Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field

Abstract

For finite dimensional Hamiltonian systems derived from 1+1 dimensional integrable systems, if they have Lax representations, then the Lax operator creates a set of conserved integrals. When these conserved integrals are in involution, it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the Liouville integrability of the Hamiltonian systems, at least for those derived from physical problems. In this paper, we give a counterexample based on the U(2) principal chiral field. It is proved that the finite dimensional Hamiltonian systems derived from the U(2) principal chiral field are Liouville integrable. Moreover, their Lax operator gives a set of involutive conserved integrals, but they are not enough to guarantee the integrability of the Hamiltonian systems.