The main goal of this talk is to demonstrate how the theory of invariants for integrable Hamiltonian  systems with two degrees of freedom created by A.T. Fomenko, H. Zieschang, and A.V. Bolsinov helps to establish Liouville and orbital equivalence of some classical integrable systems. Three such systems are treated in the talk: the Euler case in rigid body dynamics, the Jacobi problem about geodesics on the ellipsoid  and the Chaplygin  case in dynamics of a rigid body in fluid. The first two systems were known to be Liouville and even topologically  orbitally equivalent (Fomenko, Bolsinov).  Now A.T.Fomenko and S.S.Nikolaenko (Moscow State University) show that the Chaplygin system is orbitally equivalent to the Euler and Jacobi systems.

Main theorem. Let vCh (h) and vE (A, B, C, χ) be the Chaplygin and Euler systems (with parameters  h and A,B,C, χ respectively ) on regular 3-dimensional surfaces of constant energy Q = {HCh = h} and Q’ = {HE = χ}. We suppose that the energy values h and χ belong to zones with the same number. Let vJ (a, b, c) denote  the Jacobi system , i.e. geodesic flow on the ellipsoid (with parameters  a,b,c), on a non-zero level of energy H.  The triples of parameters (A, B, C) and (a, b, c) are viewed up to proportionality. Then the following statements hold.

1) If the energy value h of the Chaplygin  system vCh (h) belongs to the first zone (1)Ch , then it is Liouville equivalent to the Euler system vE (A, B, C, χ) for any A, B, C and for any χ from the first zone (1)E . There also exists a one-parameter family of the Euler systems vE (A, B(A), C = 1, χ(A)) orbitally (topologically  and smoothly) equivalent to vCh (h).  This orbital equivalence can be extended on the four-dimensional  neighbourhoods of the isoenergy surfaces.  But for any A, B, C, χ the systems vCh (h) and vE (A, B, C, χ) are not topologically conjugate.

2) If the energy value h of the Chaplygin  system vCh (h) belongs to the second zone (2)Ch , then it is Liouville equivalent to the Euler system vE (A, B, C, χ) for any A, B, C and for any χ from the second zone (2)E. There also exists a one-parameter family of the Euler systems vE (A, B(A), C = 1, χ(A)) topologically orbitally equivalent to vCh (h). But for any A, B, C, χ the systems vCh (h) and vE (A, B, C, χ) are not smoothly orbitally equivalent (even in the sense of C1 -smoothness) and are not topologically conjugate.

3) If the energy value h of the Chaplygin system vCh (h) belongs to the third zone (3)Ch , then it is Liouville equivalent to the Euler system vE (A, B, C) for any A, B, C (with the energy value from the third zone (3)E ) and to the Jacobi system vJ (a, b, c) for any a, b, c. If h is large enough, there exist unique up to proportionality triples (A, B, C) and (a, b, c) such that the system vCh (h) is topologically orbitally equivalent to vE (A, B, C) and vJ (a, b, c). But this orbital equivalence cannot be made smooth (even in the sense of C1 -smoothness). Moreover, for any A, B, C and a, b, c the system vCh (χ) is not topologically conjugate with vE (A, B, C) or vJ (a, b, c).

[1]  A. T. Fomenko and H. Zieschang, A topological  invariant  and a criterion  for the equivalence of integrable Hamiltonian  systems with two degrees of freedom. Izvest. Akad. Nauk SSSR, Ser. Matem., 54 (1990), No. 3, P. 546–575.

[2]  A. V. Bolsinov  and A. T. Fomenko, Orbital equivalence of integrable Hamiltonian  systems with two degrees of freedom. A classification theorem. I, II. Matem. Sbornik, 185 (1994), No. 4, P. 27–80; No. 5, P. 27–78.