Classical mechanical systems with one-and-a-half degrees of freedom and Vlasov kinetic equation

摘要

We consider nonstationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations, with the Hamiltonian H = p~2/2 + V(x,t), which are Liouville integrable. For this purpose we use the method of hydrodynamic reductions of the corresponding one-dimensional Vlasov kinetic equation. Also we present several examples of such systems with first integrals with nonpolynomial dependency with respect to momentum. The classes constructed in this paper of potential functions V(x, t) which give integrable systems with one-and-a-half degrees of freedom are parameterized by arbitrary number of constants.