### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 021, 37 pages      arXiv:1707.02828      https://doi.org/10.3842/SIGMA.2018.021

### Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

Stefan Klajbor-Goderich
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801 USA

Received October 31, 2017, in final form March 09, 2018; Published online March 14, 2018

Abstract
We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos's criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.

Key words: equivariant dynamics; relative equilibria; orbital stability; Hamiltonian systems.

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References

1. Abraham R., Marsden J.E., Ratiu T., Manifolds, tensor analysis, and applications, Applied Mathematical Sciences, Vol. 75, 2nd ed., Springer-Verlag, New York, 1988.
2. Chossat P., Lauterbach R., Methods in equivariant bifurcations and dynamical systems, Advanced Series in Nonlinear Dynamics, Vol. 15, World Sci. Publ. Co., Inc., River Edge, NJ, 2000.
3. Duistermaat J.J., Fourier integral operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011.
4. Duistermaat J.J., Kolk J.A.C., Lie groups, Universitext, Springer-Verlag, Berlin, 2000.
5. Field M.J., Dynamics and symmetry, ICP Advanced Texts in Mathematics, Vol. 3, Imperial College Press, London, 2007.
6. Golubitsky M., Stewart I., The symmetry perspective. From equilibrium to chaos in phase space and physical space, Progress in Mathematics, Vol. 200, Birkhäuser Verlag, Basel, 2002.
7. Guillemin V., Ginzburg V., Karshon Y., Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, Vol. 98, Amer. Math. Soc., Providence, RI, 2002.
8. Guillemin V., Sternberg S., A normal form for the moment map, in Differential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., Vol. 6, Reidel, Dordrecht, 1984, 161-175.
9. Hepworth R., Vector fields and flows on differentiable stacks, Theory Appl. Categ. 22 (2009), 542-587, arXiv:0810.0979.
10. Kolář I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
11. Krupa M., Bifurcations of relative equilibria, SIAM J. Math. Anal. 21 (1990), 1453-1486.
12. Lerman E., Invariant vector fields and groupoids, Int. Math. Res. Not. 2015 (2015), 7394-7416, arXiv:1307.7733.
13. Lerman E., Singer S.F., Stability and persistence of relative equilibria at singular values of the moment map, Nonlinearity 11 (1998), 1637-1649.
14. Marle C.-M., Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), 227-251.
15. Marsden J.E., Lectures on mechanics, London Mathematical Society Lecture Note Series, Vol. 174, Cambridge University Press, Cambridge, 1992.
16. Montaldi J., Rodríguez-Olmos M., Hamiltonian relative equilibria with continuous isotropy, arXiv:1509.04961.
17. Montaldi J., Rodríguez-Olmos M., On the stability of Hamiltonian relative equilibria with non-trivial isotropy, Nonlinearity 24 (2011), 2777-2783, arXiv:1011.1130.
18. Ortega J.-P., Ratiu T.S., Stability of Hamiltonian relative equilibria, Nonlinearity 12 (1999), 693-720.
19. Ortega J.-P., Ratiu T.S., Momentum maps and Hamiltonian reduction, Progress in Mathematics, Vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004.
20. Palais R.S., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295-323.
21. Patrick G.W., Two axially symmetric coupled rigid bodies: Relative equilibria, stability, bifurcations, and a momentum preserving symplectic integrator, Ph.D. Thesis, University of California, Berkeley, 1991.
22. Patrick G.W., Relative equilibria of Hamiltonian systems with symmetry: linearization, smoothness, and drift, J. Nonlinear Sci. 5 (1995), 373-418.
23. Roberts M., Wulff C., Lamb J.S.W., Hamiltonian systems near relative equilibria, J. Differential Equations 179 (2002), 562-604.
24. Roberts R.M., de Sousa Dias M.E.R., Bifurcations from relative equilibria of Hamiltonian systems, Nonlinearity 10 (1997), 1719-1738.
25. Wulff C., Patrick G., Roberts M., Stability of Hamiltonian relative equilibria by energy methods, in Symmetry and Perturbation Theory (Cala Gononoe, 2001), World Sci. Publ., River Edge, NJ, 2001, 214-221.