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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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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. 

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[40]

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

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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. 

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[40]

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

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