Classical Mechanical Systems with one-and-a-half Degrees of Freedom and Vlasov Kinetic Equation

TL;DR

This paper develops a method to construct Liouville integrable Hamiltonian systems with one-and-a-half degrees of freedom using hydrodynamic reductions of the Vlasov kinetic equation, providing new classes of integrable potentials.

Contribution

It introduces an algorithmic approach to generate integrable Hamiltonian systems with non-polynomial potentials using hydrodynamic reductions of the Vlasov equation.

Findings

Constructed classes of integrable systems parameterized by arbitrary constants.

Presented examples with non-polynomial first integrals.

Demonstrated the use of hydrodynamic reductions in system construction.

Abstract

We consider non-stationary 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 non-polynomial dependencies w.r.t. to momentum. The constructed in this paper classes 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.