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An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems
Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
1. | School of Mathematics, University of Manchester, Manchester, M13 9PL |
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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