Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body

Abstract The classical Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point are generalized to arbitrary Lie algebras as Hamiltonian systems on coad-joint orbits of a tangent bundle Lie group. the N-dimensional Lagrange and symmetric heavy top are thereby shown to be completely integrable.

[1]  T. Ratiu.  On the smoothness of the time t-map of the KdV equation and the bifurcation of the eigenvalues of Hill's operator , 1979 .

[2]  S. Sternberg,et al.  Coadjoint structures, solitions, and integrability , 1982 .

[3]  Shlomo Sternberg,et al.  Hamiltonian group actions and dynamical systems of calogero type , 1978 .

[4]  Y. Manin.  Algebraic aspects of nonlinear differential equations , 1979 .

[5]  S. Manakov,et al.  Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body , 1976 .

[6]  Res Jost,et al.  Poisson Brackets (An Unpedagogical Lecture) , 1964 .

[7]  S. Smale,et al.  Topology and mechanics. I , 1970 .

[8]  R. Moody.  A new class of Lie algebras , 1968 .

[9]  J. Marsden,et al.  Reduction of symplectic manifolds with symmetry , 1974 .

[10]  P. Moerbeke,et al.  Completely Integrable Systems, Euclidean Lie-algebras, and Curves , 1980 .

[11]  Alan Weinstein,et al.  Lectures on Symplectic Manifolds , 1977 .

[12]  S. Smale,et al.  Topology and mechanics. II , 1970 .

[13]  Anatoliĭ Timofeevich Fomenko,et al.  Euler equations on finite-dimensional Lie groups , 1978 .

[14]  Jerrold E. Marsden,et al.  Review: Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, and Nolan R. Wallach, Symplectic geometry and Fourier analysis , 1979 .

[15]  T. Ratiu,et al.  The Lagrange rigid body motion , 1982 .

[16]  Mark Adler,et al.  On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-devries type equations , 1978 .

[17]  B. Kostant.  On Whittaker vectors and representation theory , 1978 .

[18]  About isospectral deformations of discrete laplacians , 1979 .

[19]  M. Semenov-Tian-Shansky,et al.  Reduction of Hamiltonian systems, affine Lie algebras and Lax equations , 1979 .

[20]  Relations among generalized Korteweg–deVries systems , 1979 .

[21]  Jerrold E. Marsden,et al.  Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups , 1981 .

[22]  Jerrold E. Marsden,et al.  Symmetry and bifurcations of momentum mappings , 1981 .