Lagrangian Version of AKS systems

SANTIAGO CAPRIOTTI

San Carlos de Bariloche

Encuentro; Second Iberoamerican Meeting on Geometry, Mechanics and Control, in honor of Hernán Cendra; 2011

Instituto Balseiro

The AKS systems [BP94, Ova03] are integrable systems whose phase spaces are orbits in the dual of a Lie group. The usual constructions are hamiltonian in nature (see [RSTS79, RSTS81]), and can be related to dynamical systems on the cotangent bundle of a Lie group using reduction. Less known are the Lagrangian versions of these systems; an exception is given by the reference [FG02]. The main technique used by these authors in proving the equivalence of this lagrangian system with the usual approach is the Dirac method. In this talk we will show how to fit the AKS systems as a variational problem of more general nature and how to obtain a hamiltonian version for this variational problem. To this end we will use the notion of Lepage equivalence of variational problems as defined by M. Gotay in [Got91], allowing us to define a presymplectic manifold, where an aplication of the Gotay, Nester and Hinds algorithm [GNH78] will yield to th desired equivalence. References [BP94] F. E. Burstall y F. Pedit. Harmonic maps via Adler–Kostant–Symes theory. In A. P. Fordy and J. C. Wood, editors, Harmonic maps and integrable systems, volume 23 of Aspects of Math., pages 221-272. Vieweg, Braunschweig, Wiesbaden, 1994. [FG02] L. Fèher y A. Gabor. Adler-Kostant-Symes systems as lagrangian gauge theories. Physics Letters A, 301:58, 2002. [Got91] M.J. Gotay. An exterior differential system approach to the Cartan form. In Donato P., Duval C., Elhadad J., and G.M. Tuynman, editors, Symplectic geometry and mathematical physics. Actes du colloque de géométrie symplectique et physique mathématique en l’honneur de Jean-Marie Souriau, Aix-en-Provence, France, June 11-15, 1990., pages 160 –188. Progress in Mathematics. 99. Boston, MA, Birkhäuser, 1991. [GNH78] M.J. Gotay, J. M. Nester, y G. Hinds. Presymplectic manifolds and the Dirac-Bergmann theory of constraints. J. Math. Phys., (19):2388, 1978. [Ova03] G. Ovando. Invariant metrics and hamiltonian systems. preprint arXiv:math/0301332, 2003. [RSTS79] A. G. Reyman y M. A. Semenov-Tian-Shansky. Reduction of Hamil- tonian systems, affine Lie algebras and Lax equations. Invent. Math., 54:81–100, 1979. [RSTS81] A. G. Reyman y M. A. Semenov-Tian-Shansky. Reduction of Hamil- tonian systems, affine Lie algebras and Lax equations. II. Invent. Math., 63:423–432, 1981.