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Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies
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 |
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] | |
[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] | |
[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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