Natural boundary conditions for a free endpoint

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