# American Institute of Mathematical Sciences

July  2011, 10(4): 1257-1266. doi: 10.3934/cpaa.2011.10.1257

## On the similarity of Hamiltonian and reversible vector fields in 4D

 1 Department of Mathematics, IMECC, Unicamp, 13083-970, Campinas SP, Brazil 2 Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P., Brazil

Received  July 2010 Revised  September 2010 Published  April 2011

We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal form theory.
Citation: Ricardo Miranda Martins, Marco Antonio Teixeira. On the similarity of Hamiltonian and reversible vector fields in 4D. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1257-1266. doi: 10.3934/cpaa.2011.10.1257
##### References:
 [1] R. Abraham and J. Marsden, "Foundations of Mechanics," Benjamin Cummings, London, 1978.  Google Scholar [2] V. I. Arnold, "Arnold's Problems," Springer-Verlag, Berlin, 2004.  Google Scholar [3] V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,'' Springer-Verlag, New York, 1988. Google Scholar [4] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.2307/1988861.  Google Scholar [5] R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.2307/1997429.  Google Scholar [6] G. Gaeta, Normal Forms of reversible dynamical systems, International Journal of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar [7] A. Jacquemard, M. F. S. Lima and M. A. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. Google Scholar [8] J. S. W. Lamb, M. F. S. Lima, R. M. Martins, M. A. Teixeira and J. Yang, On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in $R^4$, IMECC/Unicamp Research Report 05/10, 2010. Available from: http://www1.ime.unicamp.br/rel_pesq/relatorio.html. Google Scholar [9] J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, Hamiltonian Structure of the Reversible Nonsemisimple $1:1$ Resonance, in "Dynamics, Bifurcation and Symmetry: New Trends and New Tools", Kluwer Academic Publishers, (1994), 221-240.  Google Scholar [10] G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.2307/1989695.  Google Scholar [11] M. A. Teixeira, Singularities of reversible vector fields, Phys. D, 100 (1997), 101-118. doi: 10.1016/S0167-2789(96)00183-2.  Google Scholar

show all references

##### References:
 [1] R. Abraham and J. Marsden, "Foundations of Mechanics," Benjamin Cummings, London, 1978.  Google Scholar [2] V. I. Arnold, "Arnold's Problems," Springer-Verlag, Berlin, 2004.  Google Scholar [3] V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,'' Springer-Verlag, New York, 1988. Google Scholar [4] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.2307/1988861.  Google Scholar [5] R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.2307/1997429.  Google Scholar [6] G. Gaeta, Normal Forms of reversible dynamical systems, International Journal of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar [7] A. Jacquemard, M. F. S. Lima and M. A. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica Pura ed Applicata, 187 (2008), 105-117. Google Scholar [8] J. S. W. Lamb, M. F. S. Lima, R. M. Martins, M. A. Teixeira and J. Yang, On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in $R^4$, IMECC/Unicamp Research Report 05/10, 2010. Available from: http://www1.ime.unicamp.br/rel_pesq/relatorio.html. Google Scholar [9] J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, Hamiltonian Structure of the Reversible Nonsemisimple $1:1$ Resonance, in "Dynamics, Bifurcation and Symmetry: New Trends and New Tools", Kluwer Academic Publishers, (1994), 221-240.  Google Scholar [10] G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.2307/1989695.  Google Scholar [11] M. A. Teixeira, Singularities of reversible vector fields, Phys. D, 100 (1997), 101-118. doi: 10.1016/S0167-2789(96)00183-2.  Google Scholar
 [1] Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97. [2] Robert Roussarie. A topological study of planar vector field singularities. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5217-5245. doi: 10.3934/dcds.2020226 [3] Nikolay A. Gusev. On the one-dimensional continuity equation with a nearly incompressible vector field. Communications on Pure & Applied Analysis, 2019, 18 (2) : 559-568. doi: 10.3934/cpaa.2019028 [4] Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020105 [5] Tomasz Kaczynski, Marian Mrozek, Thomas Wanner. Towards a formal tie between combinatorial and classical vector field dynamics. Journal of Computational Dynamics, 2016, 3 (1) : 17-50. doi: 10.3934/jcd.2016002 [6] Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763 [7] Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 [8] Biao Ou. Examinations on a three-dimensional differentiable vector field that equals its own curl. Communications on Pure & Applied Analysis, 2003, 2 (2) : 251-257. doi: 10.3934/cpaa.2003.2.251 [9] Mourad Bellassoued, Ibtissem Ben Aïcha, Zouhour Rezig. Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds. Mathematical Control & Related Fields, 2021, 11 (2) : 403-431. doi: 10.3934/mcrf.2020042 [10] Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667 [11] Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769 [12] Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375 [13] Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239 [14] Charles-Michel Marle. A property of conformally Hamiltonian vector fields; Application to the Kepler problem. Journal of Geometric Mechanics, 2012, 4 (2) : 181-206. doi: 10.3934/jgm.2012.4.181 [15] Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002 [16] Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51 [17] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [18] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [19] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [20] Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971

2020 Impact Factor: 1.916