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Fomenko invariants for the main integrable cases of the rigid body motion equations. (English) Zbl 0745.58028
Topological classification of integrable systems, Adv. Sov. Math. 6, 67-146 (1991).
[For the entire collection see Zbl 0741.00026.]
Let $$e(3)$$ be the Lie algebra of the transformation group of $$\mathbb{R}^ 3$$ and $$S_ 1,S_ 2,S_ 3,R_ 1,R_ 2,R_ 3$$ be the coordinate functions satisfying the Poisson brackets $$\{S_ i,S_ j\}=\{R_ i,R_ j\}=0$$, $$\{S_ i,R_ j\}=\{R_ i,S_ j\}=\varepsilon_{ijk}R_ k$$ where $$\varepsilon_{ijk}D$$ the unit antisymmetric tensor. For any $$H\in e^*(3)$$ consider the Hamiltonian system $$\dot s_ i=\{s_ i,H\}$$, $$\dot R_ i=\{R_ i,H\}$$. There are 8 integrable cases for various H’s: Euler’s, Lagrange’s, Kovalevskaya’s, Goryachev-Chaplygin’s (motions of a rigid body around a fixed point), Zhukovsky’s, Sretensky’s (motions of a gyrostat in a gravitational field), Clebsch’s Steklov-Lyapunov’s (motions of a rigid body in a fluid). All these systems are Hamiltonian on the symplectic manifolds $$\{f_ 1\equiv R^ 2_ 1+R^ 2_ 2+R^ 2_ 3,\;f_ 2\equiv S_ 1R_ 1+S_ 2R_ 2+S_ 3R_ 3=g\}$$ and all of them have an additional integral $$K$$. The surface $$Q_ h=\{f_ 1=1, f_ 2=g, H=h\}\subset\mathbb{R}^ 6(S,R)$$ is called the isoenergy surface of a given system and an additional integral is called Bott if the set of critical points of $$\tilde K=K|_{Q_ h}$$ is the union of smooth nondegenerate critical submanifolds.
Theorem. For each of the 8 integrable Hamiltonian systems on $$TS^ 2=\{f_ 1=1, f_ 2=g\}$$ the additional integral is Bott on all nonsingular surfaces $$Q_ h$$ with the exception, perhaps of a finite number.
The proof is obtained by constructing the Fomenko invariant for each integrable case. This invariant was introduced by A. T. Fomenko in Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 6, 1276-1307 (1986; Zbl 0619.58023), Sov. Math., Dokl. 33, 502-506 (1986); translation of Dokl. Akad. Nauk SSSR 287, 1071-1075 (1986; Zbl 0623.58009) and Funct. Anal. Appl. 22, No. 4, 286-296 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 38-51 (1988; Zbl 0697.58021).

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 70E15 Free motion of a rigid body
##### Keywords:
integrable systems; topological invariants; Morse theory