How to understand potential energy in Lagrangian

Hi guys,

So I'm trying to understand why the potential energy of a Lagrangian is the way it is.

The system I'm considering is a closed necklace of N beads, each of mass m. Each bead interacts only with its nearest neighbour.

First let me make some comments:
1) Each bead is labeled with a generalised coordinate $q_{i}$
2) there is no explicit time dependence of the generalised coordinates
3) the system is conservative, so the potential is a function only of the generalised coordinates: $V=V(q_{1},q_{2},\dots q_{N})$,

The Lagrangian for this system is

$L=\frac{1}{2}\sum_{i=1}^{N}m\dot{q}_{i}^{2}-\frac{1}{2}\sum_{i=1}^{N}hq_{i}^{2}-k(q_{i}-q_{i+1})^{2}$.

I dont understand why the potential has this form. I think i know where the second term $-k(q_{i}-q_{i+1})^{2}$ comes from - its due to the harmonic approximation. But what about the first term?

To be honest I dont think $h$ has a particular meaning - it's just a constant for dimensional consistency perhaps? something like that