Applications of the Kosambi–Cartan–Chern Theory to Hamiltonian Systems on a Cotangent Bundle: Linking Geometric Quantities to the Self-Similar Motions of Three Point Vortices

This study presents a differential geometric framework for Hamiltonian systems expressed in terms of first-order differential equations. For systems governed by second-order ordinary differential equations on tangent bundles, such as Euler–Lagrange systems, the stability of trajectories under perturbations is analyzed based on the eigenvalue of the deviation curvature tensor. Building upon this Jacobi stability analysis approach, four geometric quantities for Hamiltonian systems are derived considering perturbations to trajectories on a cotangent bundle. As a specific Hamiltonian system, a hydrodynamic three-point vortex system is examined, and its four geometric quantities are computed using the Hamiltonian equation. The eigenvalues of these geometric quantities are then used to classify the divergent and collapsing trajectories of point vortices. Specifically, for the divergent trajectories of vortices, the eigenvalues of the geometric quantities converge to zero over time. Conversely, for their collapsing trajectories, the eigenvalues increase with time. This result implies that at the point of vortex collapse, the system becomes geometrically unstable, with diverging trajectory perturbations.