On rheonomic nonholonomic deformations of the Euler equations proposed by Bilimovich

Borisov, Alexey V.; Tsiganov, A. V.

doi:10.2298/tam200120009b

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…There are very recent papers which are built on the results of Bilimović, e.g. [29,30]. We hope that the current paper will further draw attention to the heritage of the Bilimović scientific school and their contribution to nonholonomic mechanical problems.…”

Section: Letmentioning

confidence: 77%

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Dragović

Gajić

Jovanović

2020

Theor appl mech (Belgr)

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We present an integrable nonholonomic case of rolling without sliding of a gyroscopic ball over a sphere. This case was introduced and studied in detail by Vasilije Demchenko in his 1923 doctoral dissertation defended at the University of Belgrade, with Anton Bilimovic as the advisor. These results are absolutely unknown to modern researchers. The study is based on the C. Neumann coordinates and the Voronec principle. By using the involved technique of elliptic functions, a detailed study of motion is performed. Several special classes of trajectories are distinguished, including regular and pseudoregular precessions. The so-called remarkable trajectories, introduced by Paul Painlev?e and Anton Bilimovic, are described in the present case. The historical context of the results as well as their place in contemporary mechanics are outlined.

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Section: Letmentioning

confidence: 77%

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Dragović

Gajić

Jovanović

2020

Theor appl mech (Belgr)

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“…On the other hand Note that the existence of an invariant measure for nonholomic problems is well studied in many classical problems [4,6]. After Kozlov's theorem on obstruction to the existence of an invariant measure for the variant of the classical Suslov problem (e.g., see [9,14]) on Lie algebras [20], general existence statements for nonholonomic systems with symmetries are obtained in [21] and [15].…”

Section: The Reduced Systemmentioning

confidence: 99%

Spherical and planar ball bearings -- nonholonomic systems with invariant measures

Dragovic,

Gajic,

Jovanovic

2022

Preprint

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We first construct nonholonomic systems of n homogeneous balls B 1 , . . . , Bn with centers O 1 , ..., On and with the same radius r that are rolling without slipping around a fixed sphere S 0 with center O and radius R. In addition, it is assumed that a dynamically nonsymmetric sphere S of radius R + 2r and the center that coincides with the center O of the fixed sphere S 0 rolls without slipping over the moving balls B 1 , . . . , Bn. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius R tends to infinity. We obtain a corresponding planar problem consisting of n homogeneous balls B 1 , . . . , Bn with centers O 1 , ..., On and the same radius r that are rolling without slipping over a fixed plane Σ 0 , and a moving plane Σ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.

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“…At I 13 = I 23 = 0 equations (2.4) also have only two solutions. According to [9] in this case we can integrate the nonautonomous equations (2.4) in terms of hyperbolic functions at α = 0. In Fig.…”

Section: Two Critical Pointsmentioning

confidence: 99%

“…where I is the inertia tensor of the body. Thus, at α = 0 we have an integrable by quadratures system, see [2,9] and references within.…”

Section: Introductionmentioning

confidence: 99%

On inhomogeneous nonholonomic Bilimovich system

Borisov,

Mikishanina,

Tsiganov

2020

Preprint

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We consider a simple motivating example of non-Hamiltonian dynamical system with timedependent constraint obtained by imposing rheonomic nonintegrabe Bilimovich's constraint on a freely rotating rigid body. Dynamics of this low-dimensional nonlinear nonautonomous dynamic system involves different kinds of stable and unstable attractors, quasi and strange attractors, compact and noncompact invariant attractive curves, etc. We apply the Poincaré map method to study this cautionary example in order to disambiguate and discover multiscale temporal dynamics, specifically the coarse-grained dynamics which results from fast-scale nonlinear control via nonholonomic Bilimovich's constraint.

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On rheonomic nonholonomic deformations of the Euler equations proposed by Bilimovich

Cited by 8 publications

References 16 publications

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Spherical and planar ball bearings -- nonholonomic systems with invariant measures

On inhomogeneous nonholonomic Bilimovich system

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