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A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid. (English, Russian) Zbl 1336.37046
Sb. Math. 205, No. 2, 224-268 (2014); translation from Mat. Sb. 205, No. 2, 75-122 (2014).
The paper under review addresses some questions associated with the motion of a rigid body in a fluid. The operative equations are those of Kirchhoff \[ \dot s= s\times{\partial H\over\partial s}+ r\times{\partial H\over\partial r},\quad \dot r= r\times{\partial H\over\partial s}.\tag{\(*\)} \] The fluid is assumed to have infinite volume, be ideal and incompressible. Further, it is assumed that the body moves by inertia, the fluid is at rest at infinity and that the flow is irrotational.
The vector \(s=(s_1,s_2,s_3)\), is called the impulsive moment, and \(r=(r_1,r_2,r_3)\) the impulse force. The total energy is \(H\).
This system can be treated as a Hamiltonian system on the co-algebra \(e(3)^*\) endowed with a non-degenerate Lie-Poisson bracket structure. There are two Casimir functions given by \[ f_1= r^2_1+ r^2_2+ r^2_3\quad\text{and}\quad f_2= s_1r_1+ s_2r_2+ s_3r_3, \] so these are first integrals of system \((*)\).
On a symplectic fiber \(M^4_{a,g}= \{f^2_1= a,\,f_2= g\}\) the Lie-Poisson bracket defines a non-degenerate symplectic structure.
Chaplygin found an integrability condition for system \((*)\) on \(M_{a,0}\) for \[ H={1\over 2}(s^2_1, s^2_2+ 2s^2_3)+{c\over 2} (r^2_1- r^2_2) \] with the additional integral \(K= (s^2_1- s^2_2+ cr^2_3)+ 4s^2_1 s^2_2\).
In the current paper, the author develops a complete list of the topological types of Chaplygin systems for \(c\neq 0\), using Fomenko-Zieschang theory [A. T. Fomenko, Funct. Anal. Appl. 22, No. 4, 286–296 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 38–51 (1988; Zbl 0697.58021); A. T. Fomenko and H. Zieschang, Math. USSR, Izv. 36, No. 3, 567–596 (1991); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 3, 546–575 (1990; Zbl 0723.58024)] and describes the topology of the Liouville foliation in terms of the natural coordinate variables.
One conclusion is that all non-singular Chaplygin systems are Liouville-equivalent to the well-known Euler case of rigid body dynamics with a fixed point.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70E15 Free motion of a rigid body
70E40 Integrable cases of motion in rigid body dynamics
70F10 \(n\)-body problems
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