Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

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September 2014, 6(3): 373-415. doi: 10.3934/jgm.2014.6.373

Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

Lyudmila Grigoryeva ^1, , Juan-Pablo Ortega ^2, and Stanislav S. Zub ^3,

1.	Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex, France
2.	Centre National de la Recherche Scientifique, Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex
3.	Taras Shevchenko National University of Kyiv, 64, Volodymyrs'ka, 01601 Kyiv, Ukraine

Received November 2013 Revised June 2014 Published September 2014

Abstract
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This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron". We study the nonlinear stability of a branch of equatorial relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that allow us to conclude the sharpness of some of the nonlinear stability conditions obtained.

Keywords: Hamiltonian systems with symmetry, relative equilibrium, nonlinear stability/instability., momentum maps, orbitron, magnetic systems, generalized orbitron.

Mathematics Subject Classification: Primary: 37J15, 37J25; Secondary: 70E50, 70H14, 70H3.

Citation: Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Addison-Wesley, Reading, MA, 1978.
[2]	J. Cees van der Meer, The Hamiltonian Hopf Bifurcation, Springer-Verlag, 1985. doi: 10.1007/BFb0080357.
[3]	M. Dellnitz, I. Melbourne and J. E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry, Nonlinearity, 5 (1992), 979-996. doi: 10.1088/0951-7715/5/4/008.
[4]	H. R. Dullin, Poisson integrator for symmetric rigid bodies, Regular and Chaotic Dynamics, 9 (2004), 255-264. doi: 10.1070/RD2004v009n03ABEH000279.
[5]	H. R. Dullin and R. W. Easton, Stability of levitrons, Physica D, 126 (1999), 1-17. doi: 10.1016/S0167-2789(98)00251-6.
[6]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Archive for Rational Mechanics and Analysis, 158 (2001), 259-292. doi: 10.1007/PL00004245.
[7]	V. Guillemin and S. Sternberg, A normal form for the momentum map, in Differential Geometric Methods in Mathematical Physics (ed. S. Sternberg), Reidel Publishing Company, Dordrecht, (1984), 161-175.
[8]	R. Harrigan, Levitation device, US patent, 1983.
[9]	V. V. Kozorez, About a problem of two magnets, Bull. of the Ac. of Sc. of USSR. Series: Mechanics of a Rigid Body (in Russian), 3 (1974), 29-34.
[10]	V. V. Kozorez, Dynamic Systems of Free Magnetically Interacting Bodies, (Russian) Naukova dumka, Kiev, 1981.
[11]	N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, 33 (1997), 331-346. doi: 10.1016/S0005-1098(96)00176-8.
[12]	R. Krechetnikov and J. E. Marsden, On destabilizing effects of two fundamental non-conservative forces, Physica D, 214 (2006), 25-32. doi: 10.1016/j.physd.2005.12.003.
[13]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2002), 143-171. doi: 10.1088/0951-7715/15/1/307.
[14]	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.
[15]	N. E. Leonard and J. E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry, Physica D, 105 (1997), 130-162. doi: 10.1016/S0167-2789(97)83390-8.
[16]	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.
[17]	D. Lewis, Stacked Lagrange tops, Journal of Nonlinear Science, 8 (1998), 63-102. doi: 10.1007/s003329900044.
[18]	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.
[19]	C. Lim, J. A. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135. doi: 10.1016/S0167-2789(00)00167-6.
[20]	C.-M. Marle, Le voisinage d'une orbite d'une action hamiltonienne d'un groupe de Lie, Séminaire Sud-Rhodanien de Géométrie II, (1984), 19-35.
[21]	C.-M. Marle, Modéle d'action hamiltonienne d'un groupe the Lie sur une variété symplectique, Rend. Sem. Mat. Univers. Politecn. Torino, 43 (1985), 227-251.
[22]	J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics, 5 (1974), 121-130. doi: 10.1016/0034-4877(74)90021-4.
