I am currently reading part of Franz Gross' Relativistic Quantum Mechanics and Field Theory, and am getting stuck on a particular point in chapter 1 where he attempts to motivate field theory by taking the continuum limit of a series of oscillators arranged on a string.
We are given that the kinetic energy and potential energy are:
$$ KE = \frac{1}{2}m\sum_{i=0}^{N-1}{(\frac{\partial\phi}{\partial t})} $$
$$ PE = \frac{1}{2}k\sum_{i=0}^{N-1}{(\phi_{i+1}-\phi_{i})^2} $$
We are free to introduce mutually cancelling distance values $l$ as we need to take the continuum limit and form integrals, a la:
$$ PE = \frac{1}{2}kl\sum_{i=0}^{N-1}{l(\frac{\phi_{i+1}-\phi_{i}}{l})^2} \rightarrow \frac{1}{2}T\int_{0}^{L}{dz(\frac{\partial\phi}{\partial z})^2} $$
Now, the Lagrangian can be formed by taking $KE-PE$, or:
$$ L = \int_{0}^{L}{dz\frac{1}{2}\mu(\frac{\partial\phi}{\partial t})^2}- \frac{1}{2}T{(\frac{\partial\phi}{\partial z})^2} = \int_{0}^{L}{dz \textit{L(z,t)} } $$
Our next goal is to formulate the necessary partial derivatives of this expression to form the Euler-Lagrange equations of motion, solutions of the system of equations:
$$ \frac{d}{dt} \frac{\partial L}{\partial \dot\phi_i} - \frac{\partial L}{\partial \phi_i} = 0 $$
And this is where I get lost. This is page 6, and the very next step taken after mentioning the Euler-Lagrange equations is to produce the form of $ \frac{\partial L}{\partial \phi_i}$. All that is stated is:
"...in the continuum limit as the number of oscillators N $\rightarrow \infty $, $$ \frac{\partial L}{\partial \phi_i} = -k[-(\phi_{i+1}-\phi_i)+(\phi_{i}-\phi_{i-1})] $$ $$ = kl[\frac{(\phi_{i+1}-\phi_i)}{l}-\frac{(\phi_{i}-\phi_{i-1})}{l}] $$ $$ \rightarrow_{small\space\space\space l} lT\frac{1}{l}[\frac{\partial\phi(z^+,t)}{\partial z}-\frac{\partial\phi(z^-,t)}{\partial z}] $$ $$ \rightarrow_{absorb \space \sqrt{T}} -l\sqrt{T}\frac{\partial}{\partial z}\frac{\partial L(z,t)}{\partial (\frac{\partial \phi}{\partial z})} $$"
Where $L(z,t)$ is the Lagrangian density, the integrand in the original Lagrangian expression.
This first step, where we automatically jump into the expression $ \frac{\partial L}{\partial \phi_i} = -k[-(\phi_{i+1}-\phi_i)+(\phi_{i}-\phi_{i-1})]$ does not make sense to me.
I can see that the expression inside the brackets is the numerator of a second order central difference approximation, ie: $\frac{\phi_{i+1}+\phi_{i-1}-2\phi_{i}}{l^2} \rightarrow \frac{\partial^2 \phi}{\partial z^2} $, but I cannot see how we get here in the first place.
By my reckoning, the expression inside the $PE$ expression reduces to $$ \frac{1}{2}k\sum_{i=0}^{N-1}{(\phi_{i+1}^2+\phi_{i}^2-2\phi_{i+1}\phi_{i})} $$
And I can see how we might gain a term of the kind:
$$ \frac{\partial}{\partial \phi_i}\frac{1}{2}k\sum_{i=0}^{N-1}{(\phi_{i+1}^2+\phi_{i}^2-2\phi_{i+1}\phi_{i})} = \frac{2}{2}k\sum_{i=0}^{N-1}{(\phi_{i}-\phi_{i+1})} $$
But I do not see how this can then be taken forward as they do. My suspicion is that there is something simply I am missing, as other workups of similar material (as in continuous oscillations on a rod instead of a circular string) also jump directly into this simplification (see https://www.physics.rutgers.edu/grad/618/lects/classMech_2.pdf).
Can anyone explain to me how this mid-point result is achieved?
