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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 [itex]q_{i}[/itex]

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: [itex]V=V(q_{1},q_{2},\dots q_{N})[/itex],

The Lagrangian for this system is

[itex]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}[/itex].

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