Sturm theory with applications in geometry and classical mechanics

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

Linear symplectic geometry

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the Maslov index. In the section on linear complex structures particular emphasis is placed on the homotopy equivalence between the space of symplectic forms and the space of linear complex structures. The chapter includes sections on symplectic vector bundles and the first Chern class.

Ribbon graphs and bialgebra of Lagrangian subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

Principal Pivot Transforms of Quasidefinite Matrices and Semidefinite Lagrangian Subspaces

Lagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. We introduce a class of semidefinite Lagrangian subspaces and show that these subspaces can be represented by a subset I ⊆ {1, 2, . . . , n} and a Hermitian matrix X ∈ C n×n with the property that the submatrix X II is negative semidefinite and the submatrix X I c I c is positive semidefinite. A matrix X with these definiteness properties is called I-semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition we show that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence this structure is ubiquitous in control theory. The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [SIAM J. Matrix Anal. Appl., 33(2012), pp. 780–805] to convert between two different pairs (I, X) and (J , X 0 ) representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, we prove that the symmetric PPT of an I-semidefinite matrix X is a J -semidefinite matrix X 0 , and we derive an implementation of the transformation X 7→ X 0 that both makes use of the definiteness properties of X and guarantees the definiteness of the submatrices of X 0 in finite arithmetic. We use the resulting formulas to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (I opt , X opt ) with M = max i,j |(X opt ) ij | as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on M with respect to the general case.