On the stability and integration of Hamilton-Poisson systems on \(\mathfrak{so} (3)^*_{-}\). (English) Zbl 1416.17022
Summary: We consider inhomogeneous quadratic Hamilton-Poisson systems on the
Lie-Poisson space \(\mathfrak{so} (3)^*_{-}\). There are nine such systems up to affine equivalence. We investigate the stability nature of the equilibria for each of these systems. For a subclass of systems, we find explicit expressions for the integral curves in terms of Jacobi elliptic functions.
MSC:
17B80 | Applications of Lie algebras and superalgebras to integrable systems |
33E05 | Elliptic functions and integrals |
37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |
53D17 | Poisson manifolds; Poisson groupoids and algebroids |