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For a real scalar field with Lagrangian density $\mathcal{L}={\dot{\varphi }}^2/2-|\nabla \varphi {|}^2/2-V(\varphi )$, the canonical momentum is $\pi =\dot{\varphi }$. Its Hamiltonian density is the sum of momentum energy, gradient energy and potential energy. Hamilton's equations recover the Euler-Lagrange equation.

For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

A generalized coordinate absent from the Lagrangian is cyclic. Its conjugate generalized momentum is conserved by the Euler-Lagrange equations.

If a real nonzero field $\xi (x)$ appears in the Lorentzian density $-F_{ab}{F}^{ab}/4+({\partial }_a{A}^a{)}^2/(2\xi )$ and is varied, its algebraic Euler-Lagrange equation is $-({\partial }_a{A}^a{)}^2/(2{\xi }^2)=0$. Hence it imposes the Lorenz gauge condition on real classical fields. Varying $A_b$ gives ${\partial }_a{F}^{ab}-{\partial }^b[(\partial \cdot A)/\xi ]=0$, including derivatives of $\xi$. This constrained field theory differs from holding the gauge parameter fixed in a Gaussian functional integral.

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

For an autonomous Lagrangian $L(x,\dot{x})$, define its action $A=\int Ldt$ and Hamiltonian $H=\dot{x}{L}_{\dot{x}}-L$. The chain rule givesThus the identity holds along forced trajectories as well as solutions of the homogeneous Euler-Lagrange equation. Explicit time dependence adds $-{\partial }_{t}L$; stochastic work requires a consistent regularization or the Stratonovich chain rule.

An equation of motion determines how a physical state evolves. A variational action gives Euler-Lagrange equations; a phase-space action instead gives canonical Hamiltonian mechanics equations together with any imposed constraints.

Take $a=3\pi /2$, $L=10$ and $u(x)=a\exp (-|x{|}^2/L^2)$ on ${\mathbb{R}}^3$. Set $f=-\Delta u+u+\sin u$. Both functions are smooth and belong to L2 space, and $u\in H^1$. The strictly increasing reaction $h(s)=s+\sin s$ has $0\le h(u)\le h(a)=a-1$, whileConsequently $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert u{\Vert }_{\infty }$. This $u$ is a minimizer of the screened sine-Gordon energy: it solves the Euler-Lagrange equation and that energy is convex. Thus even the minimizing and smooth hypotheses do not justify the stronger source-size bound. The appropriate general estimate is maximum bound for a monotone reaction term.

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.

A Lagrangian is hyperregular when its fiber derivative, sending a velocity to its conjugate momentum, is a global diffeomorphism from the tangent bundle to the cotangent bundle. This gives a globally defined Legendre transform in mechanics and a global equivalence between the Euler-Lagrange equations and Hamilton's equations. With time dependence this requirement is imposed at each time, with a smooth inverse depending on time.

For a canonical scalar field in a spatially flat expanding universe,The field Euler-Lagrange equation is

An isoperimetric constraint fixes the value of an integral while optimizing another integral over functions. A constant Lagrange multiplier incorporates the constraint into the variational integrand, and the resulting Euler-Lagrange equation must be supplemented by the appropriate natural boundary conditions for a free endpoint. The name includes integral constraints more general than prescribed geometric length.

The Jacobi equation is the linearization of an Euler-Lagrange equation about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

Substitute a static soliton family with slowly time-dependent collective coordinates into the action. The kinetic energy induces a Riemannian metric $g$ on the family, after imposing any Gauss law constraint in gauge theory constraint and projecting out gauge directions. If its static energy is constant, the Euler-Lagrange equations are the geodesic equations of $g$. An approximately flat family instead carries a potential $V(q)$. This approximation neglects excitations of the other field modes and is justified only when their effects are small on the time and energy scales studied.

A pointwise constraint on the initial derivative need not be closed in the norm controlling the value and first-derivative square integrals. For example, minimize $b{\int }_0^T{\dot{V}}^{2}dt$ with $b,L,T>0$, $V(0)=0$ and ${\int }_0^{T}Vdt=L$. The attained minimum is $3b{L}^2/T^3$, uniquely achieved by $V_{\ast }=3Lt(2T-t)/(2T^3)$, with natural condition ${\dot{V}}_{\ast }(T)=0$. Adding $\dot{V}(0)=0$ excludes this minimizer but leaves the same infimum. Multiplying $V_{\ast }$ by $1-e^{-t/\epsilon }$ and rescaling to restore the integral gives smooth admissible approximations converging in energy. The formal quadratic from the interior Euler-Lagrange equation and two initial conditions alone is therefore not a true minimizer.

Expand a unit-mass particle's Lagrangian about an axisymmetric circular orbit of radius $r_0$ and frequency ${\Omega }_0$, using $r=r_0+x$ and $\phi ={\Omega }_{0}t+y/r_0$. Circular-orbit balance removes the linear radial term; total-time-derivative invariance of a Lagrangian removes $r_0{\Omega }_0\dot{y}$. To second order,Here $q$ is the orbital shear parameter and ${\Omega }_z$ is the vertical epicyclic frequency. The Euler-Lagrange equations retain the local Coriolis acceleration and tidal gravity.