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SYMMETRY, REDUCTION AND STABILITY OF MECHANICAL SYSTEMS

Research project of the Research Foundation - Flanders (2010 - 2013)

Description of the project

The role that symmetry properties of a dynamical system play in its reduction to a (possibly) simpler system, and in the stability properties of certain of its solutions, is an important field of research. Whereas this topic has already been extensively dealt with in the case of Hamiltonian systems in a symplectic or, more generally, a Poisson context , the Lagrangian side of the story has always received much less attention in the literature. The goal of this project therefore is to restore this balance in some sense by analyzing symmetry and reduction aspects in the context of Lagrangian dynamical systems. In particular, attention will be focused on so-called Routh reduction and Routh reduction by stages for invariant and quasi-invariant Lagrangian systems. The techniques will be illustrated on some concrete mechanical systems such as, for instance, a rigid body submerged in a fluid. The stability of relative equilibria will also be investigated in a Lagrangian setting. Finally, the extent to which the symmetry and reduction techniques for Lagrangian and Hamiltonian systems can be modified or generalized in order to apply them to systems with non-holonomic constraints and to (non-Lagrangian) systems with dissipation, will also be investigated.



Researchers involved with the project

Promotor: Frans Cantrijn (Frans.Cantrijn@UGent.be)

Co-promotor: Bavo Langerock (Bavo.Langerock@UGent.be)

Willy Sarlet (Willy.Sarlet@UGent.be)

Tom Mestdag (Tom.Mestdag@UGent.be)

Joris Vankerschaver (Joris.Vankerschaver@UGent.be)

Eduardo García-Toraño Andrés

International collaboration with:

  • Consejo Superior de Investigationes Cientificas (Madrid, Spain)

  • Universidad Complutense de Madrid (Madrid, Spain)

  • Univesidad de Zaragoza (Zaragoza, Spain)

  • University of Michigan (Michigan, USA)

  • California Institute of Technology (Pasadena, USA)

  • Waseda University (Tokio, Japan)

Papers

E. García-Toraño Andrés, T. Mestdag and H. Yoshimura, Implicit Lagrange-Routh Equations and Dirac Reduction. Journal of Geometry and Physics 104 (2016) 291-304.
[arXiv] [doi]

E. García-Toraño Andrés, E. Guzmán, J.C. Marrero and T. Mestdag, Reduced dynamics and Lagrangian submanifolds of symplectic manifolds. J Phys A: Math Theor. 47 (2014) 225203.
[arXiv] [doi]

E. García-Toraño Andrés, B. Langerock, F. Cantrijn, Aspects of reduction and transformation of Lagrangian systems with symmetry. J. Geom. Mech. 6 (2014) 1-23.
[arXiv] [doi]

B. Langerock, García-Toraño Andrés, F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems. J. Math. Phys. 53, 062902 (2012)
[arXiv] [doi]

B. Langerock, T. Mestdag and J. Vankerschaver, Routh reduction by stages. SIGMA 7 (2011), 109, 31 pages.
[arXiv] [doi]

T. Mestdag, W. Sarlet and M. Crampin, The inverse problem for Lagrangian systems with certain non-conservative forces. Diff. Geom. Appl. 29 (2011) 55-72.
[arXiv] [doi]

M. Crampin and T. Mestdag, Reduction of invariant constrained systems using anholonomic frames. J. Geometric Mechanics 3 (2011) 23 - 40.
[arXiv][doi]

M. Crampin and T. Mestdag, The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem. Int. J. Geom. Methods. Mod. Phys. 8 (2011) 897-923.
[arXiv] [doi]

T. Mestdag, W. Sarlet and M. Crampin, Second-order dynamical systems of Lagrangian type with dissipation. Diff. Geom. Appl. 29 (2011) S156-S163.
[pdf] [doi]

B. Langerock and M. Castrillón López, Routh reduction for singular Lagrangians, Int. J. Geom. Methods. Mod. Phys. 7(8), 1451--1489 (2010).
[ArXiv] [doi]

B. Langerock, F. Cantrijn, J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians. J. Math. Phys. 51 (2010) 022902.
[ArXiv] [doi]