2015/02/11 by Ender Abadoğlu, Abadoğlu, Ender, Hasan Gümral +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #math.DS
paper · pdf · doi:10.48550/arxiv.1502.03238
arxiv created 2015/02/11 · openalex publication_date 2015/02/11 · arxiv updated 2015/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Working bi-Hamiltonian structure and Jacobi identity in Frenet-Serret frame associated to a dynamical system, we proved that all dynamical systems in three dimensions possess two compatible Poisson structures. We investigate relations between geometry of surfaces defined by potential function of a gradient system and its bi-Hamiltonian structure. We show that it is possible to find Hamiltonian functions whose gradient flows have geodesic curvature zero on potential surfaces. Using this, we conclude that Hamiltonian functions are determined by distance functions on potential surfaces. We apply this technique to find conserved quantities of three dimensional gradient systems including the Aristotelian model of the three-body motion.