Bidifferential calculi, bicomplex structure and its application to bihamiltonian systems

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Abstract

In this exposition, we study the relationship between the bihamiltonian formalism of completely integrable systems using the bidifferential calculi introduced by Dimakis and Müller-Hoissen in [1] and the bihamiltonian formulation of integrable systems with a finite number of degrees of freedom via the Frölicher-Nijenhuis geometry. This pair of bidifferetial 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.

Original languageBritish English
Pages (from-to)209-232
Number of pages24
JournalInternational Journal of Geometric Methods in Modern Physics
Volume3
Issue number2
DOIs
StatePublished - Mar 2006

Keywords

  • Bidifferential calculi
  • Bihamiltonian
  • Casimirs
  • Finsler geometry
  • Frölicher-Nijenhuis
  • Nijenhuis operator
  • Poisson-Nijenhuis

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