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We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B
n, C
n, and D
n. 相似文献
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We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen. 相似文献
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We propose an integrable deformation of the known model of two interacting tops on the algebra so(p,q). We consider particular cases including the generalized Lagrange and Kovalevskaya tops. We construct the Lax matrices and the corresponding classical R-matrices. 相似文献
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Theoretical and Mathematical Physics - We discuss the properties of second-order Killing tensors in three-dimensional Euclidean space that guarantee the existence of a third integral of motion... 相似文献
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Andrey V. Tsiganov 《Regular and Chaotic Dynamics》2014,19(2):185-197
The necessary number of commuting vector fields for the Chaplygin ball in the absolute space is constructed. We propose to get these vector fields in the framework of the Poisson geometry similar to Hamiltonian mechanics. 相似文献
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A. V. Tsiganov 《Proceedings of the Steklov Institute of Mathematics》2016,295(1):243-273
We consider an algorithm for constructing auto-Bäcklund transformations for finitedimensional Hamiltonian systems whose integration reduces to the inversion of the Abel map. In this case, using equations of motion, one can construct Abel differential equations and identify the sought Bäcklund transformation with the well-known equivalence relation between the roots of the Abel polynomial. As examples, we construct Bäcklund transformations for the Lagrange top, Kowalevski top, and Goryachev–Chaplygin top, which are related to hyperelliptic curves of genera 1 and 2, as well as for the Goryachev and Dullin–Matveev systems, which are related to trigonal curves in the plane. 相似文献
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Andrey V. Tsiganov 《Regular and Chaotic Dynamics》2017,22(4):353-367
The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order. 相似文献
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Doklady Physics - In this paper, we consider the motion of a nonholonomic Chaplygin sphere on a plane in a constant magnetic field under the assumption that the sphere has dielectric and... 相似文献