Solving a differential equation involving terms depending on integral over an ImplicitRegion, which depends on the solution of the equation

Question

I have a somewhat convoluted problem and am not a Mathematica expert. I am trying to solve (numerically) the Euler-Lagrange equations for a complex Lagrangian. The difficulty lies in the fact that L(x(t), xdot(t), t) depends on the integral over a region that itself depends on x(t).

The issue is that when I supply the Euler-Lagrange equations to Mathematica, it tries to evaluate the ImplicitRegion before solving, which of course fails as the region depends on x(t).

Is there a way to formulate this problem so that Mathematica evaluates the region and integral dynamically? Defining the region analytically based on x(t) and defining the integral explicitly is not an option due to the dynamic complexity of the region.

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Added detail:

Here's the Lagrangian:

L[q_, t_] := Module[{ R, S, \[ScriptCapitalR], \[ScriptCapitalS]},
  R = ((outerL2[q, t][[1]] <= x <= innerL2[q, t][[1]]) && (y <= 
        outerL1[q, t][[2]])) || ((innerL1[q, t][[2]] <= y <= 
        outerL1[q, t][[2]]) && (x >= innerL2[q, t][[1]]));
  S = ((-1 <= x <= 0) && (-3 <= y <= 1));
  \[ScriptCapitalR] = ImplicitRegion[R, {{x, -3, 3}, {y, -3, 3}}];
  \[ScriptCapitalS] = ImplicitRegion[S, {{x, -3, 3}, {y, -3, 3}}];
  Return[4 - 
    NIntegrate[
     1, {x, y} \[Element] 
      RegionIntersection[\[ScriptCapitalR], \[ScriptCapitalS]]]]
  ]