Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2009, Article ID 987524, 21 pages doi:10.1155/2009/987524 Research Article The Meaning of Time and Covariant Superderivatives in Supermechanics Gil Salgado and Jos ´ e A. Vallejo-Rodr´ ıguez Departamento de Matem´ aticas, Facultad de Ciencias, Universidad Aut´ onoma de San Luis Potos´ ı, 78290, Mexico Correspondence should be addressed to Jos´ e A. Vallejo-Rodr´ ıguez, jvallejo@galia.fc.uaslp.mx Received 23 January 2009; Accepted 21 April 2009 Recommended by Jos´ e F. Cari˜ nena We present a review of the basics of supermanifold theory in the sense of Berezin-Kostant- Leites-Manin from a physicist’s point of view. By considering a detailed example of what does it mean the expression “to integrate an ordinary superdifferential equation” we show how the appearance of anticommuting parameters playing the role of time is very natural in this context. We conclude that in dynamical theories formulated whithin the category of supermanifolds, the space that classically parametrizes time the real line R must be replaced by the simplest linear supermanifold R 1|1 . This supermanifold admits several different Lie supergroup structures, and we analyze from a group-theoretic point of view what is the meaning of the usual covariant superderivatives, relating them to a change in the underlying group law. This result is extended to the case of N-supersymmetry. Copyright q 2009 G. Salgado and J. A. Vallejo-Rodr´ ıguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The usual interpretation of time in physics at least in classical mechanics is as follows. Consider a dynamical system described by a Hamiltonian H, which is a differentiable function defined on some symplectic manifold M, ω. Physical trajectories are then identified with integral curves of a vector field X H ∈XM such that i X ω  dH. These integral curves define a flow, that is, a differentiable mapping Φ : I × M → M given by Φt, p c p t, where c p : I → M is the maximal integral curve passing through p ∈ M at t  0, and I ⊂ R is the maximal interval of definition of the set {c p } p∈M of integral curves. Then, “time” is the space of values of the parameter t, that is, the subset I ⊂ R with a manifold structure. In some cases e.g., dynamics on compact manifolds I  R and then becomes a one-dimensional Lie group with the operation of addition although, in general, this operation is only locally defined.