How many first integrals imply integrability in infinite-dimensional Hamilton system

Abstract

The classical Liouville integrability means that there exist n independent first integrals in involution for 2 n -dimensional phase space. However, in the mfinite-dimensional case, an infinite number of independent first integrals in involution does not indicate that the system is solvable. How many first integrals do we need to make the system solvable? To answer the question, we obtain an infinite-dimensional Hamilton–Jacobi theory, and prove an infinite-dimensional Liouville theorem . Based on the theorem, we give a criterion of integrability of infinite-dimensional system. According to the criterion, we give a modified definition of the Liouville integrability in infinite dimension. This definition which dose not depend on the inverse scattering method is a natural generalization of the classical Liouville integrability. In general, an infinite number of first integrals is complete if all action variables of a Hamilton system such as KdV equation can be reconstmcted by the set of first integrals. Essentially, our theory only provides an equivalent representation to the statement that an infinite-dimensional dynamical system is integrable if it can be reformulated in terms of action-angle variables. However, our results answer the question on the relation between the first integrals and solvability of infinite-dimensional Hamilton system.