Neglecting higher order terms in a Lagrangian

Question

I have a lagrangian which is modified by variable change. I want to neglect all the 4th order and higher terms in the new lagrangian. The code being used is given below:

    lagrangian = 
      Sum[c[n]'[t]^2 - c[n][t]^2 ωsq[n], {n, {0, 1}}] + 
       11.3 c[0][t]^3 + 21.5 c[0][t] c[1][t]^2 + 
       10.7 c[0][t] c[0]'[t]^2 + 3.32 c[0][t] c[1]'[t]^2 + 
       6.64 c[0]'[t] c[1][t] c[1]'[t];
    
    c[0][t_] := C[0][t] + α1 C[0][t]^2 + α2 C[1][t]^2;
    
    c[1][t_] := C[1][t] + α3 C[0][t] C[1][t];
    
    α1 = -1.5; α2 = -0.5; α3 = -1;
    
    n = Expand[lagrangian]

I tried using this code to neglect the higher-order terms:

vars = {C[0][t], C[1][t]};
Normal[Series[n /. Thread[vars -> m*vars], {m, 0, 3}]] /. m -> 1

which does part of the job but there are still some higher-order derivative terms present. How to go about removing them too from the modified lagrangian?

Answer 1

If I understand correctly you just have to include the derivatives in the vars

vars = {C[0][t], C[1][t], C[0]'[t], C[1]'[t]};
Normal[Series[n /. Thread[vars -> m*vars], {m, 0, 3}]] /. m -> 1

Answer 2

Another approach to insert the "smallness" parameter is:

Normal[
    Series[n /. C[i_] :> Function[m C[i][#]], {m, 0, 3}]
] /. m->1

-ωsq[0] C[0][t]^2 + 11.3 C[0][t]^3 + 3. ωsq[0] C[0][t]^3 - ωsq[1] C[1][t]^2 + 21.5 C[0][t] C[1][t]^2 + 1. ωsq[0] C[0][t] C[1][t]^2 + 2 ωsq[1] C[0][t] C[1][t]^2 + C[0]'[t]^2 + 4.7 C[0][t] C[0]'[t]^2 + 2.64 C[1][t] C[0]'[t] C[1]'[t] + C[1]'[t]^2 + 1.32 C[0][t] C[1]'[t]^2

Answer 3

Another way to do this is to "tag" the expressions in the original code with some formal parameter $\epsilon$:

c[0][t_] := \[Epsilon] C[0][t] + \[Epsilon]^2 \[Alpha]1 C[0][t]^2 + \[Epsilon]^2 \[Alpha]2 C[1][t]^2;

c[1][t_] := \[Epsilon] C[1][t] + \[Epsilon]^2 \[Alpha]3 C[0][t] C[1][t];

\[Alpha]1 = -1.5; \[Alpha]2 = -0.5; \[Alpha]3 = -1;

newlagrangian = 
  Sum[c[n]'[t]^2 - c[n][t]^2 \[Omega]sq[n], {n, {0, 1}}] + 
   11.3 c[0][t]^3 + 21.5 c[0][t] c[1][t]^2 + 
   10.7 c[0][t] c[0]'[t]^2 + 3.32 c[0][t] c[1]'[t]^2 + 
   6.64 c[0]'[t] c[1][t] c[1]'[t];

newn = Expand[newlagrangian]

You can then take the coefficients order by order in $\epsilon$, either using Normal and Series or by using CoefficientList:

result = Total[Take[CoefficientList[newn, \[Epsilon]], 4]]

The 4 in there is because you want to keep the first four terms of the coefficient list, which correspond to terms of order $\epsilon^0$, $\epsilon^1$, $\epsilon^2$, and $\epsilon^3$.