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

This is an isoperimetric constraint, imposed on the whole path rather than separately at each time. Introduce a constant Lagrange multiplier $\lambda$ and replace the variational integrand by $\tilde{f}=f+\lambda u$. The modified Euler-Lagrange equations areSolve these together with the fixed endpoint conditions, treating $\lambda$ as an unknown constant, then choose its value so thatThe multiplier is constant because there is one scalar integral constraint. A condition imposed at every time would instead call for a time-dependent multiplier. As usual, this multiplier procedure assumes the constraint is regular, so its first variation does not vanish identically on all admissible variations.