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

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.
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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