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...

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained EulerLagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E = T M . 2000: 70H03, 70H25, 53D17, 17B66, 53D10. MSC

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

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.

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.

The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important feature of our approach is that we do not postulate ad hoc the ingredients of the theory, but obtain them as unavoidable consequences of the variational calculus. This picture of Field Theory is covariant and complete, containing not only the Lagrangian formalism and Euler-Lagrange equations but also the phase space, the phase dynamics and the Hamiltonian formalism. Since the configuration space turns out to be an affine bundle, we have to use affine geometry, in particular the notion of the affine duality. In our formulation, the two maps α and β which constitute the Tulczyjew triple are morphisms of double structures of affine-vector bundles. We discuss also the Legendre transformation, i.e. the transition between the Lagrangian and the Hamiltonian formulation of the first-order field theory.MSC 2010: 53D05, 58A20, 70S05, 70H03, 70H05

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain R-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The correct geometrical and frame-independent description of Newtonian Mechanics is of this type.The paper is organized as follows. In Section 2 we recall rudiments of the AV-geometry that will be needed in the sequel. Section 3 is devoted to the presentation of the concept of double affine bundle which is the fundamental concept in our approach. In Section 4 we recall the Lagrangian and Hamiltonian formalisms for general algebroids developed in [5]. On this fundamentals, special affgebroids as certain morphisms of double affine bundles and the corresponding Lagrangian and Hamiltonian formalisms are constructed and studied in Section 5. In particular, we derive analogs of Euler-Lagrange equations. We end up with some examples in Section 6. Rudiments of the AV-geometryWe refer to [4] (see also [3,29]) for a development of the AV-geometry. To make the paper selfcontained, we recall basic notions and facts here. Affine spaces and affine bundlesAn affine space is a triple (A, V, α), where A is a set, V is a vector space over a field K and α is a mapping α : A × A → V such that • α(a 3 , a 2 ) + α(a 2 , a 1 ) + α(a 1 , a 3 ) = 0;• the mapping α(·, a) : A → V is bijective for each a ∈ A.