Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry

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June 2014, 6(2): 237-260. doi: 10.3934/jgm.2014.6.237

Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry

James Montaldi ^1,

1.	School of Mathematics, University of Manchester, Manchester, M13 9PL

Received November 2013 Revised April 2014 Published June 2014

Abstract
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For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors.

Keywords: Momentum map, singularity theory., bifurcations, relative equilibria, SO(3) symmetry, symplectic reduction.

Mathematics Subject Classification: 70H33, 58F14, 37J2.

Citation: James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237

References:

[1]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978.
[2]	A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden & G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756. doi: 10.1016/0005-1098(92)90034-D.
[3]	J. W. Bruce & R. M. Roberts, Critical points of functions on analytic varieties, Topology, 27 (1988), 57-90. doi: 10.1016/0040-9383(88)90007-9.
[4]	L. Buono, F. Laurent-Polz & J. Montaldi, Symmetric Hamiltonian Bifurcations, In Geometric Mechanics and Symmetry: The Peyresq Lectures, J. Montaldi and T. S. Ratiu, eds. pp 357-402. Cambridge University Press, 2005. doi: 10.1017/CBO9780511526367.007.
[5]	J. Damon, The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcalK$, Memoirs A.M.S., 50 (1984), x+88 pp. doi: 10.1090/memo/0306.
[6]	J. Damon, Deformations of sections of singularities and Gorenstein surface singularities, Am. J. Math., 109 (1987), 695-721. doi: 10.2307/2374610.
[7]	J. Damon, $\mathcal{A}$-equivalence and the equivalence of sections of images and disriminants, In Singularity Theory and its Applications, Part I, Springer Lecture Notes in Math., 1462 (1991), 93-121. doi: 10.1007/BFb0086377.
[8]	V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511574788.
[9]	P. S. Krishnaprasad, Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035. doi: 10.1016/0362-546X(85)90083-5.
[10]	F. Laurent-Polz, J. Montaldi & M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria, J. Geom. Mech, 3 (2012), 439-486. doi: 10.3934/jgm.2011.3.439.
[11]	E. Lerman & 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.
[12]	C. Lim, J. Montaldi & M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135. doi: 10.1016/S0167-2789(00)00167-6.
[13]	J. E. Marsden, Lecture Notes in Mechanics, London Math. Soc. Lecture Notes, 174. Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.
[14]	J. E. Marsden & J. Scheurle, The reduced Euler-Lagrange equations, In Dynamics and Control of Mechanical Systems, Fields Inst. Commun., 1 (1993), 139-164.
[15]	K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd ed., Springer, New York, 2009.
[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 & M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.
[18]	J. Montaldi & 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.
[19]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, vol. 222 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.
[20]	G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, J. Geom. Phys., 9 (1992), 111-119. doi: 10.1016/0393-0440(92)90015-S.
[21]	G. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift, J. Nonlinear Sci., 5 (1995), 373-418. doi: 10.1007/BF01212907.
[22]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance, Math. Z., 232 (1999), 747-788. doi: 10.1007/PL00004782.
[23]	G. Patrick & M. Roberts, The transversal relative equilibria of a Hamiltonian system with symmetry, Nonlinearity, 13 (2000), 2089-2105. doi: 10.1088/0951-7715/13/6/311.

show all references

References:

[1]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978.
[2]	A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden & G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756. doi: 10.1016/0005-1098(92)90034-D.
[3]	J. W. Bruce & R. M. Roberts, Critical points of functions on analytic varieties, Topology, 27 (1988), 57-90. doi: 10.1016/0040-9383(88)90007-9.
[4]	L. Buono, F. Laurent-Polz & J. Montaldi, Symmetric Hamiltonian Bifurcations, In Geometric Mechanics and Symmetry: The Peyresq Lectures, J. Montaldi and T. S. Ratiu, eds. pp 357-402. Cambridge University Press, 2005. doi: 10.1017/CBO9780511526367.007.
[5]	J. Damon, The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcalK$, Memoirs A.M.S., 50 (1984), x+88 pp. doi: 10.1090/memo/0306.
[6]	J. Damon, Deformations of sections of singularities and Gorenstein surface singularities, Am. J. Math., 109 (1987), 695-721. doi: 10.2307/2374610.
[7]	J. Damon, $\mathcal{A}$-equivalence and the equivalence of sections of images and disriminants, In Singularity Theory and its Applications, Part I, Springer Lecture Notes in Math., 1462 (1991), 93-121. doi: 10.1007/BFb0086377.
[8]	V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511574788.
[9]	P. S. Krishnaprasad, Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035. doi: 10.1016/0362-546X(85)90083-5.
[10]	F. Laurent-Polz, J. Montaldi & M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria, J. Geom. Mech, 3 (2012), 439-486. doi: 10.3934/jgm.2011.3.439.
[11]	E. Lerman & 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.
[12]	C. Lim, J. Montaldi & M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135. doi: 10.1016/S0167-2789(00)00167-6.
[13]	J. E. Marsden, Lecture Notes in Mechanics, London Math. Soc. Lecture Notes, 174. Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.
[14]	J. E. Marsden & J. Scheurle, The reduced Euler-Lagrange equations, In Dynamics and Control of Mechanical Systems, Fields Inst. Commun., 1 (1993), 139-164.
[15]	K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd ed., Springer, New York, 2009.
[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 & M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.
[18]	J. Montaldi & 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.
[19]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, vol. 222 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.
[20]	G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space, J. Geom. Phys., 9 (1992), 111-119. doi: 10.1016/0393-0440(92)90015-S.
[21]	G. Patrick, Relative Equilibria of Hamiltonian Systems with Symmetry: Linearization, Smoothness, and Drift, J. Nonlinear Sci., 5 (1995), 373-418. doi: 10.1007/BF01212907.
[22]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance, Math. Z., 232 (1999), 747-788. doi: 10.1007/PL00004782.
[23]	G. Patrick & M. Roberts, The transversal relative equilibria of a Hamiltonian system with symmetry, Nonlinearity, 13 (2000), 2089-2105. doi: 10.1088/0951-7715/13/6/311.

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