I have studied the Pontryagin Maximum Principle. It is about solving a problem like this: $$ \begin{cases} \min \int_{t_0}^{t_1} f(t,x,u) dt\\ x(t_0) = x_0\\ \dot x = g(t,x,u)\\ u \in U \end{cases} $$
where $U$ is the set of admissible control. We have to find $u$ and $x$.
I am trying to link this theorem to a different problem, that is to find the min/max of the function $f(x_1, x_2) = x_1 x_2$ constrained to $g(x_1, x_2) = x_1^2+x_2^2 -1 = 0$. I used to solve this problem with Lagrange multipliers but now I'm wondering how can I use Pontryagin principle. For instance, in this last case, I noted that $f$ does not depend on $t$. Which is the functional I have to minimize?