You want to learn a function that represents the dynamics of your system, either as a function of the system state or some output like a picture of the system. If you just apply some NN technique directly, this is possible but will require a lot of data since the NN doesn't have any knowledge of physics. If you use their system, you are trying to learn the Lagrangian of the system, which contains information on e.g. its symmetries, and bakes in physical knowledge into the learning problem at hand. As a result, less data is needed to learn the system dynamics.