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September  2020, 12(3): 525-540. doi: 10.3934/jgm.2020019

Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy

Miguel Rodríguez-Olmos

Dedicated to Professor James Montaldi

Received  September 2019 Revised  June 2020 Published  September 2020 Early access  July 2020

We survey several aspects of the qualitative dynamics around Hamiltonian relative equilibria. We pay special attention to the role of continuous singularities and its effect in their stability, persistence and bifurcations. Our approach is semi-global using extensively the Hamiltonian tube of Marle, Guillemin and Sternberg.

Keywords: Relative equilibria, Hamiltonian systems, momentum maps, stability, persistence, bifurcations.
Mathematics Subject Classification: 53D20, 37J20.
Citation: Miguel Rodríguez-Olmos. Continuous singularities in hamiltonian relative equilibria with abelian momentum isotropy. Journal of Geometric Mechanics, 2020, 12 (3) : 525-540. doi: 10.3934/jgm.2020019
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley Pub. Comp. Inc., 1978.  Google Scholar

[2]

P. Chossat, J. P. Ortega and T. S. Ratiu, Hamiltonian Hopf bifurcation with symmetry, Archive for Rational Mechanics and Analysis, 63 (2002), 1-33.  doi: 10.1007/s002050200182.  Google Scholar

[3]

J. J. Duistermaat and J. A. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[4]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151.   Google Scholar

[5]

M. Fontaine and J. Montaldi, Persistence of stationary motion under explicit symmetry breaking perturbation, Nonlinearity, 32 (2019), 1999-2023.  doi: 10.1088/1361-6544/ab003e.  Google Scholar

[6]

F. Grabsi, J. Montaldi and J. P. Ortega, Bifurcation and forced symmetry breaking in Hamiltonian systems, Comptes Rendus Mathématique Académie des Sciences, 7 (2004), 565-570.  doi: 10.1016/j.crma.2004.01.029.  Google Scholar

[7]

V. Guillemin and S. Sternberg, A normal form for the moment map, in Differential Geometric Methods in Mathematical Physics, S. Sternberg ed. Mathematical Physics Studies, 6, 1984.  Google Scholar

[8]

J. L. Lagrange, Essai sur le probléme des trois corps, Oeuvres, 6 (1772), 229-334.   Google Scholar

[9]

E. Lerman and S. F. Singer, Stability and persistence of relative equilibria at singular values of the moment map, Nonlinearity, 11 (1998), 1637-1649.  doi: 10.1088/0951-7715/11/6/012.  Google Scholar

[10]

D. Lewis, T. S. Ratiu, J. C. Simo and J. E. Marsden, The heavy top: A geometric treatment, Nonlinearity, 5 (1992), 1-48.  doi: 10.1088/0951-7715/5/1/001.  Google Scholar

[11]

D. Lewis and J. C. Simo, Nonlinear stability of rotating pseudo-rigid bodies, Proc. Roy. Soc. London Ser. A, 427 (1990), 281-319.  doi: 10.1098/rspa.1990.0014.  Google Scholar

[12]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Physica D: Nonlinear Phenomena, 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.  Google Scholar

[13]

C. M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino, 43 (1985), 227-251.   Google Scholar

[14]

J. E. Marsden, Lectures on Mechanics, Lecture Note Series, 174, LMS, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar

[15]

J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[16]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  doi: 10.1088/0951-7715/10/2/009.  Google Scholar

[17]

J. Montaldi, Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry, Journal of Geometric Mechanics, 6 (2014), 237-260.  doi: 10.3934/jgm.2014.6.237.  Google Scholar

[18]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Mathematical Physics, 55 (2014), 1-14.  doi: 10.1063/1.4897210.  Google Scholar

[19]

J. Montaldi and M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar

[20]

J. Montaldi and M. Roberts, Note on semisymplectic actions of Lie groups, C. R. Acad. Sci. Paris Ser. I, 330 (2000), 1079-1084.  doi: 10.1016/S0764-4442(00)00322-0.  Google Scholar

[21]

J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc., 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar

[22]

J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.  doi: 10.1088/0951-7715/3/3/009.  Google Scholar

[23]

J. Montaldi, M. Roberts and I. Stewart, Stability of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 731-772.  doi: 10.1088/0951-7715/3/3/010.  Google Scholar

