Géométrie des systèmes bihamiltoniens. (Geometry of bi-Hamiltonian systems). (French) Zbl 0624.58009

Kosmann-Schwarzbach, Yvette

Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 102, 185-216 (1986).

Author’s summary: “Bi-Hamiltonian systems are evolution partial differential equations which are Hamiltonian systems with respect to two distinct Hamiltonian structures. They possess a recursion operator which generates a sequence of symmetries, while its adjoint generates a sequence of first integrals. Moreover, the two Hamiltonian structures are assumed to be compatible, i.e., the recursion operator has the property that its Nijenhuis torsion vanishes. This implies that the symmetries commute and that the conserved quantities are in involution, whence the complete integrability properties of the bi-Hamiltonian systems.”

Reviewer: A.P.Stone

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

Bi-Hamiltonian systems; Nijenhuis torsion