Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 16C / a

Take $T>0$ and a positive prescribed distance $L$; if $L=0$, the zero-speed journey has zero cost. Minimize $E[V]={\int }_0^T(a{V}^2+b{\dot{V}}^2)dt$ subject to $V(0)=0$ and the isoperimetric constraint ${\int }_0^{T}Vdt=L$. Introduce a constant Lagrange multiplier $\lambda$ and the integrand $a{V}^2+b{\dot{V}}^2-\lambda V$. For a variation $\eta$ with $\eta (0)=0$, integration by parts givesThe interior Euler-Lagrange equation isWhen terminal speed is free, the same calculation also imposes the natural boundary conditions for a free endpoint:One cannot drop this boundary term when interpreting an extremal as a genuine minimum. If $b=0<a$, the interior condition makes $V$ constant; with a continuous zero initial speed and $L>0$ it cannot be attained. In that case Cauchy-Schwarz inequality gives the infimum $a{L}^2/T$, approached by speeds nearly constant except in a short initial layer. For $b>0$, the derivative cost controls such layers and the natural terminal condition matters in parts (b) and (c).