[24]

J. Montaldi and M. Rodríguez-Olmos, Hamiltonian relative equilibria with continuous isotropy, arXiv: 1509.04961. Google Scholar

[25]

J. Montaldi and A. Shaddad, Generalized point vortex dynamics on CP$^2$, J. Geometric Mechanics, to appear. doi: 10.3934/jgm.2019030.  Google Scholar

[26]

J. Montaldi, A. Souliere and T. Tokieda, Vortex dynamics on cylinders, SIAM J. on Applied Dynamical Systems, 2 (2003), 417-430.  doi: 10.1137/S1111111102415569.  Google Scholar

[27]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.  Google Scholar

[28]

J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings, Springer Proceedings in Mathematics and Statistics, 35 (2013), 335-370.  doi: 10.1007/978-3-0348-0451-6_14.  Google Scholar

[29]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Communications in Pure and Applied Mathematics, 29 (1976), 727-747.  doi: 10.1002/cpa.3160290613.  Google Scholar

[30]

I. Newton, Philosophiae Naturalis Principia Mathematica, Book III, London, 1687.  Google Scholar

[31]

J. P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity, 12 (1999), 693-720.  doi: 10.1088/0951-7715/12/3/315.  Google Scholar

[32]

J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222, Birkhauser-Verlag, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[33]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[34]

G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamics interpretation of nonlinear stability on the reduced phase space, J. Geom. Phys., 9 (1992), 111-119.  doi: 10.1016/0393-0440(92)90015-S.  Google Scholar

[35]

F. Laurent-Polz, J. Montaldi and M. Roberts, Point Vortices on the Sphere: Stability of Symmetric Relative Equilibria, Journal of Geometric Mechanics, 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439.  Google Scholar

[36]

M. Roberts and M. E. Sousa-Dias, Bifurcations from relative equilibria of Hamiltonian systems, Nonlinearity, 10 (1997), 1719-1738.  doi: 10.1088/0951-7715/10/6/015.  Google Scholar

[37]

M. Roberts, C. Wulff and J. S. Lamb, Hamiltonian systems near relative equilibria, J. Differential Equations, 179 (2002), 562-604.  doi: 10.1006/jdeq.2001.4045.  Google Scholar

[38]

J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Arch. Rational Mech. Anal., 115 (1991), 15-59.  doi: 10.1007/BF01881678.  Google Scholar

[39]

S. Smale, Topology and Mechanics I, Inventiones Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar

[40]

A. Weinstein, Normal modes for nonlinear hamiltonian systems, Inventiones Mathematicae, 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley Pub. Comp. Inc., 1978.  Google Scholar

[2]

P. Chossat, J. P. Ortega and T. S. Ratiu, Hamiltonian Hopf bifurcation with symmetry, Archive for Rational Mechanics and Analysis, 63 (2002), 1-33.  doi: 10.1007/s002050200182.  Google Scholar

[3]

J. J. Duistermaat and J. A. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[4]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151.   Google Scholar

[5]

M. Fontaine and J. Montaldi, Persistence of stationary motion under explicit symmetry breaking perturbation, Nonlinearity, 32 (2019), 1999-2023.  doi: 10.1088/1361-6544/ab003e.  Google Scholar

[6]

F. Grabsi, J. Montaldi and J. P. Ortega, Bifurcation and forced symmetry breaking in Hamiltonian systems, Comptes Rendus Mathématique Académie des Sciences, 7 (2004), 565-570.  doi: 10.1016/j.crma.2004.01.029.  Google Scholar

[7]

V. Guillemin and S. Sternberg, A normal form for the moment map, in Differential Geometric Methods in Mathematical Physics, S. Sternberg ed. Mathematical Physics Studies, 6, 1984.  Google Scholar

[8]

J. L. Lagrange, Essai sur le probléme des trois corps, Oeuvres, 6 (1772), 229-334.   Google Scholar

[9]

E. Lerman and S. F. Singer, Stability and persistence of relative equilibria at singular values of the moment map, Nonlinearity, 11 (1998), 1637-1649.  doi: 10.1088/0951-7715/11/6/012.  Google Scholar

[10]

D. Lewis, T. S. Ratiu, J. C. Simo and J. E. Marsden, The heavy top: A geometric treatment, Nonlinearity, 5 (1992), 1-48.  doi: 10.1088/0951-7715/5/1/001.  Google Scholar

