Geometric Mechanics

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Proposal #0204474 PI: Jerrold E. Marsden Institution: California Institute of Technology Title: Geometric Mechanics ABSTRACT Work under this NSF grant will develop geometric mechanics and reduction theory in the context of several problems of interest: integration algorithms and reduction theory for discrete mechanics, the symplectic nature of collision algorithms, controlled Lagrangian and Hamiltonian systems with symmetry, reduction in the context of covariant field theory (such as electromagnetism and gauge fields) using covariant Poisson brackets as well as Lagrangian reduction, and bifurcations in mechanical systems with symmetry using the blowing-up technique, including the establishment of connections with singular reduction. The combination of geometric and analytical techniques continues to play an important role in many aspects of mechanics. This includes developments in computational algorithms. For instance, recent advances in variational integration algorithms have produced state of the art collision codes, for example, for collisions between elastic bodies and elastic shells. The work in this proposal further develops the mathematical framework that led to these algorithms in the first place. This includes proving, in a precise sense, that the algorithms respect the special structure of mechanics, such as conservation laws. In addition, studying the effect of symmetry on such algorithms and factoring out this symmetry directly on the discrete level is an important aspect of these algorithms which is only understood in special cases at the moment; this proposal will develop and extend these techniques. Another area where symmetry and geometric techniques are important is that of the control of mechanical systems (such as how rotors are used to control spacecraft orientation). Symmetry reduction for the control of such mechanical systems is one of the techniques that will be developed under this grant.