Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 16C (Variational Principles)

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).

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.

Here $b>0$ because $a=0$ and the two cost coefficients are not both zero. The interior Euler-Lagrange equation makes $V$ quadratic. Imposing each displayed pair of endpoint data and the distance constraint gives the following formal profiles and their costs:Thus choice (3) is the best strategy, and its true minimum cost isThe qualification from part (b) applies to row (1): its formal quadratic is not a minimizer with free terminal speed, since $\dot{V}(T)\ne 0$. Row (2) is a genuine minimizer for its fixed zero terminal speed, and row (3) is the genuine free-terminal-speed minimizer, satisfying its natural condition $\dot{V}(T)=0$.

A direct lower bound makes the comparison rigorous. For every admissible speed with $V(0)=0$,by integration by parts. The Cauchy-Schwarz inequality gives $L^2\le (T^3/3){\int }_0^T{\dot{V}}^{2}dt$. Equality requires $\dot{V}$ to be proportional to $T-t$, giving exactly row (3). Row (1)'s zero initial acceleration excludes equality; however short initial layers, as in part (b), approach that same infimum. Hence choice (1) has infimum $3b{L}^2/T^3$ but does not attain it, while choice (3) does. This distinction resolves the boundary-condition issue in the printed minimization request.