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

Put $\kappa =\sqrt{a/b}>0$. Solving the interior Euler-Lagrange equation and imposing both $V(0)=0$ and $\dot{V}(0)=0$ yields $V=C(\cosh \kappa t-1)$, with $\lambda =-2aC$. The distance constraint fixesThe denominator is positive because $\sinh z>z$ for $z>0$. This is the requested solution of the differential equation with the stated initial conditions.

As a minimum-energy problem with free terminal speed, the printed conditions do not admit a smooth minimizer. The formal profile has $\dot{V}(T)=C\kappa \sinh (\kappa T)>0$, violating the natural terminal condition from part (a). Fixing the initial acceleration restricts ${\eta }^{\prime }(0)$, but it does not restrict the independently variable $\eta (T)$ and therefore does not remove that condition.

For clarity, without the extra initial-acceleration restriction the actual unique minimizing profile isIt satisfies $V_{\ast }(0)=0$, ${\dot{V}}_{\ast }(T)=0$, the distance constraint and the interior equation, but ${\dot{V}}_{\ast }(0)>0$. For any perturbation preserving the distance and initial speed, the first variation about $V_{\ast }$ is zero, andwith equality only for $\eta =0$. Its energy is $a{L}^2/[T-\tanh (\kappa T)/\kappa ]$.

This same energy is the unattained infimum under the extra condition $\dot{V}(0)=0$. Indeed multiply $V_{\ast }$ by $1-e^{-t/\epsilon }$ and rescale by a factor tending to one to restore total distance $L$. These smooth nonnegative profiles have both initial values zero and converge to $V_{\ast }$ in value and derivative square integrals as $\epsilon \downarrow 0$. Their energies tend to $E[V_{\ast }]$, but uniqueness of $V_{\ast }$ precludes attaining it with zero initial acceleration. This is nonattainment under an initial derivative constraint.