(*For example:*) L = [Theta][t]^2 + 2 Sin[[Theta]'[t]]
D[L, [Theta]'[t] ]
D[L, [Theta][t]]

Needs["VariationalMethods`"]
ClearAll[Global];
variables = {Subscript[[Theta], 1][t], Subscript[[Theta], 2][t]};
Subscript[r, 1] =
  Subscript[l,
   1] {Sin[Subscript[[Theta], 1][t]], -Cos[
      Subscript[[Theta], 1][t]]};
Subscript[r, 2] =
  Subscript[r, 1] +
   Subscript[l,
    2] {Sin[Subscript[[Theta], 2][t]], -Cos[
       Subscript[[Theta], 2][t]]};

lagrangian =
 Subscript[m, 1]/2 D[Subscript[r, 1], t].D[Subscript[r, 1], t] +
  Subscript[m, 2]/2 D[Subscript[r, 2], t].D[Subscript[r, 2], t] -
  g {0, 1}.(Subscript[m, 1] Subscript[r, 1] +
      Subscript[m, 2] Subscript[r, 2])
eqs = EulerEquations[lagrangian, variables, t];

initial = {Subscript[[Theta], 1][0] == Pi/2,
   Subscript[[Theta], 2][0] == Pi/2, Subscript[[Theta], 1]'[0] == 0,
Subscript[[Theta], 2]'[0] == 0};

Subscript[l, 1] = 1;
Subscript[l, 2] = 1;
Subscript[m, 1] = 1;
Subscript[m, 2] = 1;
g = 9.81;
tMax = 20;

sol = First[
   NDSolve[Join[eqs, initial],
    Head /@ variables,
    {t, 0, tMax}]
   ];

D[lagrangian, [Theta]'[t]/.sol]
D[lagrangian, [Theta][t]/.sol]
D[lagrangian, [Theta]'[t]]
D[lagrangian, [Theta][t]]