g = 9.81;
a = 1.5;
Subscript[w, 0] = Sqrt[g/a]

sol1 = NDSolve[
  {x1'[t] == x2[t],
   x2'[t] ==
    [Minus]1/2  x4'[t] Cos[x1[t] [Minus] x3[t]] [Minus]
     Subscript[w, 0] ^2  Sin[x1[t]] [Minus]
     x4[t] Sin[x1[t] [Minus] x3[t]],
   x3'[t] == x4[t],
   x4'[t] == [Minus]x2'[t] Cos[x1[t] [Minus] x3[t]] [Minus]
     Subscript[w, 0] ^2 Sin[x3[t]] + x2[t]^2 Sin[x1[t] [Minus] x3[t]],
   x1[0] == 0, x2[0] == 0, x3[0] == [Pi]/9, x4[0] == 0},
  {x1, x2, x3, x4}, {t, 0, 50},
  Method -> {"EquationSimplification" -> "Residual"}];
x1sol[t_] := x1[t] /. sol1[[1]]
 x2sol[t_] := x2[t] /. sol1[[1]]
x3sol[t_] := x3[t] /. sol1[[1]]
x4sol[t_] := x4[t] /. sol1[[1]]

Plot[x1sol[t], {t, 0, 50}, Frame [RightArrow] True,
 PlotRange [RightArrow] {All, All},
 FrameLabel [Minus] > {StyleForm["time (s) ",
    FontSize [RightArrow] 14],
   StyleForm["[CurlyPhi]1 (rad)", FontSize [RightArrow] 14]}]
Plot[x3sol[t], {t, 0, 50}, Frame [RightArrow] True,
 PlotRange [RightArrow] {All, All},
 FrameLabel [Minus] > {StyleForm["time (s) ",
    FontSize [RightArrow] 14],
   StyleForm["[CurlyPhi]2 (rad)", FontSize [RightArrow] 14]}]