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- Unable to comprehend why the kinetic term of the Hamiltonian constructed from the action of perturbative string motion is not positive definite

I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation:

$$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$

where they make a statement that the action is problematic because

I am unable to comprehend the reason for it and why the variable change was needed in the first place. Any clear explanation for my doubt would be truly beneficial!

$$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$

where they make a statement that the action is problematic because

**the kinetic term of the Hamiltonian constructed from this action is not positive definite**in some regions within the trapping potential, so they apply a variable change to solve this problem.I am unable to comprehend the reason for it and why the variable change was needed in the first place. Any clear explanation for my doubt would be truly beneficial!