Geometric and Numerical analysis of nonholonomic systems - Simões, Alexandre (2021)
Geometric mechanics is a fairly recent field of mathematics lying in the intersection of at least four different scientific fields: differential geometry, physics, numerical analysis and dynamical systems. Its starting point is to shed light on the underlying geometry behind mechanics and use it to obtain new results which frequently reach a variety of different mathematical fields. One of the practical applications that was made possible by using geometric techniques was the ability to construct variational integrators, which are numerical methods reproducing the geometry of the original mechanical system such as symplecticity, conservation of momentum and energy. These methods are often computationally cheaper than standard ones while demonstrating an adequate qualitative behaviour even at low order. However, not all mechanical systems may be approximated using variational integrators. Nonholonomic mechanics is one of such cases, where we lack a variational principle, symplecticity and conservation of momentum, in general. Hence, the investigation of the geometric structure of nonholonomic mechanics must be carried out having into account its non-symplectic and non-variational nature. In this thesis, we will deduce new geometric and analytical properties of nonholonomic systems which hopefully will provide a new insight to the subject. Our main definition, which we will meet across all sections, is the nonholonomic exponential map. This map is a generalization of the well-known Riemannian exponential map and we will see that it plays a role in the description of nonholonomic trajectories as well as on the applications to numerical analysis. After introducing this new object, the thesis may be divided into two parts. In the first part, we take advantage of the nonholonomic exponential map to present new geometric properties of mechanical nonholonomic systems such as the existence of a constrained Riemannian manifold containing radial nonholonomic trajectories with fixed starting point and on which they are geodesics. This is a new and surprising result because it opens the possibility of applying variational techniques to nonholonomic dynamics, which is commonly seen to be non-variational in nature. Also, introduce the notion of a nonholonomic Jacobi field and provide a nonholonomic Jacobi equation. In the second part, which is more applied, we use the nonholonomic exponential map to characterize the exact discrete trajectory of nonholonomic systems. Then we propose a numerical method which is able to generate the exact trajectory. On the last chapter, we discuss contact systems and apply the nonholonomic exponential map to construct an exact discrete Lagrangian function for these systems.