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Nikolay Tyurin: *Integrability on compact phase spaces: standard and non standard Lagrangian tori*

We discuss classical mechanical problem for the case of compact phase spaces. The main example which we present here is the projective space, endowed with the standard Kahler structure. We describe an integrable system which live on the projective space and according to the famous Liouville theorem the system gives us a fibration by lagrangian tori.

An important problem arose both in mathematical physics and pure mathematics asks are there lagrangian tori which are not equivalent to the Liouville tori? We present examples of such non standard lagrangian tori.