Eight lectures on integrable systems

Untitled

“…Here σ k are the elementary symmetric polynomials of degree k on the eigenvalues of N. In this case relations (2.12) are the Lenard-Magri recurrence relations 14) such that the corresponding vector fields X H i are bi-hamiltonian [12,17].…”

“…The phase spaces of integrable systems will be identified with ωN-manifolds [6,7] or Poisson-Nijenhuis manifolds [14,17] endowed with a symplectic form ω and a torsion free tensor N satisfying certain compatibility conditions.…”

“…So, for a given Hamiltonian H we can look at (1.3) as one of the equations to determine torsion free tensor N and hence the separated variables. This scheme, in its basic features, has already been considered in the literature and applied to various systems, see [1,6,7,12,17,18,19,23] and references within.…”

On the Darboux-Nijenhuis Variables for the Open Toda Lattice

“…. , H n be in bi-involution is that the corresponding vector fields X H i are bi-Hamiltonian vector fields [13,17], which form a so-called anchored Lenard-Magri sequence P dH 1 = 0, X H i = P dH i = P dH i−1 , P dH n = 0.…”

“…Another useful property of N is that normalized traces of the powers of N are integrals of motion satisfying Lenard-Magri recurrent relations (1.3) [17]:…”