American Institute of Mathematical Sciences

Advanced
December  2009, 1(4): 417-444. doi: 10.3934/jgm.2009.1.417

Variational principles for spin systems and the Kirchhoff rod

François Gay-Balma 1, , Darryl D. Holm 2, and  Tudor S. Ratiu 3,

1. 

Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 911125, United States
2. 

Department of Mathematics and Institute for Mathematical Sciences, Imperial College, London, SW7 2AZ, United Kingdom
3. 

Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland

Received  April 2009 Revised  July 2009 Published  January 2010

We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.
Keywords: Kirchhoff rod., affine Euler-Poincaré equations, Lagrangian reducton, spin system, variational principles, cocycle, symmetry.
Mathematics Subject Classification: Primary: 70H33, 70G65, 70G75; Secondary: 74K10, 82D3.
Citation: François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417
[1]

Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063
[2]

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
[3]

Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1
[4]

Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008
[5]

Guohua Zhang. Variational principles of pressure. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
[6]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[7]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1431-1445. doi: 10.3934/cpaa.2021027
[8]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[9]

Harish S. Bhat, Razvan C. Fetecau. Lagrangian averaging for the 1D compressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 979-1000. doi: 10.3934/dcdsb.2006.6.979
[10]

Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure and Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845
[11]

Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399
[12]

David Kinderlehrer, Michał Kowalczyk. The Janossy effect and hybrid variational principles. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 153-176. doi: 10.3934/dcdsb.2009.11.153
[13]

Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019
[14]

Artur O. Lopes, Elismar R. Oliveira. Entropy and variational principles for holonomic probabilities of IFS. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 937-955. doi: 10.3934/dcds.2009.23.937
[15]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[16]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038
[17]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[18]

Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure and Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365
[19]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343
[20]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy