Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake" as exposed in [PeSa].

No abstract

No abstract

In this paper, we study several objects in the framework of direct limits of anchored Banach bundles over particular convenient manifolds (direct limits of Banach manifolds). In particular, we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anchor ranges.Proposition 14. Let (X n ) n∈N * be an ascending sequence of topological spaces. Equip X = n∈N * X n with the final topology with respect to the inclusion maps ε n : X n −→ X (i.e. the DL-topology). Then we have:O n is open in X and the DL-topology on O = lim − → O n coincides with the topology induced by X. 3. If each X n is locally compact, then X is Hausdorff. 4. If each X n is T 1 and K ⊂ X is compact, then K ⊂ X n for some n.Unfortunately, in general, a direct limit of Hausdorff topological spaces is not Hausdorff (see [Her] for an example of such a situation). Sufficient conditions on