Polynomial integrals of dynamical systems and the lax reduction

Abstract

Lemma 5. If the class 7) is determinate, then each space X E 7 ) \ "P2n is 7)2n_l(O)-preuniversal.Proof. We suppose that the class :P is determinate and fix spaces X E 7~\:P2n and A E 7~2,~-1(0). Let us embed the space X into the Hilbert cube Q and the space A into the Cantor cube 2 ~. Let a: 2 ~ --+ Q be an arbitrary surjective mapping. By [6, 37.1], we have a-l(X) E P. Moreover, a-l(X) r P=n (the assumption that a-l(X) E :P2n implies 2 `0 \ a-l(X) E P2n-1, and therefore, Q \ X = a(2" \ a-l(X)) E P2n-1, whence X E 7~2n, a contradiction). Since 7)2n_1 C P2n+2 and the class 7 ) is determinate,

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

“…Substitution of these expressions into (6) leads to corresponding hydrodynamic type systems (see [7]) for the hydrodynamic variables u • (x, t), v • (x, t), w • (x, t) or a • (x, t) in the aforementioned reductions. We will use for simplicity the notation u = (u 1 (x, t), .…”

“…This is nothing but the well-known dispersionless limit of the Gelfand-Dikij reduction for the remarkable Kadomtsev-Petviashvili hierarchy ( [19]). This unexpected relationship between ansatz (9) for the integrable reductions of the Vlasov kinetic equation, classical mechanical systems with one-and-a-half degrees of freedom and integrable hydrodynamic type systems was implicitly or explicitly observed in a number of publications, in particular [18,6,2,32]. Let us give the following citation from [18] 1 : "It has long been remarked that all the known first integrals of classical mechanical systems are polynomial 2 w.r.t.…”

“…possess n functions F i (q, p,t) such that In this paper we restrict our considerations to the one-dimensional non-autonomous case only. Such systems are usually called systems with one-and-a-half degrees of freedom ( [18,6,2]). Everywhere below we identify q 1 = x, p 1 = p. Definition 1 We call Hamilton's equationṡ…”

On local description of two-dimensional geodesic flows with a polynomial first integral

“…Partially this loss of interest to the local problem was not only due to the importance of the global problems; it seems reasonable to ascribe this loss of interest to absence of new ideas for the local integrability problem. The situation, in our opinion, has changed in the last decades after publications [5], [1] where the authors had remarked that the equations for the coefficients of first integrals polynomial in momenta for two specific low-dimensional cases belong to the class of diagonalizable hydrodynamic type systems integrable by differential-geometric means; the appropriate theory for such nonlinear systems of PDEs was developed in the very end of the XX-th century (cf. [4,14]).…”

“…[4,14]). In [12], developing the preliminary results of [5], we demonstrated how to apply the techniques of integrable hydrodynamic type systems to the case of so called one-and-a-halfdimensional systems (one-dimensional mechanical systems with the potential depending on time). Below we investigate (using a bit more sophisticated technologies) the problem of local description of two-dimensional Riemannian metrics with geodesic flows possessing a polynomial first integral of arbitrary high degree.…”