Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The C. Neumann problem as a completely integrable system on an adjoint orbit


Author: Tudor Raţiu
Journal: Trans. Amer. Math. Soc. 264 (1981), 321-329
MSC: Primary 58F05; Secondary 70H05
DOI: https://doi.org/10.1090/S0002-9947-1981-0603766-3
MathSciNet review: 603766
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown by purely Lie algebraic methods that the $ {\text{C}}$. Neumann problem--the motion of a material point on a sphere under the influence of a quadratic potential--is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras.


References [Enhancements On Off] (What's this?)

  • [1] R. Abraham and J. Marsden, Foundations of mechanics, 2nd ed., Benjamin, New York, 1978. MR 515141 (81e:58025)
  • [2] M. Adler and P. van Moerbeke, Completely integrable systems, Kac-Moody Lie algebras, and curves and Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Adv. in Math. (to appear). MR 597730 (83m:58042)
  • [3] R. Devaney, Transversal homoclinic orbits in an integrable system, preprint, 1978. MR 0494258 (58:13164)
  • [4] V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. of Physics (to appear). MR 576424 (81g:58011)
  • [5] A. Iacob and S. Sternberg, Coadjoint structures, solitons, and integrability, Lecture Notes in Phys., vol. 120, Springer, Berlin and New York, 1980. MR 581892 (81h:58035)
  • [6] D. Kazhdan, B. Kostant and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-508. MR 0478225 (57:17711)
  • [7] B. Kostant; The solution to a generalized Toda lattice and representation theory, M.I.T., 1979, (preprint). MR 550790 (82f:58045)
  • [8] H. McKean, Integrable systems and algebraic curves, Lecture Notes in Math., vol. 755, Springer-Verlag, Berlin and New York, 1979. MR 564904 (81g:58017)
  • [9] A. S. Mishchenko and A. T. Fomenko, Euler equations on finite dimensional Lie groups, Math. USSR-Izv. 12 (1978), 371-389.
  • [10] J. Moser, Various aspects of integrable Hamiltonian systems, Dynamical Systems (C.I.M.E., Bressanone, Italy, July 1978), Progress in Math., no. 8, Birkhäuser, Basel.
  • [11] -, Geometry of quadrics and spectral theory, Chern Symposium, Berkeley, 1979 (to appear).
  • [12] C. Neumann, De problemate quodam mechanica, quod ad primam integralium ultra-ellipticorum classem revocatur, J. Reine Angew. Math. 56 (1859), 54-66.
  • [13] T. Ratiu, The motion of the free $ N$-dimensional rigid body, Indiana J. Math. 29 (1980), 609-629. MR 578210 (81h:58032)
  • [14] -, Involution theorems, Lectures Notes in Math., vol. 755, Springer-Verlag, Berlin and New York, 1980, pp. 219-257. MR 569304 (82f:58047)
  • [15] -, Euler-Poisson equations on Lie algebras, Thesis, Univ. of California, Berkeley, 1980.
  • [16] K. Uhlenbeck, Minimal $ 2$-spheres and tori in $ {S^k}$, informal preprint, 1975.
  • [17] P. van Moerbeke and T. Ratiu, The Lagrange top (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F05, 70H05

Retrieve articles in all journals with MSC: 58F05, 70H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0603766-3
Keywords: Complete integrability, adjoint orbit, Hamiltonian system, Euler-Poisson equations, Kirillov-Kostant-Souriau symplectic structure
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society