American Institute of Mathematical Sciences

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). 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. 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. 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, (2005), 357. 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). 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. 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, 1462 (1991), 93. doi: 10.1007/BFb0086377. Google Scholar [8] V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams,, Cambridge University Press, (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. 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. 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. 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. doi: 10.1016/S0167-2789(00)00167-6. Google Scholar [13] J. E. Marsden, Lecture Notes in Mechanics,, London Math. Soc. Lecture Notes, 174 (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, 1 (1993), 139. Google Scholar [15] K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem,, 2nd ed., (2009). Google Scholar [16] J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. 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. 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. 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, (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. 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. 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. 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. doi: 10.1088/0951-7715/13/6/311. Google Scholar

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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. 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. 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, (2005), 357. 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). 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. 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, 1462 (1991), 93. doi: 10.1007/BFb0086377. Google Scholar [8] V. Guillemin, E. Lerman and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams,, Cambridge University Press, (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. 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. 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. 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. doi: 10.1016/S0167-2789(00)00167-6. Google Scholar [13] J. E. Marsden, Lecture Notes in Mechanics,, London Math. Soc. Lecture Notes, 174 (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, 1 (1993), 139. Google Scholar [15] K. Meyer, G. Hall & D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem,, 2nd ed., (2009). Google Scholar [16] J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. 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. 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. 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, (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. 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. 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. 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. doi: 10.1088/0951-7715/13/6/311. Google Scholar
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