Lie algebroid structures and Lagrangian systems on affine bundles

Abstract

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.Comment: 28 pages, Late

Variational calculus on Lie algebroids

“…We need to extend the above result to time dependent sections and flows (to ensure the existence of a solution η for an equation such as σ(t) = η(t, γ(t)) for a section σ along γ, which may not have solution for η time-independent). The best way to treat them is to move to the time-dependent setting [26]. Given the Lie algebroid τ : E → M we consider the direct product Lie algebroid of T R with E, that is, the Lie algebroidτ :Ē = T R × E →M = R × M with anchorρ(λ∂ t + a) = λ∂ t + ρ(a) and bracket determined by the bracket of biprojectable sections [α + η, β…”

Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

“…The theory reviewed in the preceding two sections has been extended (at least partially) to other dynamical systems of interest, such as time-dependent sodes (see [22]), mixed first-and second-order equations which includes the equations of non-holonomic mechanics (see [19]), and Lagrangian systems on Lie algebroids (see [14]). We limit ourselves here to a sketch of some features of the generalization to time-dependent sodes.…”

Jacobi Structures on Affine Bundles

“…The main motivation of the study of this concept was to create a geometrical model which would be a natural environment for a time-dependent version of Lagrange equations on Lie algebroids (cf. [7,8,29,32]). Now, we can decompose P = Λ + I 0 ∧ E, where Λ ∈ Γ(∧ 2 A 0 ) and E ∈ Γ(A 0 ), and we deduce…”

“…This is why frame-independent formulations require affine bundles and Jacobi structures (algebroids). We refer here to [7,8,29,32,30] (time-dependent mechanics) and [34].…”