Introduction

The Lagrangians defined on the parametrized curves (on the trajectories) and depending on higher derivatives of coordinates of the curve have been considered recently in a number of problems. These, in particular, are: the null-dimensional (particle-like) version of the rigid string [1 - 7], the model of boson-fermion transmutations in external Chern-Simons field [8 - 11], polymer theory [12]. Lagrangians of this kind also occur when applying a modified version of the space-time interval proposed in papers [13, 18] in connection with the conjuncture about existence of a limited value of acceleration. Recently these variational problems have also become interesting for mathematicians [19]. This list of references is certainly incomplete, however, it illustrates the continuing interest in the subject.

All these Lagrangians are as a rule singular. Investigation of such models in the framework of a classical variational calculus results in very complicated nonlinear differential equations of order 2p for coordinates of a curve to be found (p is the highest order of the derivatives in an initial Lagrangian function). These equations are not practically subject to analysis.

However, a considerable advance can be achieved here by applying the following basic result from the classical differential geometry [20, 21]. Any smooth curve xμ(s), μ = 0, 1, …, D - 1 in D-dimensional flat space-time is determined (up to its rotations as a whole) by specifying D - 1 principal curvatures of this curve ki(s), i = 1, 2, …, D - 1, where s is the curve length.