Classical dynamics of rigid string from Willmore functional

TL;DR

This paper introduces a novel approach to studying the classical dynamics of rigid strings by relating them to the Willmore functional in a space of constant curvature, simplifying analysis and leveraging mathematical results on Willmore surfaces.

Contribution

It establishes a connection between rigid string dynamics and the Willmore functional in constant curvature spaces, enabling new analytical methods and insights.

Findings

Rigid string dynamics can be described by the Euler-Lagrange equation for the Willmore functional.

The approach simplifies the analysis by dropping the Nambu-Goto term, focusing on the curvature term.

Mathematical results on Willmore surfaces can be applied to physical models of rigid strings.

Abstract

A new approach for investigating the classical dynamics of the relativistic string model with rigidity is proposed. It is based on the embedding of the string world surface into the space of a constant curvature. It is shown that the rigid string in flat space-time is described by the Euler-Lagrange equation for the Willmore functional in a space-time of the constant curvature K=-g/(2 a), where g and a are constants in front of the Nambu-Goto term and the curvature term in the rigid string action respectively. For simplicity the Euclidean version of the rigid string in the three-dimensional space-time is considered. The Willmore functional (the action for the "Willmore string") is obtained by dropping the Nambu-Goto term in the Polyakov-Kleinert action for the rigid string. Such "reduction" of the rigid string model would be useful, for example, by applying some results about the Numbu-Goto string dynamics in the de Sitter universe to the rigid string model in the Minkowski space-time. It allows also to use numerous mathematical results about the Willmore surfaces in the context of the physical problem.