Research Fellows Directory
Dr Luis Garcia-Naranjo
University of Manchester
Constraints in mechanics can be divided into two types. The first type restrict the possible configurations of the system, such as the fixed rod of a pendulum, and are termed "holonomic". The second type are constraints on the velocities of the system that do not arise from constraints on the configurations (they cannot be integrated). Constraints of this type are termed "nonholonomic" and often arise in rolling systems, such as a ball that rolls without slipping on a table. Nonholonomic systems have important applications in control theory and robotics.
In holonomic mechanics the equations of motion can be obtained from a variational principle and, as a consequence, they can be formulated in Hamiltonian form. This is a rich geometric structure of the equations that allows for a systematic study of different aspects of the solutions.
On the other hand, the equations of motion for a nonholonomic system do not arise from a variational principle. As a consequence, nonholonomic systems do not allow a Hamiltonian formulation and many of the properties of the solutions remain poorly understood.
A good part of my research is devoted to the use of differential geometric and analytic methods to consider various aspects of nonholonomic systems. These are powerful tools that allow me to exploit the symmetries of particular class of problems and permit a systematic treatment of various invariants, like the existence of invariant measures and first integrals. I am currently very interested in integrability and in the study of the so-called relative equilibria.