I was wondering why physical systems "like" to go to the minimum of potential energy and I found this question, that tries to justify the minumum total potential energy principle. I was also reading some notes on classical mechanics that treat oscillations in the following way:
Suppose we have a system with $n$ generalised coordinates $q^i$. The Lagrangian of the system is
$$ L=~~\frac12 g_{ij}(q)\dot q^i\dot q^j-V(q).\tag{1}$$
I know that the Euler-Lagrange equation yields a system of second order differential equations in which it can be seen that there is a conservative field that can be expressed in terms of its potential. This field would "point" towards the minumum of potential energy.
In the notes they say that $g_{ij}(q)$ is a metric, and a symmetric array which may depend upon the configuration coordinates. I know that this $g_{ij}(q)$ should be related to the masses of the particles in the system, but I sincerely don't know why the word "metric" is used there. Is there any other name for that term $g_{ij}(q)$?
Also, if we take the Taylor series expansion about $x^i:=q^i-q^i_0$, where
$$\dfrac{\partial V}{\partial q^i}(q_0)=0,\tag{2}$$
we get
$$L\approx \frac12 g_{ij}(q_0)\dot q^i\dot q^j - \frac12 \dfrac{\partial^2 V}{\partial q^i\partial q^j}(q_0)x^ix^j.\tag{3}$$
From here it is clear what I said about the Euler Lagrange equations showing the thing about the field, so this would be a justification for the minimum total potential energy principle, wouldn't it?