A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler-Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem. MSC 2000: 70G45, 70H03, 53C99, 53D17.1 with some compatibility conditions. In the classical version (E = TM ), the Klein's method is based on the vector bundle structure of TM and the existence of a vector-valued 1-form, so called the 'soldering form'. Such a form does not exist for a general Lie algebroid and the conclusion is that the immediate analogy for the Klein's approach does not exist as well. In a series of papers E. Martínez has proposed an interesting modified version of the Klein's method, in which the bundles tangent to E and E * are replaced by the prolongations (in the sense of Higgins and Mackenzie [8]) of E with respect to the vector bundle projections τ : E → M and τ * : E * → M . A similar approach for structures more general than Lie algebroids has been proposed by M. Popescu and P. Popescu [20].The ideas of J. Klein go back to 1962. Since then a lot of work has been done to get a better understanding of the geometric background for Analytical Mechanics. In the papers of Tulczyjew [25] and de León with Lacomba [12] we find another geometric constructions of the E-L equation. The starting point for the Tulczyjew's construction is the dynamics of a system, i.e. a Lagrangian submanifold of TT * M , which is the inverse image of dL(M ) with respect to the canonical diffeomorphism α M : TT * M → T * TM , where L is a function (Lagrangian) on M . The diffeomorphism α M , or its dual κ M : TTM → TTM , represent, unlike the 'soldering form', the complete structure of the tangent bundle.On the other hand, in a couple of papers [6,7], two of us have developed an approach to Lie algebroids based on the analogue of these canonical diffeomorphisms, which form a part of the so called Tulczyjew triple, and which appear to be morphisms of double vector bundles. This allowed us to introduce the notion of a (general, not necessarily Lie) algebroid as a morphism of certain double vector bundles or, equivalently, as a vector bundle equipped with a linear 2-contravariant tensor.What we propose in this paper is to adopt the Tulczyjew approach [24] (cf. also [26]) to the case of a general algebroid. In particular, we obtain a geometric construction of an equation which was suggested by A. Weinstein as a Lie algebroid version of the Euler-Lagrange equation. The main difference with the papers like [13,16,17,20] is not only that we deal with general algebroids but also tha...

Abstract. A notion of an algebroid -a generalization of a Lie algebroid structure on a vector bundle is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on TM can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid. Introduction.The classical Cartan differential calculus on a manifold M , including the exterior derivative d, the Lie derivative £, etc., can be viewed as being associated with the canonical Lie algebroid structure on TM represented by the Lie bracket of vector fields. Lie algebroids have been introduced repeatedly into differential geometry since the early 1950s, and also into physics and algebra, under a wide variety of names. They have been also recognized as infinitesimal objects for Lie groupoids ([18]). We refer to [14] for basic definitions, examples, and an extensive list of publications in these directions.Being related to many areas of geometry, like connection theory, cohomology theory, invariants of foliations and pseudogroups, symplectic and Poisson geometry, etc., Lie algebroids became recently an object of extensive studies.What we propose in this paper is to find out what are, in fact, the structures responsible for the presence of a version of the Cartan differential calculus on a vector bundle and how are they related. This leads to the notion of a general algebroid.It is well known that there exists a one-one correspondence between Lie algebroid structures on a vector bundle τ : E → M and linear Poisson structures on the dual vector bundle π: E * → M . This correspondence can be extended to much wider class of binary operations (brackets) on sections of τ on one side, and linear contravariant 2-tensor fields on E * on the other side. It is not necessary for these operations to be skew-symmetric or to satisfy the Jacobi identity. The vector bundle τ together with a bracket operation, or the equivalent contravariant 2-tensor field, will be called an algebroid. This terminology is justified by the fact that contravariant 2-tensor fields define certain binary operations on the space C ∞ (E * ). The algebroids constructed in this way include all finite-dimensional algebras over real numbers (e.g. associative, Jordan, etc.) as particular examples. The base manifold M is in these cases a single point.Searching for structures which give us differential calculi on vector bundles, we look at objects of analytical mechanics as related to the Lie algebroid structure of the tangent bundle.The tangent bundle TM is the canonical Lie algebroid associated with the canonical Poisson tensor (symplectic form) on T * M . Other canonical objects associated with TM are: the canonical isomorphism α M : TT * M −→ T * TM

Poisson-Nijenhuis structures for an arbitrary Lie algebroid are defined and studied by means of complete lifts of tensor fields.

Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems.Comment: 37 pages, minor corrections, final version to appear in J. Geom. Phy

The derivation d T on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle T M is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.