# Lagrangian for a series of $N$ one-dimensional coupled oscillators

Physics Asked by George Dixon on December 7, 2020

# The Problem

I have been given the following Lagrangian of a series of $$N$$ one-dimensional coupled oscillators, with distance a. I have also been given the boundary conditions: $$y_0=0=y_{N+1},$$ but have been given little explanation on where this came from.
$$L=sum_{i=1}^{N}frac{1}{2}mdot{y_i}^2-sum_{i=0}^Nfrac{1}{2}Tleft(frac{y_{i+1}-y_i}{a}right)^2,$$
I am having trouble understanding how this lagrangian has been formed, in particular; why there is an a on the denominator of the second squared term.

# My attempt at understanding:

I understand that the Lagrangian is expressed:
$$L=KE-U,$$
The kinetic energy contribution of each of the $$N$$ coupled oscillators is simply:
$$KE=frac{1}{2}mdot{y_i}^2,$$
Where $$y_i$$ is the displacement of the $$i$$th oscillator.
Assuming all of the oscillators have equal mass, and none are fixed we should have the total kinetic energy:
$$KE=sum_{i=0}^{N}frac{1}{2}mdot{y_i}^2,$$
Generally, the potential energy of a spring is:
$$U=frac{1}{2}kx^2,$$
Where x represents the change in displacement, thus in our case the potential ‘contributed’ by each term is simply:
$$U=frac{1}{2}k(y_{i+1}-y_i^2),$$
because the terms in the bracket represent the ‘stretch’ of each of the springs, i.e. displacement.
I assume the person who gave me the original Lagrangian has given me a spring constant of $$T$$ rather than $$k$$, but surely this should return a lagrangian:
$$L=sum_{i=0}^{N}frac{1}{2}mdot{y_i}^2-sum_{i=0}^Nfrac{1}{2}Tleft(y_{i+1}-y_iright)^2,$$
This is clearly not the same as what I was given, the first summation index is out by one place, I can assume this means that we are considering the first oscillator to be fixed, and thus the first velocity term goes to zero leaving:
$$L=sum_{i=1}^{N}frac{1}{2}mdot{y_i}^2-sum_{i=0}^Nfrac{1}{2}Tleft(y_{i+1}-y_iright)^2,$$
Dividing by a would give the ratio of the change w.r.t the original length of the spring, so I don’t understand why this would be done? If anyone can see something that I am missing/done wrong, please let me know!