Intrinsic approach to the Legendre transformation in super-mechanics

Summary

In recent years, there has been an increasing interest to provide supermechanics with a solid geometrical base. Some of the difficulties are right at the beginning: there is a general consent about the configuration space, which is taken to be a supermanifold, but there is no general agreement on what the velocity phase space should be. Naturally, the candidate for that should be a generalization of the tangent bundle in the context of supergeometry; unfortunately there are several different (and reasonable) notions that could serve that purpose. The tangent supermanifold introduced by Ibort and Marín-Solano in [8] seems to be the right candidate from the point of view of a physicist since a good deal of supermechanics can rigorously be developed; nevertheless, the main disadvantage of the tangent supermanifold is that it is not a bundle in any sense, which forces a local coordinate approach that does not give too much insight, or a purely algebraic approach, in practice difficult to handle; besides, in classical mechanics one usually takes advantage of the fact that vector fields, for instance, are, after all, sections of a bundle. In this sense, the tangent superbundle introduced by Sánchez–Valenzuela [10,11,2] seems to be quite appropriate, from a theoretical point of view, since there is a one–to–one correpondence between sections of this superbundle and supervector fields regarded as superderivations. Unfortunately, the tangent superbundle is too big, its dimension is (2m + n, 2n + m) if the dimension of the configuration space is (m,n).