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Geometric Formulation of Higher Order Lagrangian Systems in Supermechanics

Using supervector elds along a morphism of graded manifolds, we study both, the geometry of the tangent supermanifolds of an arbitrary order, and k{th order ordinary diierential su-perequations, in order to develop the higher order Lagrangian formalism of supermechanics. In particular, we obtain, in this formalism, a generalization of Noether's theorem and its converse.