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Periodic solutions of Kirchhoff equations for the free motion of a rigid body in a fluid and the extended Lyusternik-Shnirel’man-Morse theory. I. (English. Russian original) Zbl 0571.58009

Funct. Anal. Appl. 15, No. 3, 197-207 (1982); translation from Funkts. Anal. Prilozh. 15, No. 3, 54-66 (1981).
The authors construct an extended Lyusternik-Shnirel’man-Morse theory (LSM-theory) for the study of the stationary points of one- and multi- valued functionals defined by the integrals of closed l-forms on the space \({\hat \Omega}{}^+(M^ n)\) of directed closed curves in a certain manifold \(M^ n\). This theory is applied to the Kirchhoff problem concerning the free motion of a rigid body in an ideal incompressible fluid whose flow is described by a potential and which is at rest at infinity. Equations of Kirchhoff type are defined by a Hamiltonian H and the Poisson brackets \(\{\), \(\}\) for functions on a phase space which is the dual space \(L^*\) of the algebra L of the group E(3) of the motions of Euclidean space \({\mathbb{R}}^ 3\). For the classical Kirchhoff case \[ 2H=\sum^{3}_{i=1}a_{ii}M^ 2_ i+2\sum^{3}_{i,j=1}b_{ij}((p_ jM_ i=p_ iM_ j)/2)+\sum^{3}_{i,j=1}c_{ij}p_ ip_ j>0, \]
\[ \{M_ i,M_ j\}=\epsilon_{ijk}M_ k,\quad \{M_ i,p_ j\}=\epsilon_{ijk}p_ k,\quad \{p_ i,p_ j\}=0,\quad \dot M_ i=\{M_ i,H\},\quad \dot p_ i=\{p_ i,H\}. \] Here M, p are the momentum and impulse in a moving coordinate system. There are universal Kirchhoff integrals \(f_ 1=P^ 2=\sum p^ 2_ i\), \(f_ 2=ps=\sum M_ ip_ i\), \(\{f_{\ell},M_ i\}=\{f_{\ell},p_ i\}=0\), \(l=1,2\), \(i=1,2,3\). Part I contains the investigation of the case \(f_ 2=0\), while part II the completion of the extended LSM-theory and applications to the case \(f_ 2\neq 0\) (see below).

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
76B99 Incompressible inviscid fluids
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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References:

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