Attempts to relate the Navier-Stokes equations to turbulence

The present talk is designed as a survey, is slanted to my personal tastes, but I hope it is still representative. My intention is to keep the whole discussion pretty elementary by touching large numbers of topics and avoiding details as well as technical difficulties in any one of them. Subsequent talks will go deeper into some of the subjects we discuss today. The main goal is to link up the statistics, entropy, correlation functions, etc., in the engineering side with a "nice" mathematical model of turbulence

Chaos in dynamical systems by the Poincaré-Melnikov-Arnold method

Methods proving the existence of chaos in the sense of Poincaré-Birkhoff-Smale horseshoes are presented. We shall concentrate on explicitly verifiable results that apply to specific examples such as the ordinary differential equations for a forced pendulum, and for superfluid He and the partial differential equation describing the oscillations off a beam. Some discussion of the difficulties the method encounters near an elliptic fixed point is given

Remarks on Geometric Mechanics

This paper gives a few new developments in mechanics, as well as some remarks of a historical nature. To keep the discussion focused, most of the paper is confined to equations of "rigid body", or "hydrodynamic" type on Lie algebras or their duals. In particular, we will develop the variational structure of these equations and will relate it to the standard variational principle of Hamilton

Qualitative methods in bifurcation theory

No abstract

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 Hamiltonian formulation of classical field theory

In this paper I shall present some result from the theory of classical non-relativistic field theory and discuss how they might be useful in the general relativistic context. Some of the Hamiltonian formalism has already been successfully employed in the general relativistic context, but much more remains to be done in the area of dynamic stability, linearization stability, bifurcation, symmetry breaking, and covariant reduction

On the Geometry of the Liapunov-Schmidt Procedure

The lectures presented by the author are not reproduced here since that material is available in J. Marsden, Qualitative Methods in Bifurcation Theory, Bull. Am. Math. Soc. 84 (1978), 1125–1148, R. Abraham and J. Marsden, Foundations of Mechanics, Second Edition, Addison Wesley (1978), and in J. Marsden and M. McCracken, The Hopf Bifurcation and its Application

Well-posedness of the equations of a non-homogeneous perfect fluid

The Euler equations for a non-homogeneous, non-viscous compressible fluid are shown to be well-posed for a short time interval, using techniques of infinite dimensional geometry and a weighted Hodge theorem. Regularity and other properties of these solutions are pointed out as well

Non smooth geodesic flows and classical mechanics

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In face, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q -space or the configuration space), we may define a smooth function T_g on the cotangent bundle T^*M (q -p -space, or phase space). This function is the kinetic energy of q , and locally is given by T_g(q, p) = gij(q) P_i P_j Where q = (q^1,..., q^n) and p = (p_1,..., p_n) and g has components g^(ij). Using T_g as a Hamiltonian function, the associated flow (that is, the global solution of Hamiltonian's equations) is exactly the geodesic flow; geodesics are obtained by projection to M . Conversely, Hamiltonian motion in a potential V and metric g , that is, H = T_g + V , may be thought of as geodesic motion using the metric (e - V)g if e > V(q) . This new metric is called the Jacobi metric. Traditionally, the theory of classical mechanics and Riemannian geometry always assumes g and V are smooth functions. However, the most elementary examples in fact are not smooth. One of the main reasons for the smoothness assumption was to guarantee existence of the flow (geodesics). This objection has now been removed. The purpose of this note is to explain in an expository fashion what changes are necessary in the above theory to cover the non-smooth case. This new situation is quite different, although some interesting observations can be made