Euler-Lagrange equation

For $I[y]={\int }_a^{b}F(x,y,y^{\prime })dx$, stationarity under fixed-endpoint variations gives

Motion on a sphere with speed proportional to $\sin \theta$ has travel-time line element proportional to $\sqrt{d{\theta }^2/{\sin }^2\theta +d{\varphi }^2}$. With $z=\mathrm{arsinh}(\cot \theta )$, this becomes the flat cylinder metric $\sqrt{d{z}^2+d{\varphi }^2}$. Shortest lifted paths are straight lines. Nonvertical lines give $\cot \theta =\sinh (A\varphi +B)$; vertical lines are meridians of constant $\varphi$. The poles lie at infinite travel-time distance.

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.

When a variational integrand $f(z,z^{\prime })$ has no explicit dependence on the independent variable, every extremal satisfies

A catenary is a curve of the form $y=a\cosh ((x-x_0)/a)+y_0$. It is the equilibrium shape of a uniform flexible chain in a uniform gravitational field.

When the values of $y$ are free at both fixed endpoint locations, integration by parts leaves the boundary term $[F_{y^{\prime }}\eta {]}_a^b$. Since the endpoint variations are arbitrary, stationarity requires

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.

Forwith free terminal values, the Euler-Lagrange equations and natural conditions give a nonzero stationary family exactly when $x_0=(k+\frac{1}{2})\pi$. It is

Forindependent variations and integration by parts give