Research Tom Mestdag

Abstract

The mathematical models underlying physics and engineering applications are often limited by constraints. In mechanics, when the constraints are velocity-dependent (or: limiting the admissible directions), and when they can not be integrated to a merely position-depending form, they are called nonholonomic constraints. In this project we will investigate the almost-symplectic and almost- Poisson geometry behind Lagrangian systems with such constraints. In particular, we focus on the nonholonomic exponential map for general Lagrangian functions and on the curvature of nonlinear nonholonomic constraints. Chaplygin systems are a subclass of nonholonomic systems with a symmetry group. We first make for this class an in-depth study of the nonholonomic exponential map, in the case of a kinetic energy Lagrangian. Second, we use techniques from Finsler geometry to extend the results to the situation of a general Lagrangian function. Next, we consider nonlinear nonholonomic constraints. They can be represented by the horizontal manifold of a nonlinear splitting on a fibre bundle. The concept of a nonlinear splitting extends that of an Ehresmann connection. We examine the role of its curvature, in the context of nonholonomic systems.

Researcher(s)

• Promoter: Mestdag Tom
• Fellow: Belrhazi Malika

Research team(s)

• Fundamental Mathematics

Project type(s)

• Research Project

Abstract

`Geometric mechanics' usually stands for the application of differential geometric methods in the study of dynamical systems appearing in mathematical physics. The objectives in this project are all centered around such geometric techniques for reduction and unreduction of a Lagrangian system that is invariant under the action of a symmetry Lie group. One encounters such systems in the context of the calculus of variations and in Finsler geometry. On a principal fibre bundle, the terminology symmetry reduction refers to the fact that an invariant Lagrangian system on the full manifold can be reduced to a system of differential equations on the quotient manifold (the so-called Lagrange-Poincare equations). Unreduction, on the other hand, has the opposite goal: to relate a Lagrangian system on the quotient manifold to a system of differential equations on the full manifold. In this proposal we will investigate the conditions under which the unreduced system can be brought back in the form of a set of Euler-Lagrange equations, for some (yet unknown) Lagrangian on the full manifold. The main tool will be the so-called Inverse Problem of the Calculus of Variations. Besides, we will both extend and specify the method of unreduction in such a way that it fits the needs of the research on isometric submersions between two Finsler manifolds.

Researcher(s)

• Promoter: Hohloch Sonja
• Co-promoter: Mestdag Tom
• Fellow: Yasaka Kenzo

Research team(s)

• Fundamental Mathematics

Project type(s)

• Research Project

Abstract

Geometric mechanics refers to a variety of topics that lie at the intersection of differential geometry, dynamical systems, and analytical mechanics. The main idea is to identify the geometric structures underlying many classes of physical and engineering systems. These geometric structures can be useful for the qualitative study of the system and can also be used for instance in the design of control laws and geometric integrators. The property that a system can be derived from a variational principle is one of these useful structures. In this project we will use the inverse problem of the calculus of variations to find stabilizing controls for a variety of mechanical systems. One of the advantages of finding a variational structure is that we can then use energy methods to show stability, or find conditions for stability. We will also introduce more flexibility to the classical inverse problem to extend its possible applications. More precisely, for the inverse problem on a Lie algebra we will allow variable structure constants in order to have more freedom in the energy shaping step. The theory of exterior differential systems has been applied successfully to the inverse problem to identify variational cases. We will also adapt these techniques to the inverse problem for constrained systems with an eye towards the problem of Hamiltonization of nonholonomic systems. Finally we will also study geometric integrators for metriplectic and dissipative systems.

Researcher(s)

• Promoter: Mestdag Tom
• Co-promoter: Hohloch Sonja
• Fellow: Farre Puiggali Marta

Research team(s)

• Fundamental Mathematics

Project type(s)

• Research Project

Abstract

Geometric methods in the study of dynamical systems have the advantage of providing global rather than local results. Quite often, the dynamical systems that arise in theoretical physics or other sciences can be seen to be essentially defined by a geometric structure and further auxiliary data. In this project we will mainly concentrate on aspects related to symmetries of such dynamical systems. Symmetry has the defining property that it maps solutions of the system onto solutions. Since the number of dynamical systems for which we can easily write down the solutions in closed form is extremely limited, finding symmetries is a key step in the process of solving the governing differential equations for the problem at hand. Symmetries and, possibly, their associated conserved quantities can often be used to reduce the problem to a smaller system of differential equations, which may be easier to solve. Any further attempt to integrate the reduced system will rely on how much of the geometric structure of the unreduced system is transferred to the reduced one. This project has the overall goal to investigate, mainly in the context of singular Lagrangian systems, the most appropriate conditions for the existence of such structure-preserving reduction.

Researcher(s)

• Promoter: Mestdag Tom

Research team(s)

• Fundamental Mathematics

Project type(s)

• Research Project