2010/03/01 by Hasan Gümral, H. Gumral, Gumral, H. · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #3D Shape Modeling and Analysis #Computational Geometry and Mesh Generation #Dynamical Systems (math.DS) #FOS: Mathematics #Hamiltonian (control theory) #Mathematical optimization #Mathematical physics #Mathematics #Physics #Scientific Research and Discoveries #math.DS
paper · pdf · doi:10.48550/arxiv.1003.0343
12 pages
arxiv created 2010/03/01 · openalex publication_date 2010/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In three dimensions, the construction of bi-Hamiltonian structure can be reduced to the solutions of a Riccati equation with the arclength coordinate of a Frenet-Serret frame being the independent variable. Explicit integration of conserved quantities are connected with the coefficients of Riccati equation which are elements of the third cohomology class. All explicitly constructed examples of bi-Hamiltonian systems are exhausted when this class along with the first one vanishes. The latter condition provides integrating factor for explicit integration of Hamiltonian functions. For the Darboux-Halphen system, the Godbillon-Vey invariant is shown to arise as obstruction to integrability of integrating factor.