= 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 $\widetilde f=f+\lambda u$. The modified <Euler-Lagrange equations> are
$$
\boxed{\frac d{dt}f_{\dot x}-f_x-\lambda u_x=0,\qquad\frac d{dt}f_{\dot y}-f_y-\lambda u_y=0.}
$$
Solve these together with the fixed endpoint conditions, treating $\lambda$ as an unknown constant, then choose its value so that
$$
\boxed{\int_0^Tu(x(t),y(t))\,dt=0.}
$$
The 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>.
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