# Poincaré sections does not work

These are the equations for Poincaré sections:

    M = 0.5; a = 0.2; s = 0.1;M = 1/2; Δ = r^2 - 2*M*r[t] + a^2;
W = r[t]^2 + a^2*Cos[θ[t]]^2;
h = s*M^3*r[t]/W^2;
gtt = -Simplify[(1 - 2*M*r[t]/W)*(1 + h)];
grr = Simplify[W (1 + h)/(Δ + a^2*Sin[θ[t]]^2*h)];
gss = W;
gff = Simplify[(r[t]^2 + a^2 + 2 a^2*M*r[t]*Sin[θ[t]]^2/W)*
Sin[θ[t]]^2 + h*a^2*Sin[θ[t]]^4*(1 + 2*M*r[t]/W)]
gtf = -2*a*M*r[t]*Sin[θ[t]]^2*(1 + h)/W
L = 1/2 (gtt*en^2 + grr*D[r[t], t]^2 + gss*D[θ[t], t]^2 +
gff*ln^2 + 2 gtf*en*ln);


I would like to plot the Poincaré section for collection of (r[t], Derivative[1][r][t]), I wrote:

   date = Block[{en = 0.95, ln = 1.5},
Reap[NDSolve[{grr*(r^′′)[t] ==
D[L, r[t]] - D[grr, r[t]]*Derivative[1][r][t]^2 -
D[grr, θ[t]]*Derivative[1][r][t]*
Derivative[1][θ][t],
gss*(θ^′′)[t] ==
D[L, θ[t]] -
D[gss, θ[t]]*Derivative[1][θ][t]^2 -
D[gss, r[t]]*Derivative[1][r][t]*Derivative[1][θ][t],
r[0] == 8, θ[0] == Pi/2,
WhenEvent[Derivative[1][r][0] == 0,
Sow[{r[t], Derivative[1][r][t]}]]}, {}, {t, 100000},
MaxSteps -> ∞]]][[-1, 1]]; ListPlot[data,
ImageSize -> Medium, PlotRange -> {{0, 20}, All},
PlotStyle -> PointSize[0.0025]]


But it does not work.

The code in the question contain (r^′′)[t] and (θ^′′)[t], which are, I presume, transcription errors. They should be r''[t] and θ''[t]. With these corrections, the code contains two errors. First, Δ should be given by

Δ = r[t]^2 - 2*M*r[t] + a^2


Second, values for r'[0]and θ'[0] must be provided to NDSolve. For all values of r'[0]and θ'[0] that I have tried, r[t] grows exponentially, and at most one value of {r[t], r'[t]} is returned by WhenEvent. So, for instance, with r'[0] == -10, θ'[0] == -2, one obtains (with {} replaced by {r[t], θ[t]} in NDSolve and [[-1, 1]] removed),

LogPlot[Evaluate[{r[t], θ[t]} /. date[[1, 1]]], {t, 0, 10}]


shows that r[t] grows exponentially and θ[t] becomes constant after only a single oscillation. Thus,

date[[-1, 1]]
(* {{6.96449, 1.87522*10^-12}} *)


Perhaps a better choice for r'[0]and θ'[0] would yield an oscillatory solution, but finding it is not trivial.