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

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

Received  February 2010 Revised  June 2011 Published  July 2011

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.
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., (1978).   Google Scholar

[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.  doi: 10.1109/9.895562.  Google Scholar

[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.   Google Scholar

[4]

H. Cendra, J. E. Marsden and T. Ratiu, "Lagrangian Reduction by Stages,", Memoirs of the American Mathematical Society, 152 (2001).   Google Scholar

[5]

R. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser Verlag, (1997).  doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[6]

V. Guillemin and S. Sternberg, A normal form for the moment map,, In, 6 (1984).   Google Scholar

[7]

J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000).  doi: 10.1007/978-3-642-56936-4.  Google Scholar

[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 (2009).   Google Scholar

[9]

E. Lerman and 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

[10]

D. Lewis, Lagrangian block diagonalization,, Journal of Dynamics and Differential Equations, 4 (1992), 1.  doi: 10.1007/BF01048153.  Google Scholar

[11]

C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique,, Rendiconti del Seminario Matematico, 43 (1985), 227.   Google Scholar

[12]

J. E. Marsden, "Lectures on Mechanics,", London Mathematical Society Lecture Note Series, 174 (1992).   Google Scholar

[13]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition., Texts in Applied Mathematics, 17 (1999).   Google Scholar

[14]

J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry,, Rep. Math. Phys., 5 (1974), 121.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[15]

J. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553.  doi: 10.1016/S0764-4442(99)80389-9.  Google Scholar

[16]

J. Montaldi and R. M. Roberts, Relative Equilibria of Molecules,, J. Nonlinear Sci., 9 (1999), 53.  doi: 10.1007/s003329900064.  Google Scholar

[17]

J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria,, Nonlinearity, 12 (1999), 693.  doi: 10.1088/0951-7715/12/3/315.  Google Scholar

[18]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).   Google Scholar

[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.  doi: 10.1016/0393-0440(92)90015-S.  Google Scholar

[20]

R. Palais, On the existence of slices for actions of non-compact Lie groups,, Ann. Math., 73 (1961), 295.  doi: 10.2307/1970335.  Google Scholar

[21]

M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems,, Nonlinearity, 19 (2006), 853.  doi: 10.1088/0951-7715/19/4/005.  Google Scholar

[22]

R. M. Roberts and M. E. R. de Sousa Dias, Bifurcations from relative equilibria of Hamiltonian systems,, Nonlinearity, 10 (1997), 1719.  doi: 10.1088/0951-7715/10/6/015.  Google Scholar

[23]

M. Roberts, T. Schmah and C. Stoica, Relative equilibria in systems with configuration space isotropy,, J. Geom. Phys., 56 (2006), 762.  doi: 10.1016/j.geomphys.2005.04.017.  Google Scholar

[24]

R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction,, Ann. of Math., 134 (1991), 375.  doi: 10.2307/2944350.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition,, Benjamin/Cummings Publishing Co., (1978).   Google Scholar

[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.  doi: 10.1109/9.895562.  Google Scholar

[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.   Google Scholar

[4]

H. Cendra, J. E. Marsden and T. Ratiu, "Lagrangian Reduction by Stages,", Memoirs of the American Mathematical Society, 152 (2001).   Google Scholar

[5]

R. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser Verlag, (1997).  doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[6]

V. Guillemin and S. Sternberg, A normal form for the moment map,, In, 6 (1984).   Google Scholar

[7]

J. J. Duistermaat and J. A. C. Kolk, "Lie Groups,", Springer-Verlag, (2000).  doi: 10.1007/978-3-642-56936-4.  Google Scholar

[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 (2009).   Google Scholar

[9]

E. Lerman and 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

[10]

D. Lewis, Lagrangian block diagonalization,, Journal of Dynamics and Differential Equations, 4 (1992), 1.  doi: 10.1007/BF01048153.  Google Scholar

[11]

C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique,, Rendiconti del Seminario Matematico, 43 (1985), 227.   Google Scholar

[12]

J. E. Marsden, "Lectures on Mechanics,", London Mathematical Society Lecture Note Series, 174 (1992).   Google Scholar

[13]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 2nd edition., Texts in Applied Mathematics, 17 (1999).   Google Scholar

[14]

J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry,, Rep. Math. Phys., 5 (1974), 121.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[15]

J. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553.  doi: 10.1016/S0764-4442(99)80389-9.  Google Scholar

[16]

J. Montaldi and R. M. Roberts, Relative Equilibria of Molecules,, J. Nonlinear Sci., 9 (1999), 53.  doi: 10.1007/s003329900064.  Google Scholar

[17]

J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria,, Nonlinearity, 12 (1999), 693.  doi: 10.1088/0951-7715/12/3/315.  Google Scholar

[18]

J.-P. Ortega and T. S. Ratiu, "Momentum Maps and Hamiltonian Reduction,", Progress in Mathematics, 222 (2004).   Google Scholar

[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.  doi: 10.1016/0393-0440(92)90015-S.  Google Scholar

[20]

R. Palais, On the existence of slices for actions of non-compact Lie groups,, Ann. Math., 73 (1961), 295.  doi: 10.2307/1970335.  Google Scholar

[21]

M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems,, Nonlinearity, 19 (2006), 853.  doi: 10.1088/0951-7715/19/4/005.  Google Scholar

[22]

R. M. Roberts and M. E. R. de Sousa Dias, Bifurcations from relative equilibria of Hamiltonian systems,, Nonlinearity, 10 (1997), 1719.  doi: 10.1088/0951-7715/10/6/015.  Google Scholar

[23]

M. Roberts, T. Schmah and C. Stoica, Relative equilibria in systems with configuration space isotropy,, J. Geom. Phys., 56 (2006), 762.  doi: 10.1016/j.geomphys.2005.04.017.  Google Scholar

[24]

R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction,, Ann. of Math., 134 (1991), 375.  doi: 10.2307/2944350.  Google Scholar

[1]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

[2]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[3]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

[4]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

[5]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[7]

Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394

[8]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[9]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

[10]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[11]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[12]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135

[13]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[14]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[15]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[16]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[17]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[18]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[19]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[20]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (1)

[Back to Top]