# Finding Coefficients in a Perturbation Problem related to Chaotic Dynamics

I am trying to reproduce the results from this paper: Chaotic dynamics of a suspended string in a gravitational background with magnetic field

I am currently stuck with finding the coefficients. First I define all the variables, as given in the paper:


gtt[r_] := (-r^2)*f[r];
g11[r_] := r^2*h[r];
grr[r_] := 1/(r^2*f[r]);
Derivative[xb][l_] := Sqrt[(-gtt[r0])*g11[r0]]/
(Sqrt[-gtt[rb[l]]]*g11[rb[l]]);
Derivative[rb][l_] :=
Sqrt[(-gtt[rb[l]])*g11[rb[l]] + gtt[r0]*g11[r0]]/
Sqrt[(-gtt[rb[l]])*g11[rb[l]]*grr[rb[l]]];
nx[l_] := Sqrt[grr[rb[l]]/g11[rb[l]]]*Derivative[rb][l];
nr[l_] := (-Sqrt[g11[rb[l]]/grr[rb[l]]])*Derivative[xb][l];
r[t_, l_] := rb[l] + \[Alpha]*\[Xi][t, l]*nr[l];
x[t_, l_] := xb[l] + \[Alpha]*\[Xi][t, l]*nx[l];
rd[t_, l_] := D[r[t, l], t];
xd[t_, l_] := D[x[t, l], t];
rp[t_, l_] := D[r[t, l], l];
xp[t_, l_] := D[x[t, l], l];



Then I expand the metric function around the static solution rb[l]to the third order in \[Xi][t, l] :

GTT[r_] := Normal[Series[gtt[r], {r, rb[l], 3}]] /.
r -> r[t, l];
G11[r_] := Normal[Series[g11[r], {r, rb[l], 3}]] /.
r -> r[t, l];
GRR[r_] := Normal[Series[grr[r], {r, rb[l], 3}]] /.
r -> r[t, l];


Then I find the corresponding NG action, comprising of third-order terms, and collect their coefficients:


Gab := {{GTT[r] + G11[r]*xd^2 + GRR[r]*rd^2, G11[r]*xd*xp +
GRR[r]*rd*rp}, {G11[r]*xd*xp + GRR[r]*rd*rp,
G11[r]*xp^2 + GRR[r]*rp^2}} /. xd -> xd[t, l] /.
rd -> rd[t, l] /. xp -> xp[t, l] /. rp -> rp[t, l];
g = Sqrt[-Det[Gab]];
Action = -Cancel[PowerExpand[Collect[Coefficient[
Normal[Series[g, {\[Alpha], 0, 3}]] /. \[Xi][t, l] -> \[Xi] /.
Derivative[1, 0][\[Xi]][t, l] -> OverDot[\[Xi], 1] /.
Derivative[0, 1][\[Xi]][t, l] -> Derivative[\[Xi]], \[Alpha]^3],
{\[Xi]*Derivative[\[Xi]]^2, \[Xi]^3, \[Xi]*OverDot[\[Xi], 1]^2}]]];



where specifically, the coefficient of D2[l]is found by:


D2[l_] = Simplify[PowerExpand[Coefficient[Action,
\[Xi]*OverDot[\[Xi], 1]^2]]]



What modifications shall I make in the code to get matching results?

• Please see How to copy code from Mathematica so it looks good on this site for guidance on how to include complex traditional form expressions in your post. Feb 10 at 12:34
• @MarcoB thanks for the advice. I will edit my question using the steps mentioned in the meta site. Feb 10 at 12:39