Today I’ll say a few words on completely integrable systems. There is a nice result of Liouville-Arnold that describes how these are fibered, at least when the fibers are compact. It will provide a useful illustration of the ideas discussed already.

Charles Siegel of Rigorous Trivialties has a post on this topic at a much more sophisticated level. There was also a successful MO question by Gil Kalai about them.

A completely integrable system is a symplectic manifold (with symplectic form ) of dimension together with smooth functions in involution, i.e.

for all , and with the differentials independent at each cotangent space . Physically, this corresponds to a set of conservation laws. For instance, if is energy, then a particle goes through the integral curves of . Thus because , is constant on these integral curves.

Now, fix a point . We consider the map