[23]	K. R. Meyer, Symmetries and integrals in mechanics, in Dynamical Systems (ed. M. M. Peixoto), Academic Press, (1973), 259-273.
[24]	J. A. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553-558. doi: 10.1016/S0764-4442(99)80389-9.
[25]	J. A. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466. doi: 10.1088/0951-7715/10/2/009.
[26]	J. A. Montaldi and R. M. Roberts, Relative equilibria of molecules, Journal of Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.
[27]	J. A. Montaldi, R. M. Roberts and I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237-293. doi: 10.1098/rsta.1988.0053.
[28]	J. Montaldi and M. Rodríguez-Olmos, On the stability of Hamiltonian relative equilibria with non-trivial isotropy, Nonlinearity, 24 (2011), 2777-2783. doi: 10.1088/0951-7715/24/10/007.
[29]	J.-P. Ortega, Symmetry, Reduction, and Stability in Hamiltonian Systems, Ph.D thesis, University of California, Santa Cruz, 1998.
[30]	J.-P. Ortega and T. S. Ratiu, A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems, Journal of Geometry and Physics, 32 (1999), 131-159. doi: 10.1016/S0393-0440(99)00025-X.
[31]	J.-P. Ortega and T. S. Ratiu, Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry, Journal of Geometry and Physics, 32 (1999), 160-188. doi: 10.1016/S0393-0440(99)00024-8.
[32]	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.
[33]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhauser Verlag, 2004. doi: 10.1007/978-1-4757-3811-7.
[34]	J.-P. Ortega and T. S. Ratiu, The reduced spaces of a symplectic Lie group action, Annals of Global Analysis and Geometry, 30 (2006), 335-381. doi: 10.1007/s10455-006-9017-9.
[35]	J.-P. Ortega and T. S. Ratiu, The stratified spaces of a symplectic Lie group action, Reports on Mathematical Physics, 58 (2006), 51-75. doi: 10.1016/S0034-4877(06)80040-6.
[36]	G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, Journal of Geometry and Physics, 9 (1992), 111-119. doi: 10.1016/0393-0440(92)90015-S.
[37]	G. W. Patrick, Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift, Journal of Nonlinear Science, 5 (1995), 373-418. doi: 10.1007/BF01212907.
[38]	G. W. Patrick, Two Axially Symmetric Coupled Rigid Bodies: Relative Equilibria, Stability, Bifurcations, and a Momentum Preserving Symplectic Integrator, Ph.D thesis, University of California, Berkeley, 1995.
[39]	G. W. Patrick, M. Roberts and C. Wulff, Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods, Archive for Rational Mechanics and Analysis, 174 (2004), 301-344. doi: 10.1007/s00205-004-0322-9.
[40]	S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria, Journal of Mathematical Physics, 39 (1998), 5894-5907. doi: 10.1063/1.532602.
[41]	M. Roberts, C. Wulff and J. S. W. Lamb, Hamiltonian systems near relative equilibria, Journal of Differential Equations, 179 (2002), 562-604. doi: 10.1006/jdeq.2001.4045.
[42]	M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems, Nonlinearity, 19 (2006), 853-877. doi: 10.1088/0951-7715/19/4/005.
[43]	M. Rodríguez-Olmos and M. E. Sousa-Dias, Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations, Journal of Nonlinear Science, 19 (2009), 179-219. doi: 10.1007/s00332-008-9032-z.
[44]	J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Archive for Rational Mechanics and Analysis, 115 (1991), 15-59. doi: 10.1007/BF01881678.
[45]	R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Annals of Mathematics, 134 (1991), 375-422. doi: 10.2307/2944350.
[46]	W. R. Smythe, Static and Dynamic Electricity, McGraw-Hill, New York, 1939.
[47]	S. S. Zub, Research into orbital motion stability in system of two magnetically interacting bodies, preprint, arXiv:1101.3237, 2012.
[48]	S. S. Zub, Stable orbital motion of magnetic dipole in the field of permanent magnets, Physica D, 275 (2014), 67-73. doi: 10.1016/j.physd.2014.02.007.

