Journal of Geometric Mechanics
June 2009 , Volume 1 , Issue 2
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We use Frölicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is a Lagrangian vector field. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.
Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems [18, 19]. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles . Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.
We generalize the notion of submersive second-order differential equations by relaxing the condition that the decoupling stems from the tangent lift of a basic distribution. It is shown that this leads to adapted coordinates in which a number of first-order equations decouple from the remaining second-order ones.
We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabeling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics.
The first Part of the two Geometric Mechanics books focuses on dynamics in geometric mechanical systems and using Lie symmetry reductions to analyse the dynamics. The second Part looks in depth at translational and rotational motions of rigid bodies in the geometric context of Lie symmetry groups and sets up the Euler-Poincare framework. It finishes with a chapter on rolling motion as an example of applications of the framework to other problems in geometric mechanics. The aim of the Geometric Mechanics books is to make the reader familiar with the concepts of geometric mechanics and the power of symmetry reduction. Mathematical rigour is not an aim, though details of the proofs of most statements in Part I are provided. As pointed out in its preface, Part II has a more inquiry based approach and doesn't focus on mathematical rigour. Instead many references are given and there are two appendices providing more mathematical background. In both Parts, physical examples play an important role. Indeed, all concepts and theory are motivated by examples. There are also excellent references to recent literature as well as nice historic contexts. The level of the books is aimed at advanced undergraduate students and beginning graduate students. The assumed background consists of first courses in classical mechanics, standard linear algebra, vector calculus and ordinary differential equations. No knowledge of differential geometry or Lie groups is assumed, these topics are introduced when needed in the books.
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