Boris Kolev

On the geometric approach to the motion of inertial mechanical systems

According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L 2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C 1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H 1 rightinvariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a C 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D ,t heycan be joined by a unique length-minimizing geodesic—a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy. We also analyse for both metrics the breakdown of the geodesic flow.

The Degasperis-Procesi equation as a non-metric Euler equation

In this paper we present a geometric interpretation of the Degasperis–Procesi (DP) equation as the geodesic flow of a right-invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the DP and the Camassa–Holm equation as an ODE on the Fréchet space of all smooth functions on the circle.

Integrability of Invariant Metrics on the Diffeomorphism Group of the Circle

AbstractEach Hk Sobolev inner product (k ≥ 0) defines a Hamiltonian vector field Xk on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X0 and X1 are bi-Hamiltonian nrelative to a modified Lie-Poisson structure.

Lie Groups and Mechanics: An Introduction

Abstract The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a third section we apply these methods for the diffeomorphism group of the circle with two particular examples: the Burger equation and the Camassa-Holm equation.

Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie–Poisson structure and the second one is constant. These structures, called affine or modified Lie–Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

Poisson brackets in Hydrodynamics

This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a valid and usable bracket to study rotational fluid flows with a free boundary. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open.

The geometry of a vorticity model equation

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

mathrm{Diff}_{1}^{infty}(mathbb{S})

Two-component equations modelling water waves with constant vorticity

of orientation-preserving diffeomorphisms

Geodesic completeness for Sobolev H s -metrics on the diffeomorphism group of the circle

varphi in mathrm{Diff}^{infty}(mathbb{S})