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I have this code:

Gtt = gtt[rb] + (gtt')[rb]*(ζ*nr[l]) + (1/2)*(gtt'')[rb]*
     (ζ*nr[l])^2 + (1/6)*(gtt''')[rb]*(ζ*nr[l])^3; 
G11 = g11[rb] + (g11')[rb]*(ζ*nr[l]) + (1/2)*(g11'')[rb]*
     (ζ*nr[l])^2 + (1/6)*(g11''')[rb]*(ζ*nr[l])^3; 
Grr = grr[rb] + (grr')[rb]*(ζ*nr[l]) + (1/2)*(grr'')[rb]*
     (ζ*nr[l])^2 + (1/6)*(grr''')[rb]*(ζ*nr[l])^3; 
gtt[r_] = (-r^2)*Exp[2*A[r]]*g[r]; 
g11[r_] = r^2*Exp[2*A[r]]; 
grr[r_] = Exp[2*A[r]]/(r^2*g[r]); 
X = {t, r[t, l], x1[t, l], x2, x3}; 
χ = {t, l}; 
r[t_, l_] = rb[l] + ζ[t, l]*nr[l]; 
x1[t_, l_] = xb[l] + ζ[t, l]*nx[l]; 
rl = D[r[t, l], l]; 
rt = D[r[t, l], t]; 
x1l = D[x1[t, l], l]; 
x1t = D[x1[t, l], t]; 
rb'[l] = Sqrt[1 - (g11[r0]*gtt[r0])/(g11[rb[l]]*gtt[rb[l]])]/
    Sqrt[grr[rb[l]]]; 
xb'[l] = (Sqrt[-gtt[rb[l]]]*Sqrt[(-g11[r0])*gtt[r0]])/
    (g11[rb[l]]*gtt[rb[l]]); 
nx[l_] = (Sqrt[grr[rb[l]]]*(rb')[l])/Sqrt[g11[rb[l]]]; 
nr[l_] = -((Sqrt[g11[rb[l]]]*(xb')[l])/Sqrt[grr[rb[l]]]); 
Huv = {{Gtt + Grr*rt^2 + G11*x1t^2, Grr*rt*rl + G11*x1t*x1l}, 
    {Grr*rt*rl + G11*x1t*x1l, Grr*rl^2 + G11*x1l^2}}; 
S = Sqrt[-Det[Huv]] /. Derivative[1, 0][ζ][t, l] -> OverDot[ζ, 1] /. 
     Derivative[0, 1][ζ][t, l] -> Derivative[1][ζ] /. rb[l] -> rb /. ζ[t, l] -> ζ

I am having trouble finding the coefficients of quadratic and cubic terms of ζ and its derivatives. Can somebody please help me find the coefficients?

In the code, I am figuring out the action of a string after it is perturbed slightly by ζ. The action is given by S at the end of the code. I must isolate the quadratic and the cubic terms separately from the S itself.

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  • $\begingroup$ Please see How to copy code from Mathematica so it looks good on this site $\endgroup$
    – MarcoB
    Commented Sep 22, 2023 at 16:03
  • $\begingroup$ Thank you. I'll try to do better. $\endgroup$
    – justJasper
    Commented Sep 22, 2023 at 16:07
  • $\begingroup$ Go ahead and format the existing code according to the suggestions in that post. At the moment your code is unreadable. It is unlikely you will get help in this form. $\endgroup$
    – MarcoB
    Commented Sep 22, 2023 at 18:12
  • $\begingroup$ Thanks for the help. It's much better now. $\endgroup$
    – justJasper
    Commented Sep 22, 2023 at 20:14

1 Answer 1

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We can use a trick here: Instead of writing just

r[t_, l_] = rb[l] + 𝜁[t, l]*nr[l]; 
x1[t_, l_] = xb[l] + 𝜁[t, l]*nx[l]; 

I used

r[t_, l_] = rb[l] + 𝛼*𝜁[t, l]*nr[l]; 
x1[t_, l_] = xb[l] + 𝛼*𝜁[t, l]*nx[l];

I introduced a new constant term $\alpha$.

Now I can easily distinguish all the quadratic and cubic terms using

action = Cancel[PowerExpand[
    -Collect[Coefficient[Normal[Series[S, {𝛼, 0, 3}]] /. Derivative[1, 0][𝜁][t, l] -> 
           OverDot[𝜁, 1] /.𝜁'[t, l] -> 𝜁' /. 
        𝜁[t, l] -> 𝜁, 𝛼^2], {(𝜁')^2, 𝜁^2, OverDot[𝜁, 1]^2}]]]
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