We prove that $\W_3$ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The $\W_3$ generators $T$ and $W$ then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the ``zero-curvature gauge'', and how to rewrite them in terms of the $\KdV$ operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the $\SKdV$ operator. We also partially extend our analysis in the supersymmetric domain to the scale invariant rigid particle by explicitly constructing a supercovariant version of its action. Comment: This is an expanded version of hep-th/9406072 (to be published in the Proceedings of the Workshop on the Geometry of Constrained Dynamical Systems, held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, June 14-18, 1994.).Comment: 14 pages, plain TeX (macros included). QMW-PH-94-2

We prove that particle models whose action is given by the integrated n-th curvature function over the world line possess n + 1 gauge invariances. A geometrical characterization of these symmetries is obtained via Frenet equations by rephrasing the n-th curvature model in R d in terms of a standard relativistic particle in S d−n . We "prove by example" that the algebra of these infinitesimal gauge invariances is nothing but W n+2 , thus providing a geometrical picture of the W-symmetry for these models. As a spin-off of our approach we give a new global invariant for four-dimensional curves subject to a curvature constraint. †

Classical W 3 transformations are discussed as restricted diffeomorphism transformations (W-Diff) in two-dimensional space. We formulate them by using Riemannian geometry as a basic ingredient. The extended W 3 generators are given as particular combinations of Christoffel symbols. The defining equations of W-Diff are shown to depend on these generators explicitly. We also consider the issues of finite transformations, global SL(3) transformations and W-Schwarzians.

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