A gauge theory approach to the swimming in curved spaces problem

Author(s)

Danilo Borim do Nascimento

• Document type: Master's Dissertation
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2010-05-20

Examining board members

Ricardo Antonio Mosna; Marcos Benevenuto Jardim; George Emanuel Avraam Matsas

• Advisor: Ricardo Antonio Mosna

Abstract

In Euclidean space, cyclic deformations of quasi-rigid bodies can lead to net global rotations even though they satisfy, at each moment, the angular momentum conservation law (the falling cat problem is an example). In curved spaces, cyclic changes in the body shape can also lead to rotations, but also to global translations. This phenomenon is known as the swimming effect. In a recent work, Avron and Kenneth developed a formalism to describe this phenomenon in the non-relativistic context [Avron JE, Kenneth O, New J. Phys. 8, 68 (2006)], which may be used to calculate the net displacement caused by an infinitesimal cycle of deformations of a given body. This displacement is then related, for small swimmers, to the curvature of the ambient space. In the present work, we propose a new formulation for the swimming effect in terms of principal bundles and connections. The configuration space of the system is described by the total space of a principal bundle, whose base space is given by the space of shapes of the body and whose structural group is (essentially) given by the isometries of the ambient manifold. A given deformation cycleof the body then corresponds to a loop in the base space. By defining a connection in this bundle which conveys the physical conservation laws of the system, the corresponding physical motion of the body is then given by the horizontal lift of this curve in the base space, while the net displacement of the body is given by the holonomy associated with this loop. As a result we obtain, in a systematical way, the displacement generated by arbitrary deformation cycles and we get, for each instant of time, the time evolution of the system analytically (AU)

• FAPESP's process: 07/06098-1 - Geometric phases via concrete models of Gauge theories
• Grantee: Danilo Borim do Nascimento
• Support Opportunities: Scholarships in Brazil - Master