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

2013/06/17 by Maxim V. Pavlov, Pavlov, Maxim V., Sergey P. Tsarev +1
Mathematics · Physics and Astronomy · #35L40 #35Q70 #35Q83 #37B55 #37K10 #70H05 #70H06 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:35L40 #msc:35Q70 #msc:35Q83 #msc:37B55 #msc:37K10 #msc:70H05 #msc:70H06 #nlin.SI

paper · pdf · doi:10.48550/arxiv.1306.3737

32 pages, standard LaTeX, in the second version: misprints corrected, Section "Multi-Time Generalization. Explicit Solutions" added

arxiv created 2013/08/05 · arxiv updated 2013/08/06

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=p²/2+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.

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