Euler-Poincaré reduction for systems with configuration space isotropy

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June 2011, 3(2): 261-275. doi: 10.3934/jgm.2011.3.261

Euler-Poincaré reduction for systems with configuration space isotropy

Jeffrey K. Lawson ^1, , Tanya Schmah ^2, and Cristina Stoica ^3,

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

Received February 2010 Revised June 2011 Published July 2011

Abstract
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This paper concerns Lagrangian systems with symmetries, near points with configuration space isotropy. Using twisted parametrisations corresponding to phase space slices based at zero points of tangent fibres, we deduce reduced equations of motion, which are a hybrid of the Euler-Poincaré and Euler-Lagrange equations. Further, we state a corresponding variational principle.

Keywords: Euler-Poincaré-Lagrange bundle equations., Euler-Poincaré reduction, Symmetric Lagrangian, slice theorem.

Mathematics Subject Classification: Primary: 70H03, 70H4.

Citation: Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261

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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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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