The aim of this paper is to discuss the relations between the bi-Hamiltonian formulation and the Lax representation (with spectral parameter) of an integrable system. In the Introduction, the author recalls, with the help of many examples, the notions of Lax pair and discusses the importance of having a Lax representation with a spectral parameter. Then, in Section 2, he gives an account of the theory of bi-Hamiltonian manifolds and of $r$-matrix theory. The third section is devoted to the study of a special class of bi-Hamiltonian manifolds, admitting an interpretation also in terms of linear $r$-matrices. They are the product of $n$ copies of (the dual of) a finite-dimensional Lie algebra and actually possess an $(n+1)$-dimensional family of compatible Poisson brackets. Then it is shown how to construct bi-Hamiltonian hierarchies for some of these manifolds, using a kind of rule of fractional powers. The last section contains some finite- and infinite-dimensional examples to which the results of the paper are applied. For the $n$-dimensional rigid body, the Manakov integrals and the Mishchenko integrals are obtained. The Lagrange top is interpreted as a bi-Hamiltonian system on ${\germ s}{\germ o}(3)\times {\germ s}{\germ o}(3)$, clarifying a result due to T. Raţiu [Amer. J. Math. 104 (1982), no. 2, 409--448]. The KdV and Boussinesq hierarchies are treated as in [G. Magnano and F. Magri, Rev. Math. Phys. 3 (1991), no. 4, 403--466], avoiding the introduction of pseudodifferential operators.
Bihamiltonian approach to Lax equations with spectral parameter
MAGNANO, Guido
1996-01-01
Abstract
The aim of this paper is to discuss the relations between the bi-Hamiltonian formulation and the Lax representation (with spectral parameter) of an integrable system. In the Introduction, the author recalls, with the help of many examples, the notions of Lax pair and discusses the importance of having a Lax representation with a spectral parameter. Then, in Section 2, he gives an account of the theory of bi-Hamiltonian manifolds and of $r$-matrix theory. The third section is devoted to the study of a special class of bi-Hamiltonian manifolds, admitting an interpretation also in terms of linear $r$-matrices. They are the product of $n$ copies of (the dual of) a finite-dimensional Lie algebra and actually possess an $(n+1)$-dimensional family of compatible Poisson brackets. Then it is shown how to construct bi-Hamiltonian hierarchies for some of these manifolds, using a kind of rule of fractional powers. The last section contains some finite- and infinite-dimensional examples to which the results of the paper are applied. For the $n$-dimensional rigid body, the Manakov integrals and the Mishchenko integrals are obtained. The Lagrange top is interpreted as a bi-Hamiltonian system on ${\germ s}{\germ o}(3)\times {\germ s}{\germ o}(3)$, clarifying a result due to T. Raţiu [Amer. J. Math. 104 (1982), no. 2, 409--448]. The KdV and Boussinesq hierarchies are treated as in [G. Magnano and F. Magri, Rev. Math. Phys. 3 (1991), no. 4, 403--466], avoiding the introduction of pseudodifferential operators.
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