Maslov index and symplectic Sturm theorems (Q1289327)

The paper is devoted to a generalization of symplectic Sturm theorems. Recall that the classical Sturm theorems for a second order ODE describe the rotation of a line in the phase space of the equation. In the symplectic version lines are replaced by Lagrangian planes and the instants of intersection with a given line are replaced by instants of nontransversality with a given plane. To be more precise, the train of a given Lagrangian plane is defined to be a hypersurface in Lagrangian Grassmannian generated by all Lagrangian planes not transversal to the given one, the Maslov index is defined to be an index of intersection of a curve on a Lagrangian Grassmannian with a given train, and the symplectic Sturm theorems describe some properties of this index [see \textit{V. I. Arnold}, Funkts. Anal. Prilozh. 19, 1-10 (1985; Zbl 0606.58017)]. In the present paper the generalization of the notion of the train for a given Lagrangian plane is suggested and the analogues of symplectic Sturm theorems for the generalized trains are obtained.