Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle. They are examples of double structures, graded-linear (GL) bundles, including double vector bundles as a particular case. On GL-bundles we define what we shall call weighted algebroids, which are to be understood as algebroids in the category of graded bundles. They can be considered as a geometrical framework for higher order Lagrangian mechanics. Canonical examples are reductions of higher tangent bundles of Lie groupoids. Weighted algebroids represent also a generalisation of VB-algebroids as defined by Gracia-Saz & Mehta and the LA-bundles of Mackenzie. The resulting structures are strikingly similar to Voronov's higher Lie algebroids, however our approach does not require the initial structures to be defined on supermanifolds.

Abstract. We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1, R)-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

We extend the notion of super-Minkowski space-time to include Z n 2 -graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of Z n 2 -manifolds understood as locally ringed spaces. The formalism we present resembles N -extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed.

In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie groupoids. MSC (2010): 53C10; 53D17; 53Z05; 55R10; 70H03; 70H50.