The relationship between Lax and bi-Hamiltonian formulations of dynamical systems on finite-or infinite-dimensional phase spaces is investigated. The Lax-Nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher-order Hamiltonian structures for the Toda system, a second Hamiltonian structure of the Euler equation for a rigid body in n-dimensional space, and the quadratic Adler-Gelfand-Dickey structure for the KdV hierarchy are derived using the Lax-Nijenhuis equation. © 1996 American Institute of Physics.
Kosmann Schwarzbach, Y., Magri, F. (1996). Lax-Nijenhuis operators for integrable systems. JOURNAL OF MATHEMATICAL PHYSICS, 37(12), 6173-6197 [10.1063/1.531771].
Lax-Nijenhuis operators for integrable systems
MAGRI, FRANCO
1996
Abstract
The relationship between Lax and bi-Hamiltonian formulations of dynamical systems on finite-or infinite-dimensional phase spaces is investigated. The Lax-Nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher-order Hamiltonian structures for the Toda system, a second Hamiltonian structure of the Euler equation for a rigid body in n-dimensional space, and the quadratic Adler-Gelfand-Dickey structure for the KdV hierarchy are derived using the Lax-Nijenhuis equation. © 1996 American Institute of Physics.
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