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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Damon, The unfolding and determinacy theorems for subgroups of $\mathcalA$ and $\mathcalK$, Memoirs A.M.S., 50 (1984), x+88 pp. doi: 10.1090/memo/0306.  Google Scholar

[6]

J. Damon, Deformations of sections of singularities and Gorenstein surface singularities, Am. J. Math., 109 (1987), 695-721. doi: 10.2307/2374610.  Google Scholar

[7]

J. Damon, $\mathcalA$-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.  Google Scholar

[8]

V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511574788.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. E. Marsden, Lecture Notes in Mechanics, London Math. Soc. Lecture Notes, 174. Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar

[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.  Google Scholar

[15]

K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd ed., Springer, New York, 2009.  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 & M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Damon, The unfolding and determinacy theorems for subgroups of $\mathcalA$ and $\mathcalK$, Memoirs A.M.S., 50 (1984), x+88 pp. doi: 10.1090/memo/0306.  Google Scholar

[6]

J. Damon, Deformations of sections of singularities and Gorenstein surface singularities, Am. J. Math., 109 (1987), 695-721. doi: 10.2307/2374610.  Google Scholar

[7]

J. Damon, $\mathcalA$-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.  Google Scholar

[8]

V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511574788.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. E. Marsden, Lecture Notes in Mechanics, London Math. Soc. Lecture Notes, 174. Cambridge University Press, 1992. doi: 10.1017/CBO9780511624001.  Google Scholar

[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.  Google Scholar

[15]

K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd ed., Springer, New York, 2009.  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 & M. Roberts, Relative equilibria of molecules, J. Nonlinear Science, 9 (1999), 53-88. doi: 10.1007/s003329900064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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