equations for fields are derived by requiring the variation of the action \delta S = 0, leading to the field equations

\frac{\partial L}{\partial \phi} - \partial_\mu \left( \frac{\partial L}{\partial (\partial_\mu \phi)} \right) = 0.

This covariant form ensures Lorentz invariance, with the divergence \partial_\mu acting on the canonical momentum conjugate to the field derivatives.^[31] For multiple fields or vector/tensor fields, the equation generalizes accordingly, maintaining the structure of balancing the explicit field dependence against the spacetime variation of its gradients.^[33] A canonical example is the real scalar field theory, where the Lagrangian density is

L = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2.

Applying the Euler–Lagrange equation yields the Klein–Gordon equation,

(\partial_\mu \partial^\mu + m^2) \phi = 0,

which describes massive spin-0 particles in relativistic quantum field theory.^[32] This equation captures the relativistic wave propagation with a mass term, foundational for models like the Higgs mechanism. For vector fields, the electromagnetic field is described by the Lagrangian density

L = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu},

where F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu is the field strength tensor constructed from the four-potential A_\mu. The Euler–Lagrange equations produce the sourceless inhomogeneous Maxwell equations,

\partial_\mu F^{\mu\nu} = 0,

while the Bianchi identity \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 encodes the homogeneous Maxwell equations.^[34] Including sources J^\mu modifies the first to \partial_\mu F^{\mu\nu} = J^\nu, fully recovering Maxwell's equations.^[35] The covariant structure of these equations facilitates the derivation of conserved quantities via Noether's theorem. For translation invariance, the stress-energy tensor emerges as

T^\mu{}_\nu = \frac{\partial L}{\partial (\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_\nu L,

whose divergence \partial_\mu T^{\mu\nu} = 0 encodes energy-momentum conservation in flat spacetime.^[33] This tensor couples to gravity in general relativity, providing the source for spacetime curvature. In gauge theories like electromagnetism, the Lagrangian exhibits local gauge invariance under transformations A_\mu \to A_\mu + \partial_\mu \Lambda, which leaves F_{\mu\nu} unchanged and preserves the action.^[34] This redundancy implies the Euler–Lagrange equations are underdetermined, requiring gauge-fixing constraints to select physical degrees of freedom, often leading to non-variational subsidiary conditions such as the Lorenz gauge \partial_\mu A^\mu = 0.^[35] Such invariances underpin broader frameworks like the Standard Model, where similar structures appear in non-Abelian gauge fields.

Advanced Extensions

Manifold generalization

The manifold generalization of the Euler–Lagrange equation provides a coordinate-invariant framework for variational problems on curved spaces, extending the classical theory to mappings between Riemannian manifolds. Consider a smooth map f: (M, g) \to (N, h), where M and N are smooth Riemannian manifolds equipped with metrics g and h, respectively. The associated functional is defined as J = \int_M L(f, df) \, \vol_g, where L: \TN \times f^* TN \to \mathbb{R} is a smooth Lagrangian density depending on the map f and its differential df, and \vol_g denotes the volume form induced by the metric g on M. To find the critical points of J, consider variations of f given by sections of the pullback bundle f^* TN \to M. The first variation \delta J = 0 yields the intrinsic Euler–Lagrange equation in local coordinates (x^\mu) on M and (y^a) on N:

\frac{\partial L}{\partial f^a} - \nabla_\mu \left( \frac{\partial L}{\partial (\partial_\mu f^a)} \right) = 0,

where \nabla is the Levi-Civita covariant derivative on M coupled with the connection on N via the pullback, and indices follow Einstein summation convention. This form replaces ordinary partial derivatives with covariant ones to ensure tensorial invariance under coordinate changes.^[36] The derivation proceeds by expressing the variation as f_t = f + t \xi, with \xi a smooth section of f^* TN vanishing on the boundary if M has one, and computing \frac{d}{dt} \big|_{t=0} J[f_t] = 0. The resulting integral involves the Lie derivative of L along \xi, which, after integration by parts, uses the Stokes theorem on manifolds to transfer the derivative term: \int_M \div(\cdot) \, \vol_g = \int_{\partial M} (\cdot), yielding the Euler–Lagrange equation with no remainder when \xi vanishes on \partial M. A prominent special case arises when the Lagrangian is the kinetic energy L = \frac{1}{2} g(df, df) = \frac{1}{2} g^{\mu\nu} h_{ab}(f) \partial_\mu f^a \partial_\nu f^b, corresponding to the energy functional for maps. The Euler–Lagrange equation simplifies to the geodesic equation \nabla_{df} df = 0, or in coordinates, \frac{d^2 f^a}{d\tau^2} + \Gamma^a_{bc}(f) \frac{df^b}{d\tau} \frac{df^c}{d\tau} = 0, where \Gamma are the Christoffel symbols of h and \tau parameterizes curves on M; here, f restricted to curves describes geodesics in N. If the domain M has a nonempty boundary and variations \xi do not vanish there, natural boundary conditions emerge from the boundary term in the integration by parts, requiring \frac{\partial L}{\partial (\partial_\mu f^a)} n^\mu \xi^a = 0 on \partial M, where n is the outward normal; this is known as the variational completion, enforcing transversality for free boundaries.^[36]

Infinite-dimensional variants

In infinite-dimensional settings, the Euler–Lagrange framework extends to functionals defined on spaces of functions, such as Sobolev spaces W^{1,p}(\Omega), where minimizers are sought for variational problems involving partial differential equations (PDEs). These spaces provide the appropriate topology for handling weak solutions, ensuring compactness via embeddings like the Rellich-Kondrachov theorem, which facilitates the direct method in the calculus of variations for existence proofs.^[37]^[38] The Euler–Lagrange equation in this context adopts a weak form, derived from the first variation of the functional J(u) = \int_\Omega L(x, u, \nabla u) \, dx = 0 for test functions \delta u with compact support. This yields the integral condition

\int_\Omega \left( \frac{\partial L}{\partial u} \delta u + \frac{\partial L}{\partial (\nabla u)} \cdot \nabla (\delta u) \right) dx = 0,

which holds for all admissible variations and corresponds to the PDE in the distributional sense when integrated by parts, assuming sufficient regularity. Such formulations are essential for problems where classical solutions may not exist, allowing solutions in the sense of distributions.^[39]^[40] In optimal control theory, the Pontryagin maximum principle serves as a counterpart to the Euler–Lagrange equation for infinite-dimensional systems with control variables, maximizing the Hamiltonian along optimal trajectories in function spaces like those of state and adjoint variables. This principle provides necessary conditions for optimality in problems such as distributed parameter systems, where the control enters the dynamics nonlinearly, analogous to how the Euler–Lagrange equation governs unconstrained variational paths.^[41]^[42] A prominent example is the Plateau problem in infinite-dimensional spaces, where minimal surfaces spanning a given boundary are sought in Banach spaces using the theory of currents. Here, integral currents generalize the notion of oriented surfaces to infinite dimensions, and the mass-minimizing current solving the variational problem satisfies an Euler–Lagrange-type equation in the weak sense, ensuring existence via compactness in the space of bounded mass currents. Regularity theory for these weak solutions employs bootstrapping techniques to upgrade regularity: starting from u \in W^{1,2}(\Omega), higher integrability or Schauder estimates are applied iteratively using the structure of the Euler–Lagrange operator, often assuming ellipticity or growth conditions on L, to obtain classical C^\infty solutions in the interior. This process relies on elliptic regularity results tailored to the variational setting, confirming that weak minimizers are smooth where the data permits.^[43]^[44]

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