Lagrangian foliations and Lax equations
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.
scientific article; zbMATH DE number 3925832