ar X iv : d g - ga / 9 70 30 18 v 1 2 4 M ar 1 99 7 Higher order Lagrangian supermechanics

Generalized mechanics where the Lagrangian describing the system depends on higher order derivatives has been investigated in connection with relativistic dynamics of particles, supersymmetry, polymer physics, string models and gravity theory, among other problems (see, e.g., [11]). For instance, it has been proposed to add to the point particle action a term proportional to acceleration, or the inclusion of higher order terms in superstring action. The mathematical theory of such generalized Lagrangians was started by Ostrogradski [9], and it was more recently analysed, from a geometric perspective, in [5]. The study of classical higher order Lagrangian systems was considered in [6]. Our aim is to provide the appropriate mathematical framework to deal with such systems, along similar lines to those developed in [1]. The first order Lagrangian formalism of supermechanics takes place on the tangent supermanifold TM of a given graded manifold M = (M,A), [1, 7]. As in the non–graded case, using the geometric structure of the tangent supermanifold, which is encoded in the vertical superendomorphism S, one can construct a graded symplectic structure ΩL = −dΘL = −d(dL ◦ S), out of a regular Lagrangian superfunction L ∈ TM. The dynamics of such systems is governed by a second order differential superequation Γ ∈ X (TM), which is the unique solution of the dynamical equation iXΩL = dEL, where the superenergy is defined, in terms of the Liouville supervector field ∆, by EL := ∆L− L. In analogy to the classical case, to establish a one–to–one correspondence between symmetries of L and constants of motion it is necessary to generalize the concept of symmetry so as to include “non–point transformations”. These symmetries turn out NOT to be supervector fields but rather supervector fields along a morphism (see [2] for a detailed discussion of supervector fields and graded forms along a morphism). Moreover, from a technical point of