CLASSICAL DYNAMICS OF THE RIGID STRING FROM THE WILLMORE FUNCTIONAL

A new approach for investigating the classical dynamics of the relativ istic string model with rigidity is proposed. It is based on the embed ding of the string world surface into a space of constant curvature. I t is shown that the rigid string in flat space-time is described by th e Euler-Lagrange equation for the Willmore functional in a space-time of constant curvature K = -gamma/(2 alpha), where gamma and alpha are constants in front of the Nambu-Goto term and the curvature term in th e rigid string action, respectively. For simplicity the Euclidean vers ion of the rigid string in three-dimensional space-time is considered. The Willmore functional (the action for the ''Willmore string'') is o btained by dropping the Nambu-Goto term in the Polyakov-Kleinert actio n for the rigid string. Such a ''reduction'' of the rigid string model would be useful, for example, by applying some results about the Namb u-Goto string dynamics in the de Sitter universe to the rigid string m odel in the Minkowski space-time. It also allows us to use numerous ma thematical results about Willmore surfaces in the context of the physi cal problem.