Abstract: In this talk, which reports on the joint work arXiv:math.DG/1006.4227v2 with J.W.van de Leur, we extend a standard geometric object for smooth manifolds (appearing in relation to ordinary differential equations in mechanics) to its proper analogue for vector bundles over manifolds (which is relevant for partial differential equations in mathematical physics). Namely, we define Lie algebroids over infinite jet spaces for vector bundles and establish their equivalent representation through homological evolutionary vector fields. Lie algebroids over smooth manifolds are a well-known and convenient construction: the tangent bundle to a manifold or the cotangent bundle to a Poisson manifold are examples, and Lie algebras are toy examples of Lie algebroids over a point. It is readily seen that a literal transfer of the definition over manifolds to the infinite jet bundles is impossible, because the most essential axiom is lost ab initio. We now guess which consequence of the classical definition should be taken for the new one over jet spaces. Then, miraculously, the entire construction works: we recover the alternative definition through the odd (homological) vector fields on infinite jet super-spaces, and the equivalence proof does hold, although its main ingredient has vanished. The resulting picture, which we develop in full parallel with Vaintrob (1997), more fully grasps the dynamics of strings in comparison with the study of QP-manifolds by Aleksandrov, Kontsevich, Zaboronsky, and Schwarz (1997). We consider a clarifying example of the variational Poisson algebroids (the simplest model is the Korteweg-de Vries equation). In this case, the odd field Q is Hamiltonian with respect to the canonical symplectic structure and the variational Poisson bi-vector P as the Hamiltonian. The Hamiltonian P of Q then amounts to the bilinear kinetic term that equals the Lagrangian. This is a legitimate reason for us to regard the variational Poisson formalism (e.g., for KdV-like hierarchies) as a remote part of the theory of uniform rectilinear motion.
