Sturm theory with applications in geometry and classical mechanics
Autor: | Li Wu, Vivina Barutello, Alessandro Portaluri, Daniel Offin |
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Jazyk: | angličtina |
Rok vydání: | 2020 |
Předmět: |
Mathematics - Differential Geometry
Pure mathematics 53D12 70F05 70F10 General Mathematics 010102 general mathematics Dynamical Systems (math.DS) Phase plane Space (mathematics) 01 natural sciences Linear subspace Celestial mechanics 010101 applied mathematics Differential Geometry (math.DG) Mathematics - Classical Analysis and ODEs Line (geometry) Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics Mathematics - Dynamical Systems Rotation (mathematics) Mathematics::Symplectic Geometry Distribution (differential geometry) Symplectic geometry Mathematics |
Popis: | 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. 38 pages, no figures |
Databáze: | OpenAIRE |
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