Euler–Lagrange Equations

The Euler–Lagrange equations are differential equations that characterize stationary functions of integral functionals: functions for which every admissible infinitesimal variation produces zero first-order change in the functional. They are central to the calculus of variations. In classical mechanics, they express the equations of motion through stationarity of the action; in field theory, they give the corresponding equations for fields. Stationarity is a necessary condition for a smooth extremum, but does not by itself establish a minimum or maximum. (damtp.cam.ac.uk)

Mathematical formulation

Consider the functional

J[y]=\int_a^b F(x,y(x),y'(x))\,dx,

with prescribed endpoint values $y(a)$ and $y(b)$ . Here $F$ is a sufficiently smooth integrand, and $y'=dy/dx$ . A sufficiently smooth stationary function satisfies

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

The partial derivatives treat $x,y,y'$ as independent arguments of $F$ , whereas $d/dx$ differentiates the resulting expression along the function $y(x)$ . (math.mit.edu)

For several dependent functions $y_1,\ldots,y_n$ , there is one equation for each:

\frac{\partial F}{\partial y_i} -\frac{d}{dx}\left(\frac{\partial F}{\partial y_i'}\right)=0, \qquad i=1,\ldots,n.

These equations generally contain second derivatives, although special or degenerate integrands can reduce their order. (web.mit.edu)

Derivation from the first variation

Replace $y$ by

y_\varepsilon(x)=y(x)+\varepsilon\eta(x),

where $\eta(a)=\eta(b)=0$ preserves the endpoint values. The first variation is

\delta J[y;\eta] =\left.\frac{d}{d\varepsilon}J[y+\varepsilon\eta]\right|_{\varepsilon=0} =\int_a^b\left(F_y\eta+F_{y'}\eta'\right)\,dx.

Integration by parts gives

\delta J =\left[F_{y'}\eta\right]_a^b +\int_a^b\left(F_y-\frac{d}{dx}F_{y'}\right)\eta\,dx.

The boundary term vanishes. Requiring the remaining integral to vanish for every admissible $\eta$ forces its coefficient to vanish, yielding the Euler–Lagrange equation. This implication is the fundamental lemma of the calculus of variations. (damtp.cam.ac.uk)

Classical mechanics

For independent generalized coordinates $q_i(t)$ , define the action using a Lagrangian $L(q,\dot q,t)$ :

S[q]=\int_{t_1}^{t_2}L(q,\dot q,t)\,dt.

Stationarity under fixed-endpoint variations gives

\boxed{\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right) -\frac{\partial L}{\partial q_i}=0.}

The formulation retains its form under invertible changes of generalized coordinates, allowing positions, angles, and other suitable variables to describe motion. (damtp.cam.ac.uk)

For a particle of constant mass $m$ moving in one dimension under a position-dependent potential,

L=\frac12m\dot x^2-V(x).

The first term is kinetic energy, and $V$ is potential energy. Substitution gives

m\ddot x=-\frac{dV}{dx},

recovering Newton’s second law for a conservative force. (damtp.cam.ac.uk)

Conserved quantities

The conjugate momentum is

p_i=\frac{\partial L}{\partial\dot q_i}.

If $L$ is independent of $q_i$ , that coordinate is cyclic or ignorable, and its equation implies $\dot p_i=0$ . If $L$ has no explicit time dependence, then

E=\sum_i\dot q_i p_i-L

is conserved. For the usual time-independent mechanical Lagrangian with quadratic kinetic energy and velocity-independent potential, this equals the total energy. These results are examples of Noether’s theorem, which relates continuous symmetries of the action to conservation laws. (damtp.cam.ac.uk)

Geometric example: shortest curves

The length of a plane curve represented as $y(x)$ is

J[y]=\int_a^b\sqrt{1+(y')^2}\,dx.

Because the integrand is independent of $y$ , its Euler–Lagrange equation becomes

\frac{d}{dx}\left(\frac{y'}{\sqrt{1+(y')^2}}\right)=0.

Thus $y'$ is constant, giving the straight segment between fixed endpoints. This illustrates how optimization over entire functions becomes a differential equation. (math.mit.edu)

On curved spaces, analogous variational formulations lead to geodesic equations. For example, a kinetic-type Lagrangian

L=\frac12g_{ij}(q)\dot q^i\dot q^j

built from a metric tensor yields the equations for affinely parametrized geodesics. (damtp.cam.ac.uk)

Fields and several independent variables

For fields $\phi^A(x)$ on a region $\Omega$ , consider

S[\phi]=\int_\Omega \mathcal L(x,\phi^A,\partial_\mu\phi^A)\,d^dx.

With suitable boundary conditions, stationarity gives

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

Repeated $\mu$ indices are summed. Here $\mathcal L$ is the Lagrangian density. These are generally partial differential equations, and provide the variational framework for classical field theories underlying quantum field theory. (damtp.cam.ac.uk)

Boundary conditions and scope

The interior equations must be accompanied by appropriate boundary conditions. If an endpoint’s independent-variable position is fixed but its function value is free, the boundary term requires

\frac{\partial F}{\partial y'}=0

there, provided the functional has no additional endpoint contribution. Such requirements are called natural boundary conditions. Moving endpoints produce further conditions. (web.mit.edu)

For an integrand depending on derivatives through order $r$ , the higher-order equation is

\sum_{k=0}^{r}(-1)^k \frac{d^k}{dx^k} \left(\frac{\partial F}{\partial y^{(k)}}\right)=0, \qquad y^{(0)}=y.

Its derivation requires repeated integration by parts and appropriate endpoint restrictions. (web.mit.edu)

The equations identify stationary candidates rather than prove optimality or existence. Additional analysis is needed to distinguish minima, maxima, and saddle points. When solutions lack the smoothness required for the classical differential equation, stationarity can instead be expressed in a weak integral formulation, connecting variational methods with weak solutions. (math.mit.edu)

References

Calculus of Variations — Gilbert Strangmath.mit.edu
A Mathematical Primer — Rohan Abeyaratneweb.mit.edu