Liouville

\[Theta][t_, \[Theta]0_, \[Theta]d0_] := (-(\[Theta]0^2/
      Sqrt[\[Theta]0^2 + \[Theta]d0^2]) - \[Theta]d0^2/
     Sqrt[\[Theta]0^2 + \[Theta]d0^2] ) Cos[
    2 Pi t +
     ArcTan[-(\[Theta]0/Sqrt[\[Theta]0^2 + \[Theta]d0^2]), \[Theta]d0/
      Sqrt[\[Theta]0^2 + \[Theta]d0^2]]];
\[Theta]d[
   t_, \[Theta]0_, \[Theta]d0_] := -(-(\[Theta]0^2/
       Sqrt[\[Theta]0^2 + \[Theta]d0^2]) - \[Theta]d0^2/
      Sqrt[\[Theta]0^2 + \[Theta]d0^2]) Sin[
    2 Pi t +
     ArcTan[-(\[Theta]0/Sqrt[\[Theta]0^2 + \[Theta]d0^2]), \[Theta]d0/
      Sqrt[\[Theta]0^2 + \[Theta]d0^2]]];
n = 100;
p0s = RandomVariate[
   MultinormalDistribution[{0, 0}, DiagonalMatrix[{0.05, 0.001}]],
   n];
p0s = Sort[p0s]; cols =
 ColorData["DarkRainbow"] /@ (Range[1/n, 1, 1/n]);

frame[t_] := (Show[
    Graphics[
     {
      PointSize[Medium],
      Point[{0, 0}],
      Opacity[0.7],
      Table[
       {
        cols[[i]],
        With[{p = {Sin[\[Theta][t, p0s[[i, 1]],
              p0s[[i, 2]]]], -Cos[\[Theta][t, p0s[[i, 1]],
               p0s[[i, 2]]]]},

          p2 = {Sin[\[Theta][t, p0s[[i, 1]], p0s[[i, 2]]]],
            1 + \[Theta]d[t, p0s[[i, 1]], p0s[[i, 2]]]}},
         {
          {Black, Opacity[0.1], Line[{p, p2}]},
          {Black, Opacity[0.1], Line[{{0, 0}, p}]},
          Point[p],
          Point[p2]
          }
         ]
        },
       {i, n}
       ],
      {White, Rectangle[{-0.2, 0.15}, {0.2, 0.25}]},
      Text[Style["position", Medium], {0, 0.2}],
      Rotate[Text[Style["momentum", Medium], {0, 0.2}],
       Pi/2, {0, 1}],
      Scale[
       Inset[
        Histogram[
         Table[Sin[\[Theta][t, p0s[[i, 1]], p0s[[i, 2]]]], {i, n}],
         {0.05},
         "PDF", PlotRange -> {{-1, 1}, {0, 10}},
         ChartStyle -> Directive[cols[[3]], Opacity[0.5]],
         Axes -> None]
        ,
        {0, 0}, {Center, Bottom}, {2, Automatic}
        ],
       {1, 0.5}
       ],
      Rotate[
       Scale[
        Inset[
         Histogram[
          Table[\[Theta]d[t, p0s[[i, 1]], p0s[[i, 2]]], {i, n}],
          {0.05},
          "PDF", PlotRange -> {{-1, 1}, {0, 10}},
          ChartStyle -> Directive[cols[[3]], Opacity[0.5]],
          Axes -> None]
         ,
         {0, 0}, {Center, Bottom}, {2, Automatic}
         ],
        {1, 0.5}
        ],
       Pi/2,
       {0, 1}
       ]
      },
     PlotRange -> {{-1, 1}, {-1.1, 1.5}}
     ]]);
Manipulate[
 frame[t],
 {t, 0, 1}]