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In particular, say we seek a sufficiently smooth function $ u : [a,b] \to \mathbb{R} $ such that the solution $x$ to the differential equation with given initial conditions

$$ G(x, x', \dots, x^{(n)} ; u, u', \dots, u^{(m)}) = 0, \quad x^{(k)}(0) = \alpha_k, k=0,\dots n $$

extremizes (or at least is a critical point of) the functional

$$ \int_a^b F(x, x', \dots, x^{(n)} ; u, u', \dots, u^{(m)}) dt $$

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Yes you can. I think the particular theory you are looking for is discussed in the section "holonomic and non-holonomic constraints" of the following book :

The Calculus of Variations

This method can be used, for example, in order to compute the geodesics on a surface such as the sphere. The condition would simply be $g(x,y,z)=x^2+y^2+z^2-1=0$ and the functional optimize would be arc length.

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