\(\(\(
\)\(\(rs = 0;\)

    \(h[x_, 
          p_] := \((1/2)\)*\((p^2)\) + \((1/
                2)\)*\((x^2*\((1 - rs*x)\))\);\)

    \(g = 1;\)

    \(v[t_] := D[h[x, p], p] /. {x -> x[t], p -> p[t]};\)

    \(w[t_] := \(-D[h[x, p], x]\) /. {x -> x[t], p -> p[t]};\)

    {v[t], w[t]}\)\)\)

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(\(rs = 0;\)

    \(tmax = 2.95;\)

    \(acgoal = 10;\)

    \(precgoal = 16;\)

    \(xin = 0.01;\)

    \(pin = 0.1;\)

    \(aa = 
        NDSolve[{\(x'\)[t] == v[t], \(p'\)[t] == w[t], x[0] == xin, 
            p[0] == pin}, {x, p}, 
	{t, 0, tmax}, MaxSteps -> 10000, 
	
          AccuracyGoal -> acgoal, PrecisionGoal -> precgoal];\)

    ParametricPlot[{\((x[t] /. aa)\)[\([1]\)], \((p[t] /. 
              aa)\)[\([1]\)]}, {t, 0, tmax}, PlotPoints -> 2000, 
      AxesLabel -> {x, p}, PlotRange -> All]


    \)\)\)

fig = 
      ParametricPlot[{Cos[t]/((x[t] /. aa)[[1]]), 
          Sin[t]/((x[t] /. aa)[[1]])}, {t, 0, tmax}, 
        PlotPoints -> 2000, PlotRange -> All]

\(\(\( (*\ 
      Integrating\ the\ differential\ equation\ *) \)\(
\)\(
\)\(\(rs = 
        0.1;\)

    \(tmax = 3;\)

    \(acgoal = 10;\)

    \(precgoal = 16;\)

    \(xin = 0.01;\)

    \(pin = 0.11;\)

    \(aa = 
        NDSolve[{\(x'\)[t] == v[t], \(p'\)[t] == w[t], x[0] == xin, 
            p[0] == pin}, {x, p}, 
	{t, 0, tmax}, MaxSteps -> 10000, 
	
          AccuracyGoal -> acgoal, PrecisionGoal -> precgoal];\)

    ParametricPlot[{\((x[t] /. aa)\)[\([1]\)], \((p[t] /. 
              aa)\)[\([1]\)]}, {t, 0, tmax}, PlotPoints -> 2000, 
      AxesLabel -> {x, p}, PlotRange -> All]


    \)\)\)

fig2 = 
      ParametricPlot[{Cos[t]/((x[t] /. aa)[[1]]), 
          Sin[t]/((x[t] /. aa)[[1]])}, {t, 0, tmax}, 
        PlotPoints -> 2000, PlotRange -> All, 
        PlotStyle -> RGBColor[1, 0, 0]]

Show[fig, fig2, PlotRange -> All]

K = pin^2 + xin^2*(1 - rs*xin)

((x[1] /. aa)[[1]])

Integrate[(K - u^2 - rs*u^3)^(-1/
            2), {u, 0.01, 0.0981412}]