Dynamical systems with homogeneous configuration spaces

Part of the Mathematics and Its Applications book series (MAIA, volume 443)


Let G be a connected semi-simple Lie group and K a closed reductive subgroup. In this chapter we enumerate all the homogeneous spaces G/K for which arbitrary Hamiltonian systems on T*(G/K) with G-invariant Hamiltonians are integrable within the class of Noether integrals (see Section 1 for definition). It is known that all symmetric spaces G/K of semi-simple groups G possess this property (see (Timm, 1988), (Mishchenko, 1982), (Mykytiuk, 1983) and (Ii, 1982)). It will also be proved here that if, in addition, the groups G and K have a complex structure or are compact, then the following conditions are equivalent:
  1. (1)

    All G-invariant Hamiltonian systems on T*(G/K) are integrable within the class of Noether integrals.

  2. (2)

    The subgroup K of G is spherical; i.e., the quasiregular representation of G on the space C[G/K] of regular functions on the affine algebraic variety G/K has a simple spectrum if G is complex, and likewise on L 2(G/K) if G is compact. In (Guillemin et al, 1984a) it was shown that a subgroup K of a compact Lie group G is spherical if and only if

  3. (3)

    The algebra of G-invariant functions on T*(G/K) is commutative with respect to the standard Poisson bracket.



Poisson Bracket Poisson Structure Borel Subalgebra Zariski Open Subset Hamiltonian Vector Field 
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Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of Mining and MetallurgyCracowPoland
  2. 2.Institute for Applied Problems of Mechanics and Mathematics of the NASLvivUkraine
  3. 3.Lviv Polytechnic State UniversityLvivUkraine

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