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

SIGMA 11 (2015), 089, 11 pages      arXiv:1506.08675      https://doi.org/10.3842/SIGMA.2015.089
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems

Manuele Santoprete
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

Received June 30, 2015, in final form November 03, 2015; Published online November 07, 2015

Abstract
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fassò and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis coordinates and symplectic connections.

Key words: bi-Hamiltonian systems; Lagrangian foliation; bott connection; symplectic connections.

pdf (318 kb)   tex (17 kb)

References

  1. Bieliavsky P., Cahen M., Gutt S., Rawnsley J., Schwachhöfer L., Symplectic connections, Int. J. Geom. Methods Mod. Phys. 3 (2006), 375-420, math.SG/0511194.
  2. Bogoyavlenskij O.I., Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures, Comm. Math. Phys. 180 (1996), 529-586.
  3. Brouzet R., Systèmes bihamiltoniens et complète intégrabilité en dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 895-898.
  4. Falqui G., Pedroni M., Poisson pencils, algebraic integrability, and separation of variables, Regul. Chaotic Dyn. 16 (2011), 223-244.
  5. Fassò F., Ratiu T., Compatibility of symplectic structures adapted to noncommutatively integrable systems, J. Geom. Phys. 27 (1998), 199-220.
  6. Fernandes R.L., Completely integrable bi-Hamiltonian systems, J. Dynam. Differential Equations 6 (1994), 53-69.
  7. Forger M., Yepes S.Z., Lagrangian distributions and connections in multisymplectic and polysymplectic geometry, Differential Geom. Appl. 31 (2013), 775-807, arXiv:1202.5054.
  8. Gel'fand I.M., Dorfman I.Ja., Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979), 248-262.
  9. Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Vol. 218, Springer-Verlag, New York, 2003.
  10. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
  11. Magri F., Casati P., Falqui G., Pedroni M., Eight lectures on integrable systems, in Integrability of Nonlinear Systems, Lecture Notes in Phys., Vol. 638, Editors Y. Kosmann-Schwarzbach, K.M. Tamizhmani, B. Grammaticos, Springer, Berlin, 2004, 209-250.
  12. Magri F., Morosi C., A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderni del Dipartimento di Matematica, Università di Milano, 1984.
  13. Olver P.J., Canonical forms and integrability of bi-Hamiltonian systems, Phys. Lett. A 148 (1990), 177-187.
  14. Tondo G., Generalized Lenard chains and separation of variables, Quad. Mat. Univ. Trieste 573 (2006), 1-27.
  15. Turiel F.-J., Classification locale simultanée de deux formes symplectiques compatibles, Manuscripta Math. 82 (1994), 349-362.

Previous article  Next article   Contents of Volume 11 (2015)