Dynamics of mechanical systems with symmetry

点击翻译按钮获取中文摘要

Prediction and control of the future are central themes of science and technology. We all want to know what the future will bring. We also want to control the future so that either undesired states are eliminated (e.g., preventing the spreading of a disease), or desired states are attained (e.g., redirecting the camera of a satellite towards a target). *** This proposal concerns the dynamics of mechanical systems, that is, how they move or evolve over time. While the future evolution of a system can to some degree be predicted using computer algorithms, a theoretical understanding of possible dynamical features can improve prediction and also lay the groundwork for strategies to control it.  *** Many systems may be analyzed together, if they display similar characteristics. For mechanical systems, these characteristics may refer to the geometry of the environment (e.g., motion on a flat surface as opposed to motions in space), the physical laws (e.g., motions where the energy is conserved, as opposed to motions with friction), or symmetries (e.g. a perfectly round ball, without markings on it, looks the same from all directions). Systems are then organized in specific classes, and within each one, we attempt to describe the possible dynamical behaviours. Of particular interest are the steady states (in which the system remains "frozen") and other special trajectories, their stability, and the changes ("bifurcations'') that may occur as certain physical features or parameters vary.*** My aim is to develop methods and techniques useful for capturing the common dynamical features of energy conserving mechanical systems with symmetry. The proposal considers two sub-classes: 1) unconstrained mechanical systems such as n-particle models of molecules, or rigid bodies in gravitational interactions (for example, asteroids); and 2) constrained mechanical systems, such as sleighs, rolling spheres, and finned arrows or underwater vehicles modelled under the assumption that dissipative forces (for instance, friction) are neglected. Within each subclass, I propose to develop methods for studying bifurcation and stability. I will 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).*** My research adds to the fundamental knowledge of dynamical systems and has applications in mechanical engineering (control and stabilization of mechanical devices), space science (e.g., stability of asteroidal systems, spin-orbit coupling of moon-planet systems) and physical chemistry (e.g., roto-vibrational states of polyatomic molecules). **