Integrable Homogeneous Dissipative Dynamical Systems of an Arbitrary Odd Order
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
INTEGRABLE HOMOGENEOUS DISSIPATIVE DYNAMICAL SYSTEMS OF AN ARBITRARY ODD ORDER M. V. Shamolin Lomonosov Moscow State University, Institute of Mechanics 1, Michurinskii pr., Moscow 119192, Russia [email protected]
UDC 517.01+531.01
We establish the integrability of homogeneous (in some variables) dynamical systems with dissipation in the case of an arbitrary odd order and thereby generalize the results earlier obtained by the author in particular cases of such systems. Bibliography: 6 titles. This paper is devoted to integrable homogeneous dynamical systems with dissipation and generalizes the results earlier obtained by the author (cf., for example, [1]–[4] and the references therein). In particular, based on the study of particular cases of systems of the third and fifth order [3] and systems of seventh and ninth order [4], it becomes possible to develop a general approach to the study of such systems in the case of an arbitrary odd order.
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Systems Without External Forces
Let v, α, β = (β1 , . . . , βn−1 ), z = (z1 , . . . , zn ) be the phase variables in a system the righthand side of which is a homogeneous polynomial of degree 2 in the variables v, z with coefficients depending on α and β. Taking the variable q for the time (dq = vdt, d/dq = , v = 0), we consider the following (2n + 1)th order system as a system without external forces: v = Ψ(α, Z)v, ⎧ α = −Zn + b(Z12 + . . . + Zn2 )δ(α), ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ Zn = Γα11 (α, β)f12 (α)Zn−1 + Γα22 (α, β)f22 (α)g12 (β1 )Zn−2 + ... ⎪ ⎪ ⎪ ⎪ α 2 2 2 2 ⎪ + Γn−1,n−1 (α, β)fn−1 (α)gn−2 (β1 )hn−3 (β2 ) . . . i1 (βn−2 )Z12 − Zn Ψ(α, Z), ⎪ ⎪ ⎪ ⎨ f22 (α) 2 1 1 2 Z = [2Γ (α, β) + Df (α)]Z Z − Γ (α, β) − ... g1 (β1 )Zn−2 1 n−1 n n−1 α1 22 ⎪ f (α) ⎪ 1 ⎪ ⎪ ⎪ 2 (α) ⎪ fn−1 ⎪ 1 ⎪ (α, β) g 2 (β1 )h2n−3 (β2 ) . . . i21 (βn−2 )Z12 − Zn−1 Ψ(α, Z), − Γ ⎪ n−1,n−1 ⎪ ⎪ f1 (α) n−2 ⎪ ⎪ ⎩ ........................................................................
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 179-189. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0760
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⎧ n−2 n−2 (α, β) + Dfn−2 (α)]Z2 Zn − [2Γ1,n−2 (α, β) + Dgn−3 (β1 )]f1 (α)Z2 Zn−1 − . . . Z2 = [2Γα,n−2 ⎪ ⎪ ⎪ ⎪ ⎪ n−2 ⎪ − [2Γn−3,n−2 (α, β) + Dr1 (βn−3 )]fn−3 (α)gn−4 (β1 )hn−5 (β2 ) . . . s1 (βn−4 )Z2 Z3 ⎪ ⎪ ⎪ ⎪ 2 (α) g 2 (β ) h2 ⎪ fn−1 ⎪ r22 (βn−3 ) 2 n−2 1 n−3 (β2 ) n−2 ⎪ ⎪ . . . i1 (βn−2 )Z12 − Z2 Ψ(α, Z), − Γ (α, β) ⎪ n−1,n−1 ⎪ f (α) g (β ) h (β ) r (β ) ⎪ n−2 n−3 1 n−4 2 1 n−3 ⎪ ⎪ ⎪ n−1 n−1 ⎪ (α, β) + Dfn−1 (α)]Z1 Zn − [2Γ1,n−1 (α, β) + Dgn−2 (β1 )]f1 (α)Z1 Zn−1 Z1 = [2Γα,n−1 ⎪ ⎪ ⎪ ⎪ n−1 n−1 ⎨ − [2Γ2,n−1 (α, β) + Dhn−3 (β2 )]f2 (α)g1 (β1 )Z1 Zn−2 − . . . − [2Γn−2,n−1 (α, β) ⎪ + Di1 (βn−2 )]fn−2 (α)gn−3 (β1 )hn−4 (β2 ) . . . r1 (βn−3 )Z1 Z2 − Z1 Ψ(α, Z), ⎪ ⎪ ⎪ ⎪ ⎪ β = Zn−1 f1 (α), ⎪ 1 ⎪ ⎪ ⎪ ⎪ β = Z ⎪ n−2 f2 (α)g1 (β1 ), ⎪ 2 ⎪ ⎪ ⎪ ⎪ β3 = Zn−3 f3 (α)g2 (β1 )h1 (β2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ........................................................................ ⎪ ⎪ ⎪ ⎩ βn−1 = Z1 fn−1 (α)gn−2 (β1 )hn−3
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