2018/10/06 by Andrei Agrachev, Agrachev, Andrei, Ivan Beschastnyi +1 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Mathematical Biology Tumor Growth #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1810.02960
openalex publication_date 2018/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.