Quadratic Hamilton-Poisson systems in three dimensions: equivalence, stability, and integration. (English) Zbl 1406.70024
In this paper the authors consider Hamilton-Poisson quadratic systems on three-dimensional Lie-Poisson spaces. Positive (semi-definite) quadratic Hamilton-Poisson systems in Lee-Poisson spaces arise when invariant optimal control problems are studied on Lie groups [A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint. Berlin: Springer (2004; Zbl 1062.93001), V. Jurdjevic, Geometric control theory. Cambridge: Cambridge Univ. Press (1997; Zbl 0940.93005), R. Biggs and C. C. Remsing, Mediterr. J. Math. 11, No. 1, 193–215 (2014; Zbl 1305.93007)].
Only homogeneous (positive) semidefinite systems are investigated in this article. First, the systems are classified: an exhaustive and nonredundant list of of 23 normal forms is presented.
The classification is done in two steps: the authors classified the systems on each three-dimensional Lie-Poisson space and subsequently, equivalences of systems on non-isomorphic Lie-Poisson spaces (- on each three-dimensional Lie-Poisson space) is consider. Second, for each normal form, the stability of the equilibria is investigated and explicit expressions for the integral curves of all but three families of systems are obtained.
Only homogeneous (positive) semidefinite systems are investigated in this article. First, the systems are classified: an exhaustive and nonredundant list of of 23 normal forms is presented.
The classification is done in two steps: the authors classified the systems on each three-dimensional Lie-Poisson space and subsequently, equivalences of systems on non-isomorphic Lie-Poisson spaces (- on each three-dimensional Lie-Poisson space) is consider. Second, for each normal form, the stability of the equilibria is investigated and explicit expressions for the integral curves of all but three families of systems are obtained.
Reviewer: Irina V. Konopleva (Ul’yanovsk)
MSC:
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |
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