Many Body Systems are everywhere (matter physics, superconductivity, nanoscience, in general any category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting entities).
Imagine a ball moving in a force field spatially and temporally varying. If we know the force on the ball at every point in space and time then we can easily calculate the trajectory of the ball by using Newton’s laws. Problem is easy even if the field is varying, as long as the field variation does not depend upon the position or velocity of the ball. Now let’s attach a second ball to the first one by means of a spring. In general the spring will either be stretched or compressed the balls will exert a force with each other. The motion of the balls is still described by Newton’s classical laws of motion, but now the motion of the first ball A linked with the motion of the second one. That is, we cannot compute the trajectories of the balls separately. If there are many balls then the problem rapidly becomes complex. It should be stressed that the balls are still subject to Newton’s laws, but the difficulty of solving the motion equations increases rapidly because of the coupling. This is, in essence, a Many Body System : a sort of network of interacting entities in a field.
Now imagine “replacing” the ball with a neuron and the force field (spatially and temporally varying) with the chemical field of neurotransmitters. Now add a second neuron, attached to the first one with some synapsis and so on… that’s modelling brain’s neural network as a Many Body System. Neurons might be simple, but complexity (with its wonderful properties) rapidly emerges because of their coupling.
Also future networks and even more social networks could be seen, in principle, as Many Body Systems (think about people behaviors coupling in certain socio-cultural fields): amazingly concepts and mathematical models being used in many-body mechanics will be valuable instruments to describe and master complexity of networks problems.