In my lecturer's notes on Lagrangian mechanics in the chapter on normal modes they state the following:
Instead of rigid constraints, let us now consider a situation where the constraints are flexible so that the particles can move around their equilibrium positions. We assume that the system is described by $N$ generalised coordinates $q_i$. We also assume that it is natural, which means that the kinetic energy is a quadratic homogeneous function of the generalised velocities. We can write this as $$T = \frac{1}{2}\sum_{ij}a_{}(q_1, \ldots,q_N)\dot{q_i}\dot{q_j},$$ where the coefficients a_{ij} can depend on the coordinates $q_i$ but not on velocities $\dot{q_i}$. They can be chosen to be symmetric $(a_{ji}=a_{ij})$ without any loss of generality.
I am curious about two things here:
1) How can we just assume this form of the kinetic energy, what is the basis of this assumption? If I were to look at this subject myself with no notes, I would hardly just come up with this form.
2) $a_{ij}$ can be "chosen" to be symmetric, is this not necessary true from the commutative property of multiplication?
