NUMERICAL INTEGRATION OF NONHOLONOMIC SYSTEMS

 

December 21, 2006

CSIC, Madrid

 

.

Hernan Cendra

      

GENERAL INFORMATION

The meeting will begin at 9:15am (first talk at 9:30am) and continue through to 7:00pm.  There will be 6 talks during the day.

There is no registration fee, and pre-registering is strongly encouraged. To pre-register   and information  consult www.simumat.es

 

CONFERENCE ROOM

 C/Jorge Manrique 27, Madrid 28006 (Map)

 

LOCAL ORGANIZER

David Martín de Diego

Departamento de Matemáticas

Instituto de Matemáticas y Física Fundamental, CSIC,

Madrid d.martin@imaff.cfmac.csic.es

 

 

PROGRAMME

 

9:15-9:30 Introduction (David Martín de Diego)
   
9:30-10:30 An introduction to noholonomic mechanics and its applications
  Hernán Cendra, University Nacional del Sur, ARGENTINA
 

Abstract: The principle of D'Alembert share some of the advantages of Hamilton's principle in classical mechanics, among them, the immediate covariance of equations of motion and the existence of convenient integrators. I will comment on this, with examples of interest, and also on some generalizations of D'Alembert's principle. The latter include, for instance, problems in control theory and some systems with friction, presented as generalized nonholonomic systems. The same advantages mentioned before seem to be preserved for this kind of systems


[PDF file]

 

 

10:30-11:30 An introduction to nonholonomic integrators
  Jorge Cortés, University of California, Santa Cruz, USA
 

Abstract: This talk is an introduction to nonholonomic integrators for mechanical systems subject to nonholonomic constraints. These integrators are derived from a discrete Lagrange-d’Alembert principle, in a way parallel to how variational integrators for mechanical systems are derived from a discrete variational principle. After discussing in detail the discrete Lagrange-d'Alembert principle, we will examine the geometric properties of nonholonomic integrators regarding the preservation of the symplectic form and the conservation of energy and momentum. In the presence of symmetry, we will derive a discrete momentum equation, and establish various parallelisms with variational integrators by considering the reduced system

[PDF file]

 

 

12:00-13:00 Structure-preserving nonholonomic integrators
  Dmitry Zenkov, North Carolina State University, USA
 

Abstract: It is well known that in the absence of nonholonomic constraints, variational integrators are structure-preserving, e.g. they conserve momentum of mechanical systems with symmetry. In this lecture we will discuss conservation of momentum, of energy, and of invariant measure by nonholonomic integrators on Lie groups characterized by left-invariant Lagrangian and left-invariant constraints. We will discuss how this structure-preservation can be broken by an inappropriate choice of discrete constraints

[PDF file]


 

15:30-16:30 Invariant objects in discrete nonholonomic systems
  Yuri Fedorov, Technique University of Catalonia, SPAIN
 

Abstract: Recently the formalizm of variational integrators (discrete Lagrangian systems) was extended to systems with nonholonomic constraints. We apply it to the case when the configuration space is a Lie group G and the discrete Lagrangian is left-invariant.
As examples, we construct discretizations of several classical integrable nonholonomic systems with an invariant measure, in particular, of the celebrated Chaplygin nonholonomic sphere problem. It appears that the resulting discrete dynamics is similar to that of the continuous models.
We then propose a method of choosing discrete nonholonomic constraints that ensures preservation of the energy integral in the discretizations. The conservation of an invariant measure in the discrete systems will also be discussed.

[PDF file]

 

 

17:00-18:00 Construction of variational integrators on Lie groupoids
  Eduardo Martínez, University of Zaragoza
 

Abstract: During the last decade, a lot of effort has been devoted to the construction of geometric integrators for Lagrangian systems using a discrete variational principle. In particular, this effort has been concentrated for the case of discrete Lagrangian functions L on the cartesian product QxQ of a differentiable manifold. This cartesian product plays the role of a “discretized version” of the standard velocity phase space TQ. These variational integrators inherit some of the geometric properties of the continuous Lagrangian (symplecticity, momentum preservation).

The purpose of this talk is to describe Lagrangian and Hamiltonian Mechanics on a Lie groupoid, which is a structure that includes, as particular examples, the case of cartesian products QxQ, quotients by Lie groups (QxQ)/G as well as Lie groups G.  From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations, we define the Hamiltonian evolution operator, which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we obtain include, as particular cases, the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations. Our results may be important for the construction of geometric integrators for continuous Lagrangian systems with a Lie group of symmetries.

[PDF file]


 

18:00-19:00 Discrete nonholonomic Lagrangian systems on Lie groupoids 
  Juan Carlos Marrero, University of La Laguna
 

Abstract: In this talk, I will present some ideas about the geometric formulation of discrete nonholonomic Lagrangian systems on Lie groupoids.  First of all, from a discrete generalized Holder's principle we will derive the so-called discrete nonholonomic Euler-Lagrange equations for a constrained Lagrangian system on a Lie groupoid. Then, we will introduce the discrete nonholonomic Legendre transformations associated with the system  and we will obtain diferent characterizations of the regularity. We will also discuss the reduction theory and the nonholonomic momentum map in this setting. Finally, we will exhibit some examples.

[PDF file]

 

 

 

 

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