[11]

D. Lewis and J. C. Simo, Nonlinear stability of rotating pseudo-rigid bodies, Proc. Roy. Soc. London Ser. A, 427 (1990), 281-319.  doi: 10.1098/rspa.1990.0014.  Google Scholar

[12]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Physica D: Nonlinear Phenomena, 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.  Google Scholar

[13]

C. M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino, 43 (1985), 227-251.   Google Scholar

[14]

J. E. Marsden, Lectures on Mechanics, Lecture Note Series, 174, LMS, Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar

[15]

J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[16]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  doi: 10.1088/0951-7715/10/2/009.  Google Scholar

[17]

J. Montaldi, Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry, Journal of Geometric Mechanics, 6 (2014), 237-260.  doi: 10.3934/jgm.2014.6.237.  Google Scholar

[18]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Mathematical Physics, 55 (2014), 1-14.  doi: 10.1063/1.4897210.  Google Scholar

[19]

J. Montaldi and M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar

[20]

J. Montaldi and M. Roberts, Note on semisymplectic actions of Lie groups, C. R. Acad. Sci. Paris Ser. I, 330 (2000), 1079-1084.  doi: 10.1016/S0764-4442(00)00322-0.  Google Scholar

[21]

J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc., 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar

[22]

J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.  doi: 10.1088/0951-7715/3/3/009.  Google Scholar

[23]

J. Montaldi, M. Roberts and I. Stewart, Stability of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 731-772.  doi: 10.1088/0951-7715/3/3/010.  Google Scholar

[24]

J. Montaldi and M. Rodríguez-Olmos, Hamiltonian relative equilibria with continuous isotropy, arXiv: 1509.04961. Google Scholar

[25]

J. Montaldi and A. Shaddad, Generalized point vortex dynamics on CP$^2$, J. Geometric Mechanics, to appear. doi: 10.3934/jgm.2019030.  Google Scholar

[26]

J. Montaldi, A. Souliere and T. Tokieda, Vortex dynamics on cylinders, SIAM J. on Applied Dynamical Systems, 2 (2003), 417-430.  doi: 10.1137/S1111111102415569.  Google Scholar

[27]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.  Google Scholar

[28]

J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings, Springer Proceedings in Mathematics and Statistics, 35 (2013), 335-370.  doi: 10.1007/978-3-0348-0451-6_14.  Google Scholar

[29]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Communications in Pure and Applied Mathematics, 29 (1976), 727-747.  doi: 10.1002/cpa.3160290613.  Google Scholar

[30]

I. Newton, Philosophiae Naturalis Principia Mathematica, Book III, London, 1687.  Google Scholar

[31]

J. P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity, 12 (1999), 693-720.  doi: 10.1088/0951-7715/12/3/315.  Google Scholar

[32]

J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222, Birkhauser-Verlag, 2004. doi: 10.1007/978-1-4757-3811-7.  Google Scholar

[33]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[34]

G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamics interpretation of nonlinear stability on the reduced phase space, J. Geom. Phys., 9 (1992), 111-119.  doi: 10.1016/0393-0440(92)90015-S.  Google Scholar

[35]

F. Laurent-Polz, J. Montaldi and M. Roberts, Point Vortices on the Sphere: Stability of Symmetric Relative Equilibria, Journal of Geometric Mechanics, 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439.  Google Scholar

[36]

M. Roberts and M. E. Sousa-Dias, Bifurcations from relative equilibria of Hamiltonian systems, Nonlinearity, 10 (1997), 1719-1738.  doi: 10.1088/0951-7715/10/6/015.  Google Scholar

[37]

M. Roberts, C. Wulff and J. S. Lamb, Hamiltonian systems near relative equilibria, J. Differential Equations, 179 (2002), 562-604.  doi: 10.1006/jdeq.2001.4045.  Google Scholar

[38]

J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Arch. Rational Mech. Anal., 115 (1991), 15-59.  doi: 10.1007/BF01881678.  Google Scholar

[39]

S. Smale, Topology and Mechanics I, Inventiones Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar

[40]

A. Weinstein, Normal modes for nonlinear hamiltonian systems, Inventiones Mathematicae, 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

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