Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems
| Autor: | Pietro-Luciano Buono, Daniel Offin |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Control and Optimization Computer Science::Information Retrieval Applied Mathematics 010102 general mathematics Mathematical analysis Symmetry group 01 natural sciences Linear subspace Hamiltonian system 010101 applied mathematics symbols.namesake Mechanics of Materials Homogeneous space symbols Geometry and Topology Boundary value problem 0101 mathematics Hamiltonian (quantum mechanics) Subspace topology Mathematics Linear stability |
| Zdroj: | Journal of Geometric Mechanics. 9:439-457 |
| ISSN: | 1941-4897 |
| DOI: | 10.3934/jgm.2017017 |
| Popis: | We consider the question of linear stability of a periodic solution \begin{document}$z(t)$\end{document} with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that \begin{document}$z(t)$\end{document} is unstable if a subspace \begin{document}$W$\end{document} associated with the boundary conditions of the minimizing problem is a Lagrangian subspace with no focal points on the time interval defined by the boundary conditions and the second variation restricted to the subspace \begin{document}$W$\end{document} at the minimizer has positive directions. We show that the conditions of our theorem are always met for a class of minimizing periodic orbits with the standard mechanical reversing symmetry. Comparison theorems for Lagrangian subspaces and the use of time-reversing symmetries are essential tools in constructing stable and unstable subspaces for \begin{document}$z(t)$\end{document} . In particular, our results are complementary to the recent paper of Hu and Sun Commun. Math. Phys . 290 , (2009). |
| Databáze: | OpenAIRE |
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