Geometric Mechanics: Differential geometry applied to problems in cassical mechanics, mathematical physics, and to the study of dynamical systems. Lagrangian and Hamiltonian systems and Finsler geometry.
AbstractGeometric 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.
AbstractGeometric 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.
- Promotor: Mestdag Tom