# Static solutions of a series of coupled pendulums

## Homework Statement

The equation of motions of a series of pendulums coupled by a torsion spring is this:
##\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})##, where k is the torsion spring constant, m is the mass of a single pendulum, and l is the length of a single pendulum. We have N pendulums and we define ##\Phi_0 = \Phi_1##, and ##\Phi_{N+1}=\Phi_N##, in order to avoid the indices to go out of range at the boundaries. I'm looking for static solutions of this system and I'm expecting to get something like a static soliton.

## The Attempt at a Solution

First, for all ##i##, ##\ddot{\Phi_i} = 0##
Then, to make it a bit simpler in the beginning I tried with N = 3, and all constants are 1.
The equations are:
##\Phi_2-\Phi_1-\sin(\Phi_1)=0##
##\Phi_1+\Phi_3-2\Phi_2-\sin(\Phi_2) = 0##
##\Phi_2-\Phi_3-\sin(\Phi_3)=0##
If i add these equations, I get ##\sum_i \sin(\Phi_i)=0##, which is a nicer neccessary condition.

I'm struggling to see through this problem. There has to be infinite solutions. I can't give any random value to any ##\Phi## because if I put it into the equations and calculate the rest, my condition for the sums of the sines doesn't work. What tools/tricks can I use to narrow the domain?
I also tried ##\Phi_3=\arcsin(-\sin(\Phi_1)-\sin(\Phi_1+\sin(\Phi_1))## which is a very messy function.

The equation of motions of a series of pendulums coupled by a torsion spring is this:
¨Φi=−kml2(2Φi−Φi−1−Φi+1)
Ops, I made a mistake there. I left out the term from the gravitational potential. The correct equation of motion is:
##\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})-\frac{g}{l}\sin(\Phi_i)##

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