Lagrangian foliations and Lax equations
type: Item
instance of: scholarly article
title: Lagrangian foliations and Lax equations
zbMATH Open document ID: 0578.58017
DOI: 10.1007/BF00400715
author: Yvette Kosmann-Schwarzbach
published in: Letters in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: Let M be a finite dimensional manifold with two different symplectic structures \(\omega\) and \({\bar \omega}\). If a vector field X on M can be considered as locally Hamiltonian with respect to both \(\omega\) and \({\bar \omega}\), it is known that the equation \(\dot x=X(x)\) gives rise to, in each local coordinate system, a local matrix equation \(\dot R=[A,R]\) which expresses the invariance with respect to X of the (1,1) tensor field R. The order of these matrices are the dimension of the symplectic manifold. In a recent paper \textit{J. F. Cariñena} and \textit{L. A. Ibort} [ibid. 8, 21-26 (1984; Zbl 0526.58018)] showed that if \({\mathcal F}\) is a foliation on M that is Lagrangian with respect to both symplectic structures \(\omega\) and \({\bar \omega}\), the (1,1) tensor field R preserves the foliation \({\mathcal F}\) and, if X is tangent to \({\mathcal F}\), the restriction L of R to \({\mathcal F}\) is parallel along X with respect to the canonical flat connection on \({\mathcal F}\) defined by either of the symplectic forms. Moreover, this last condition, written in local coordinates adapted to the foliation, yields an equation \(\dot L=[L,B]\) in which the order of the matrices L and B is half the dimension of the symplectic manifold. For this reason, Cariñena and Ibort call this equation a ''local Lax equation''. The author of the present paper disagrees. She believes that the term ''Lax equation'' should be reserved for the case where \(\tilde L=[L,B]\) is equivalent to the dynamical system \(\dot x=X(x)\). She explains geometrically how such equations arise, obtains their form in arbitrary local coordinates adapted to the foliation, and shows that in any system of adapted local coordinates which are canonical for one of the symplectic structures, the field of matrices B vanishes identically. Thus, in this special case, the ''Lax equation'' reduces to \(\dot L=0\), and, therefore, yields no information regarding the dynamical vector field X.
reviewed by: Chung-Wu Ho
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 3925832
zbMATH Keywords: Hamiltonian vector field
zbMATH Keywords: Lagrangian foliation
zbMATH Keywords: Lax equation
zbMATH Keywords: symplectic structures
zbMATH Keywords: locally Hamiltonian
MaRDI profile type: Publication
cites work: On Lax equations arising from Lagrangian foliations
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cites work: Q3908200
description: scientific article; zbMATH DE number 3925832