Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 13D (Variational Principles)

Let $y_{\epsilon }=y+\epsilon \eta$, where the variation $\eta$ satisfies $\eta (a)=\eta (b)=0$. Differentiating the functional at $\epsilon =0$ gives its first variationUsing integration by parts and the fixed endpoint conditions,If this vanishes for every admissible $\eta$, the fundamental lemma of the calculus of variations gives the Euler-Lagrange equation

Forone has $L_y=0$ andA circle of radius $R$ centred at $(0,c)$ on the $y$-axis satisfiesOn any arc that does not cross $y=c$,which is constant. Hence $d{L}_{y^{\prime }}/dx=0=L_y$, so the Euler-Lagrange equation holds and $\delta I=0$. These circles are the coordinate-swapped form of a Geodesic in the Poincare half-plane model.

Here the integrandhas no explicit $x$-dependence. The Beltrami identity therefore givesfor a positive constant $R$. Thuswhose nonvertical solutions are the semicircles

The endpoint $(a,a)$ and the prescribed value at $x=b$ both lie onbecauseThe required positive arc is consequentlyThe condition $b<2a$ keeps the endpoint above the $x$-axis. The sketch is the descending part of the upper semicircle of radius $a$ centred at $(a,0)$, from $(a,a)$ toThis is precisely a Geodesic in the Poincare half-plane model.