Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 9 / d / Solution

The needed partial derivatives areSubstitution into the Euler-Lagrange equations givesThe ${\gamma }_1$ terms cancel in the first equation, leavingThese are accompanied by $x(0)=y(0)=0$ and $x(T)=y(T)=S$. The cancellation is consistent with part (b): the term proportional to ${\gamma }_1$ is the total derivative of $\alpha {\gamma }_1(x+\beta x^2/2)$.