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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[1][xb][l_] := Sqrt[(-gtt[r0])*g11[r0]]/
        (Sqrt[-gtt[rb[l]]]*g11[rb[l]]); 
    Derivative[1][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[1][rb][l]; 
    nr[l_] := (-Sqrt[g11[rb[l]]/grr[rb[l]]])*Derivative[1][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[1][\[Xi]], \[Alpha]^3], 
         {\[Xi]*Derivative[1][\[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?

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