Bidifferential calculi, bicomplex structure and its application to bihamiltonian systems. (English) Zbl 1134.37354
Summary: We study the relationship between the bihamiltonian formalism of completely integrable systems using the bidifferential calculi introduced by A. Dimakis and F. Müller-Hoissen [J. Phys. A, Math. Gen. 33, 957–974 (2000; Zbl 1043.37508)] and the bihamiltonian formulation of integrable systems with a finite number of degrees of freedom via the Frölicher–Nijenhuis geometry. This pair of bidifferential operators are used to construct alternative Lie algebroids as shown by Camacaro and Carinena. We find its connection to Finsler geometry. We also find the dispersionless integrable hierarchies using the bidifferential ideals. Finally, we lay out its connection to Gelfand–Zakharevich bihamiltonian geometry.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
Poisson–Nijenhuis; Frölicher–Nijenhuis; bidifferential calculi; bihamiltonian; Casimirs; Nijenhuis operator; Finsler geometryCitations:
Zbl 1043.37508References:
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