Linearization of Lagrange equations

[stefano77]

TL;DR Summary

linearization lagrange equation

l am italian student from Milan university, so sorry for my bad english.
l am studying lagrange meccanics. We are linearizating lagrange equations. Here l don't understand how you can expand A matrix, how the function f is derivable, how the inverse matrix A is expanded? l am expanding with q0 center, x is the small displacement . G is a quadratic form.
O(|x|) is order of magnitude

 

[Paul Colby]

Do you mean, $$G(q,\dot q)=\sum_{ij}A_{ij}~q_i\dot q_j,$$ is linear in $\dot{q}_i$? Or,
$$G=\sum A_{ij}(q)\dot{q}_i\dot{q}_j$$

 

[PhDeezNutz]

Can we assume anything about $A$? Is it symmetric?

 

[stefano77]

L = T - V $T = \sum a_{ij}(q) q'_i q'_j$ $$ A= [a_{ij}] $$

$G_h(q,q')= \sum_{jl} \frac {\partial\a_{hj}} {\partial q_l} q'_l q'_j + \sum_{ij} \frac {\partial\a_{ij}} {2 \partial q_h} q'_l q'_j$
but l am interestin how to expand $[A(q)]^{-1}$ with $q_0$ as center

 

[stefano77]

A is symmetri...
it is kinetic energy matrix
L is a natural lagrangian

 

[PhDeezNutz]

A is symmetri...
it is kinetic energy matrix
L is a natural lagrangian

I think it's also diagonalizable. In which case

$A^{-1}_{ii} = \frac{1}{A_{ii}}$