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Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover
Euler-Poincaré reduction for systems with configuration space isotropy
1. | Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723, United States |
2. | Department of Computer Science, University of Toronto, 10 King’s College Road, Room 3302, Toronto, ON, M5S 3G4, Canada |
3. | Department of Mathematics, Wilfrid Laurier University, 75 University Ave West, Waterloo, ON, N2L 3C5, Canada |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. on Systems and Control, 45 (2000), 2253-2270.
doi: 10.1109/9.895562. |
[3] |
A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. on Automatic Control, 46 (2001), 1556-1571 |
[4] |
H. Cendra, J. E. Marsden and T. Ratiu, "Lagrangian Reduction by Stages," Memoirs of the American Mathematical Society, 152, 2001. |
[5] |
R. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[6] |
V. Guillemin and S. Sternberg, A normal form for the moment map, In "Differential Geometric Methods in Mathematical Physics," S. Sternberg ed., Mathematical Physics Studies, 6, Reidel, Dordrecht, 1984. |
[7] |
J. J. Duistermaat and J. A. C. Kolk, "Lie Groups," Springer-Verlag, Berlin, 2000.
doi: 10.1007/978-3-642-56936-4. |
[8] |
D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: from Finite to Infinite Dimensions," Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009. |
[9] |
E. Lerman and 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. |
[10] |
D. Lewis, Lagrangian block diagonalization, Journal of Dynamics and Differential Equations, 4 (1992), 1-41.
doi: 10.1007/BF01048153. |
[11] |
C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rendiconti del Seminario Matematico, Università e Politecnico, Torino, 43 (1985), 227-251. |
[12] |
J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Note Series, 174, Cambridge Univ. Press, Cambridge, 1992. |
[13] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition. Texts in Applied Mathematics, 17, Springer, 1999. |
[14] |
J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.
doi: 10.1016/0034-4877(74)90021-4. |
[15] |
J. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553-558.
doi: 10.1016/S0764-4442(99)80389-9. |
[16] |
J. Montaldi and R. M. Roberts, Relative Equilibria of Molecules, J. Nonlinear Sci., 9 (1999), 53-88.
doi: 10.1007/s003329900064. |
[17] |
J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity, 12 (1999), 693-720.
doi: 10.1088/0951-7715/12/3/315. |
[18] |
J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. |
[19] |
G. W. 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. |
[20] |
R. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. Math., 73 (1961), 295-323.
doi: 10.2307/1970335. |
[21] |
M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems, Nonlinearity, 19 (2006), 853-877.
doi: 10.1088/0951-7715/19/4/005. |
[22] |
R. M. Roberts and M. E. R. de Sousa Dias, Bifurcations from relative equilibria of Hamiltonian systems, Nonlinearity, 10 (1997), 1719-1738.
doi: 10.1088/0951-7715/10/6/015. |
[23] |
M. Roberts, T. Schmah and C. Stoica, Relative equilibria in systems with configuration space isotropy, J. Geom. Phys., 56 (2006), 762-779.
doi: 10.1016/j.geomphys.2005.04.017. |
[24] |
R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math., 134 (1991), 375-422.
doi: 10.2307/2944350. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. |
[2] |
A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. on Systems and Control, 45 (2000), 2253-2270.
doi: 10.1109/9.895562. |
[3] |
A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. on Automatic Control, 46 (2001), 1556-1571 |
[4] |
H. Cendra, J. E. Marsden and T. Ratiu, "Lagrangian Reduction by Stages," Memoirs of the American Mathematical Society, 152, 2001. |
[5] |
R. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[6] |
V. Guillemin and S. Sternberg, A normal form for the moment map, In "Differential Geometric Methods in Mathematical Physics," S. Sternberg ed., Mathematical Physics Studies, 6, Reidel, Dordrecht, 1984. |
[7] |
J. J. Duistermaat and J. A. C. Kolk, "Lie Groups," Springer-Verlag, Berlin, 2000.
doi: 10.1007/978-3-642-56936-4. |
[8] |
D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: from Finite to Infinite Dimensions," Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009. |
[9] |
E. Lerman and 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. |
[10] |
D. Lewis, Lagrangian block diagonalization, Journal of Dynamics and Differential Equations, 4 (1992), 1-41.
doi: 10.1007/BF01048153. |
[11] |
C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rendiconti del Seminario Matematico, Università e Politecnico, Torino, 43 (1985), 227-251. |
[12] |
J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Note Series, 174, Cambridge Univ. Press, Cambridge, 1992. |
[13] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition. Texts in Applied Mathematics, 17, Springer, 1999. |
[14] |
J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.
doi: 10.1016/0034-4877(74)90021-4. |
[15] |
J. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553-558.
doi: 10.1016/S0764-4442(99)80389-9. |
[16] |
J. Montaldi and R. M. Roberts, Relative Equilibria of Molecules, J. Nonlinear Sci., 9 (1999), 53-88.
doi: 10.1007/s003329900064. |
[17] |
J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria, Nonlinearity, 12 (1999), 693-720.
doi: 10.1088/0951-7715/12/3/315. |
[18] |
J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction," Progress in Mathematics, 222, Birkhäuser Boston, Inc., Boston, MA, 2004. |
[19] |
G. W. 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. |
[20] |
R. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. Math., 73 (1961), 295-323.
doi: 10.2307/1970335. |
[21] |
M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems, Nonlinearity, 19 (2006), 853-877.
doi: 10.1088/0951-7715/19/4/005. |
[22] |
R. M. Roberts and M. E. R. de Sousa Dias, Bifurcations from relative equilibria of Hamiltonian systems, Nonlinearity, 10 (1997), 1719-1738.
doi: 10.1088/0951-7715/10/6/015. |
[23] |
M. Roberts, T. Schmah and C. Stoica, Relative equilibria in systems with configuration space isotropy, J. Geom. Phys., 56 (2006), 762-779.
doi: 10.1016/j.geomphys.2005.04.017. |
[24] |
R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math., 134 (1991), 375-422.
doi: 10.2307/2944350. |
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