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

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 nonpolynomial dependency 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.

The Gibbons–Tsarev equation: symmetries, invariant solutions, and applications

“…we obtain a compatible system for S. This system has the following solutions that correspond to the invariant solutions (11), (15), (16), (17) of the Gibbons-Tsarev equation, respectively:…”

“…where τ = x + t y and function v(τ, y) is a solution of equation (1) with x replaced by τ . Since we know four explicit solutions (11), (15), (16), (17) of the Gibbons-Tsarev equation, after substituting them into (18) we obtain four explicit solutions of equation (4). They are, respectively,…”

“…In this section we study solutions of system (6) that correspond to the obtained solutions (11), (15), (16), (17) of the Gibbons-Tsarev equation. Each solution of (1) yields by substituting (5) into (6) a linear equation for w. Any linear equation admits trivial symmetries, that is, symmetries of the form w 0 ∂ ∂w , where w 0 is a (fixed) arbitrary solution of the equation.…”

“…Therefore for each choice of β there exists an infinite number of values for λ such that equation (19) is integrable in quadratures. (15) Without loss of generality it is possible to put β = 0 in solution (15). Then we have…”

Classical and Stochastic Laplacian Growth

“…For every t ≥ 0, the function f (z, t) has a continuous extension on the closure of H, and the extended function denoted also by f (z, t) satisfies equation (8.26) at least almost everywhere. Namely, for a given solution f (z, x, s) with the normalization (8.25), choose a solution φ(z, x, s) = ϕ(f (z, x, s)) where ϕ is an appropriate rapidly decreasing at infinities z → ±∞ function, see, e.g., [427]. where 0 ≤ τ ≤ t < ∞.…”