Steve Smale and Geometric Mechanics

Thus, one can say-perhaps with only a slight danger of oversimplification- that reduction theory synthesises the work of Smale, Arnold (and their predecesors of course) into a bundle, with Smale as the base and Arnold as the fiber. This bundle has interesting topology and carries mechanical connections (with associated Chern classes and Hannay-Berry phases) and has interesting singularities (Arms, Marsden, and Moncrief, Guillemin and Sternberg, Atiyab, and otbers). We will describe some of these features later

The Orbit Bundle Picture of Cotangent Bundle Reduction

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T*Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T^*Q/G, decomposed as a Whitney sum bundle, T^*⊕(Q/G)g^* over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found

Geometric Mechanics, Stability and Control

This paper gives an overview of selected topics in mechanics and their relation to questions of stability, control and stabilization. The mechanical connection, whose holonomy gives phases and that plays an important role in block diagonalization, provides a unifying theme

On some aspects of the geometry of differential equations in physics

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. Furthermore, research to be developed in these areas is also commented.Comment: 21 page

Generalized Hamiltonian mechanics

Our purpose is to generalize Hamiltonian mechanics t the case in which the energy function (Hamiltonian), H , is a distribution (generalized function) in the sense of Schwartz. We follow the same general program as in the smooth case. Familiarity with the smooth case is helpful, although we have striven to make the exposition self-contained, starting from calculus on manifold

A List of References on Spacetime Splitting and Gravitoelectromagnetism

Revised version to be published in the Proceedings of the Encuentros Relativistas Espa\~noles, September, 2000 [ http://hades.eis.uva.es/EREs2000 ]Comment: 22 pages, LaTeX article style. Please send corrections and additions to mailto:[email protected]

A Nonlinear Analysis of the Averaged Euler Equations

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $\alpha$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $\alpha$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.Comment: 25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of his 60th birthday, Arnold Festschrift Volume 2 (in press