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Addison-Wesley, Reading, MA, 1978.
[2]	J. Cees van der Meer, The Hamiltonian Hopf Bifurcation, Springer-Verlag, 1985. doi: 10.1007/BFb0080357.
[3]	M. Dellnitz, I. Melbourne and J. E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry, Nonlinearity, 5 (1992), 979-996. doi: 10.1088/0951-7715/5/4/008.
[4]	H. R. Dullin, Poisson integrator for symmetric rigid bodies, Regular and Chaotic Dynamics, 9 (2004), 255-264. doi: 10.1070/RD2004v009n03ABEH000279.
[5]	H. R. Dullin and R. W. Easton, Stability of levitrons, Physica D, 126 (1999), 1-17. doi: 10.1016/S0167-2789(98)00251-6.
[6]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids, Archive for Rational Mechanics and Analysis, 158 (2001), 259-292. doi: 10.1007/PL00004245.
[7]	V. Guillemin and S. Sternberg, A normal form for the momentum map, in Differential Geometric Methods in Mathematical Physics (ed. S. Sternberg), Reidel Publishing Company, Dordrecht, (1984), 161-175.
[8]	R. Harrigan, Levitation device, US patent, 1983.
[9]	V. V. Kozorez, About a problem of two magnets, Bull. of the Ac. of Sc. of USSR. Series: Mechanics of a Rigid Body (in Russian), 3 (1974), 29-34.
[10]	V. V. Kozorez, Dynamic Systems of Free Magnetically Interacting Bodies, (Russian) Naukova dumka, Kiev, 1981.
[11]	N. E. Leonard, Stability of a bottom-heavy underwater vehicle, Automatica, 33 (1997), 331-346. doi: 10.1016/S0005-1098(96)00176-8.
[12]	R. Krechetnikov and J. E. Marsden, On destabilizing effects of two fundamental non-conservative forces, Physica D, 214 (2006), 25-32. doi: 10.1016/j.physd.2005.12.003.
[13]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2002), 143-171. doi: 10.1088/0951-7715/15/1/307.
[14]	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.
[15]	N. E. Leonard and J. E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry, Physica D, 105 (1997), 130-162. doi: 10.1016/S0167-2789(97)83390-8.
[16]	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.
[17]	D. Lewis, Stacked Lagrange tops, Journal of Nonlinear Science, 8 (1998), 63-102. doi: 10.1007/s003329900044.
[18]	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.
[19]	C. Lim, J. A. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135. doi: 10.1016/S0167-2789(00)00167-6.
[20]	C.-M. Marle, Le voisinage d'une orbite d'une action hamiltonienne d'un groupe de Lie, Séminaire Sud-Rhodanien de Géométrie II, (1984), 19-35.
[21]	C.-M. Marle, Modéle d'action hamiltonienne d'un groupe the Lie sur une variété symplectique, Rend. Sem. Mat. Univers. Politecn. Torino, 43 (1985), 227-251.
[22]	J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics, 5 (1974), 121-130. doi: 10.1016/0034-4877(74)90021-4.
[23]	K. R. Meyer, Symmetries and integrals in mechanics, in Dynamical Systems (ed. M. M. Peixoto), Academic Press, (1973), 259-273.
[24]	J. A. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553-558. doi: 10.1016/S0764-4442(99)80389-9.
[25]	J. A. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466. doi: 10.1088/0951-7715/10/2/009.
[26]	J. A. Montaldi and R. M. Roberts, Relative equilibria of molecules, Journal of Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.
[27]	J. A. Montaldi, R. M. Roberts and I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237-293. doi: 10.1098/rsta.1988.0053.
[28]	J. Montaldi and M. Rodríguez-Olmos, On the stability of Hamiltonian relative equilibria with non-trivial isotropy, Nonlinearity, 24 (2011), 2777-2783. doi: 10.1088/0951-7715/24/10/007.
[29]	J.-P. Ortega, Symmetry, Reduction, and Stability in Hamiltonian Systems, Ph.D thesis, University of California, Santa Cruz, 1998.
[30]	J.-P. Ortega and T. S. Ratiu, A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems, Journal of Geometry and Physics, 32 (1999), 131-159. doi: 10.1016/S0393-0440(99)00025-X.
[31]	J.-P. Ortega and T. S. Ratiu, Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry, Journal of Geometry and Physics, 32 (1999), 160-188. doi: 10.1016/S0393-0440(99)00024-8.
[32]	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.
[33]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhauser Verlag, 2004. doi: 10.1007/978-1-4757-3811-7.
[34]	J.-P. Ortega and T. S. Ratiu, The reduced spaces of a symplectic Lie group action, Annals of Global Analysis and Geometry, 30 (2006), 335-381. doi: 10.1007/s10455-006-9017-9.
[35]	J.-P. Ortega and T. S. Ratiu, The stratified spaces of a symplectic Lie group action, Reports on Mathematical Physics, 58 (2006), 51-75. doi: 10.1016/S0034-4877(06)80040-6.
[36]	G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, Journal of Geometry and Physics, 9 (1992), 111-119. doi: 10.1016/0393-0440(92)90015-S.
[37]	G. W. Patrick, Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift, Journal of Nonlinear Science, 5 (1995), 373-418. doi: 10.1007/BF01212907.
[38]	G. W. Patrick, Two Axially Symmetric Coupled Rigid Bodies: Relative Equilibria, Stability, Bifurcations, and a Momentum Preserving Symplectic Integrator, Ph.D thesis, University of California, Berkeley, 1995.
[39]	G. W. Patrick, M. Roberts and C. Wulff, Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods, Archive for Rational Mechanics and Analysis, 174 (2004), 301-344. doi: 10.1007/s00205-004-0322-9.
[40]	S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria, Journal of Mathematical Physics, 39 (1998), 5894-5907. doi: 10.1063/1.532602.
[41]	M. Roberts, C. Wulff and J. S. W. Lamb, Hamiltonian systems near relative equilibria, Journal of Differential Equations, 179 (2002), 562-604. doi: 10.1006/jdeq.2001.4045.
[42]	M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems, Nonlinearity, 19 (2006), 853-877. doi: 10.1088/0951-7715/19/4/005.
[43]	M. Rodríguez-Olmos and M. E. Sousa-Dias, Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations, Journal of Nonlinear Science, 19 (2009), 179-219. doi: 10.1007/s00332-008-9032-z.
[44]	J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method, Archive for Rational Mechanics and Analysis, 115 (1991), 15-59. doi: 10.1007/BF01881678.
[45]	R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Annals of Mathematics, 134 (1991), 375-422. doi: 10.2307/2944350.
[46]	W. R. Smythe, Static and Dynamic Electricity, McGraw-Hill, New York, 1939.
[47]	S. S. Zub, Research into orbital motion stability in system of two magnetically interacting bodies, preprint, arXiv:1101.3237, 2012.
[48]	S. S. Zub, Stable orbital motion of magnetic dipole in the field of permanent magnets, Physica D, 275 (2014), 67-73. doi: 10.1016/j.physd.2014.02.007.

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