The Routh theorem for mechanical systems with unknown first integrals

In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane.

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Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses

Journal of Applied Mathematics and Mechanics ◽

10.1016/j.jappmathmech.2008.04.013 ◽

2008 ◽

Vol 72 (2) ◽

pp. 126-135 ◽

Cited By ~ 45

Author(s):

N.N. Bolotnik ◽

T.Yu. Figurina

Keyword(s):

Optimal Control ◽

Rigid Body ◽

Rectilinear Motion ◽

Rough Plane

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On the first integrals of the problem of jumping a massive rigid body on a smooth horizontal plane

Doklady Physics ◽

10.1134/1.1774062 ◽

2004 ◽

Vol 49 (6) ◽

pp. 366-368

Author(s):

A. A. Burov ◽

D. P. Chevallier

Keyword(s):

Rigid Body ◽

Horizontal Plane ◽

First Integrals

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On First Integrals and Invariant Manifolds in the Generalized Problem of the Motion of a Rigid Body in a Magnetic Field

Computer Algebra in Scientific Computing - Lecture Notes in Computer Science ◽

10.1007/978-3-030-85165-1_10 ◽

2021 ◽

pp. 157-173

Author(s):

Valentin Irtegov ◽

Tatiana Titorenko

Keyword(s):

Magnetic Field ◽

Rigid Body ◽

Invariant Manifolds ◽

First Integrals

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Complete list of first integrals of dynamic equations of motion of a 4D rigid body in a nonconservative field under the assumption of linear damping

Doklady Physics ◽

10.1134/s1028335813040022 ◽

2013 ◽

Vol 58 (4) ◽

pp. 143-146

Author(s):

M. V. Shamolin

Keyword(s):

Rigid Body ◽

Equations Of Motion ◽

First Integrals ◽

Dynamic Equations ◽

Complete List ◽

Linear Damping

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On the optimal regulation of an axi-symmetric rigid body with two controls

10.2514/6.1996-3791 ◽

1996 ◽

Cited By ~ 5

Author(s):

Panagiotis Tsiotras

Keyword(s):

Rigid Body ◽

Optimal Regulation ◽

Symmetric Rigid Body

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On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

ITM Web of Conferences ◽

10.1051/itmconf/20192901015 ◽

2019 ◽

Vol 29 ◽

pp. 01015 ◽

Cited By ~ 1

Author(s):

Cristian Lăzureanu ◽

Ciprian Hedrea ◽

Camelia Petrişor

Keyword(s):

Incompressible Fluid ◽

Mechanical System ◽

Integrable Systems ◽

Degrees Of Freedom ◽

Mechanical Systems ◽

Ideal Incompressible Fluid ◽

First Integrals ◽

Integrable Deformation ◽

Integrable Deformations ◽

The Given

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems too.

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