Almost Lie structures on an anchored Banach bundle
Several geometric structures on anchor bundles are generalized to the setting of Banach manifolds. In order to adapt results from the finite-dimensional case to this Banach framework, additional natural assumptions are imposed. The following are results distinguished by the authors: the equivalence between almost Lie-Banach algebroid structure on a bundle and Poisson structure on the dual bundle; the construction of a unique bracket and a Banach anchored bundle associated to a given derivation, and the affine structure of the set of all almost Lie brackets on an anchored Banach bundle. As applications of these results, the classical formalism for mechanical systems is also adapted. This formalism includes Hamiltonian systems, Hamilton-Jacobi equation, Lagrangian system and Euler-Lagrange equation. Finally, an interesting example, the problem of the Hilbert snake, shows how the results on Riemannian almost Lie algebroids can be applied.
scientific article; zbMATH DE number 6086945
scientific article