Routh reduction and the class of magnetic Lagrangian systems

In this paper, some new aspects related to Routh reduction of Lagrangian systems with symmetry are discussed. The main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after Routh reduction of Lagrangian systems with symmetry, so-called magnetic Lagrangian systems. We use these transformations in order to show that, under suitable conditions, the reduction with respect to a (full) semi-direct product group is equivalent to the reduction with respect to an Abelian normal subgroup. The results in this paper are closely related to the more general theory of Routh reduction by stages.Comment: 23 page

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

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

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

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

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