Grant: Discovery Grants Program - Individual

Virtually all systems in nature are subject to dynamics, that is, to change. The ultimate goal of the study of a system's dynamics is the prediction of the system's state (in space and time), when some information on the initial state is given. Systems displaying similar characteristics may be analyzed together. The information obtained is then put into use by control methods, or adds to our understanding of the universe. The research proposed concerns two topics: 1) Dynamics of unconstrained mechanical systems such as n-particle systems (e.g. interacting molecules or planetary systems), and 2) Dynamics of constrained mechanical systems such as sleighs, rolling spheres, and finned arrows or underwater vehicles modeled under the assumption that dissipative forces (e.g. friction) are neglected. A common feature of these systems is that nearby initial states, in systems that are almost alike, may lead to very different dynamical evolutions. To make progress, one has to use the mathematical formalism and methods of differential equations (to describe the evolution of the system), differential geometry and topology (to describe the space) and bifurcation theory (to describe the possible branches of the dynamics). A specific problem I want to study is how molecules change their shape when subject to spins of different orientations. Also, I want to investigate dynamical bifurcations for the class of constrained mechanical systems as described above.