Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 070, 23 pages      arXiv:0705.3628      https://doi.org/10.3842/SIGMA.2007.070
Contribution to the Vadim Kuznetsov Memorial Issue

Hamilton-Jacobi Theory and Moving Frames

Joshua D. MacArthur a, Raymond G. McLenaghan b and Roman G. Smirnov a
a) Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5
b) Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Received February 03, 2007, in final form May 14, 2007; Published online May 24, 2007

Abstract
The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.

Key words: Hamilton-Jacobi theory; orthogonal separable coordinates; Killing tensors; group action; moving frame map; regular foliation.

pdf (432 kb)   ps (299 kb)   tex (224 kb)

References

  1. Benenti S., Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578-6602.
  2. Eisenhart L.P., Separable systems of Stäckel, Ann. Math. 35 (1934), 284-305.
  3. Olver P.J., Moving coframes. I. A practical algorithm, Acta. Appl. Math. 51 (1998), 161-213.
  4. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta. Appl. Math. 55 (1999), 127-208.
  5. Horwood J.T., McLenaghan R.G., Smirnov R.G., Invariant classification of orthogonal separable Hamiltonian systems in Euclidean space, Comm. Math. Phys. 221 (2005), 679-709, math-ph/0605023.
  6. Kuznetsov V.B., Simultaneous separation for the Kowalevski and Goryachev-Chaplygin gyrostats, J. Phys. A: Math. Gen. 35 (2002), 6419-6430, nlin.SI/0201004.
  7. MacArthur J.D., The equivalence problem in differential geometry, MSc thesis, Dalhousie University, 2005.
  8. McLenaghan R.G., Smirnov R.G., The D., Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the O(4)-symmetric Yang-Mills theories of Yatsun, J. Math. Phys. 43 (2002), 1422-1440.
  9. McLenaghan R.G., Smirnov R.G., The D., An extension of the classical theory of invariants to pseudo-Riemannian geometry and Hamiltonian mechanics, J. Math. Phys. 45 (2004), 1079-1120.
  10. Olver P.J., Classical invariant theory, Student Texts, Vol. 44, London Mathematical Society, Cambridge University Press, 1999.
  11. Palais R.S., A global formulation of the Lie theory of transportation groups, Memoirs Amer. Math. Soc., no. 22, AMS, Providence, R.I., 1957.
  12. The D., Notes on complete sets of group-invariants in K2(R2) and K3(R2), 2004, unpublished.


Previous article   Next article   Contents of Volume 3 (2007)