ASPHODELUS · Aspects of holonomy, decoupling and reduction of Lagrangian systems

Final Activity Report Summary - ASPHODELUS (Aspects of holonomy, decoupling and reduction of Lagrangian systems)

The mathematical theory of dynamical systems provides models for a large variety of phenomena in different areas of science. In many cases the solution of these systems cannot be obtained analytically and one can only analyse them by means of their qualitative and geometric features. The term 'geometric mechanics' usually refers to research on mechanical systems in which differential geometric models and techniques are being used. The enormous importance of the concept of symmetry in a great number of applications in physics is beyond any doubt. Symmetry properties of mechanical systems in particular have been studied intensively during the last decades. The bulk of literature, however, concentrates on the Hamiltonian description of symmetric systems in which the theory of Poisson manifolds plays an important role. The process of symmetry reduction for Lagrangian systems is less known. When the Lagrangian is invariant under the action of a Lie group, so that we are dealing with a symmetry group of the Euler-Lagrange equations, the equations of motion can be reduced to a new set of equations with fewer unknowns, which are therefore easier to solve. In fact, there exist many methods for reducing equations of motion, such as the Lagrange-Poincare method, the Routh method, etc. We showed, for a few of these methods, that the reduced equations could be derived in a relatively straightforward fashion by choosing a suitably adapted, anholonomic, frame, or equivalently by making use of well-chosen quasi-velocities. We extended literature-known results to non-Abelian symmetry groups and arbitrary Lagrangians. We also investigated the inverse process of reconstruction by means of a generalisation of the so-called mechanical connection. Moreover, we proved a criterion for characterising relative equilibria in this context. Next, for the case that the configuration space of the system is a Lie group, we investigated the conditions for the existence of a regular Lagrangian whose Euler-Lagrange equations were equivalent to a given system of, invariant, second-order differential equations. Many interesting mechanical systems are subject to additional velocity-dependent, i.e. nonholonomic, constraints. Typical engineering problems that involve nonholonomic constraints arise for example in robotics, where the wheels of a mobile robot are often required to roll without slipping, as well as in many applications in aerodynamics such as the reorientation of a satellite using internal rotors. In a second line of research we investigated the derivation of the equations of motion of a nonholonomic system from an unconstrained Hamiltonian or Lagrangian function, which amounted to setting up a kind of inverse problem. That is to say, for a certain class of nonholonomic systems, we searched among all possible second-order systems which restricted to the given dynamics on the constraints, for an appropriate one which was variational. We also investigated whether this type of Hamiltonisation could be useful from the viewpoint of numerical integrators that preserved the underlying geometric structure of a system. Finally, we extended the applicability of a second Hamiltonisation procedure, namely that of Chaplygin 'reducibility trick', to a more general class of nonholonomic systems. The fellow Tom Mestdag executed this project in collaboration with Anthony M Bloch and Oscar E Fernandez from the University of Michigan and Michael Crampin and Willy Sarlet from Ghent University.

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