In the present paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found. We prove that the nonlinear connection induced by a regular Hamiltonian on a Lie algebroid is the unique connection which is compatible with the metric and symplectic structures

In this paper the problem of compatibility between a nonlinear connection and other geometric structures on Lie algebroids is studied. The notion of dynamical covariant derivative is introduced and a metric nonlinear connection is found in the more general case of Lie algebroids. We prove that the canonical nonlinear connection induced by a regular Lagrangian on a Lie algebroid is the unique connection which is metric and compatible with the symplectic

In this paper we generalize the linear contravariant connection on Poisson manifolds to Lie algebroids and study its tensors of torsion and curvature. A Poisson connection which depends only on the Poisson bivector and structural functions of the Lie algebroid is given. The notions of complete and horizontal lifts are introduced and their compatibility conditions are pointed out.

In this paper we study the distributional systems (driftless control affine systems) with positive homogeneous cost, using the framework of Lie algebroids. The method consists of applying the Pontryagin Maximum Principle at the level of Lie al- gebroid built for both holonomic and nonholonomic distributions. This simplified the approaches and reveals certain important connections.

In this paper we study on the prolongation of Lie algebroid the notions such as: nonlinear connection, related connections, torsion and curvature, semispray and complex structures. The case of homogeneous connections and some examples are presented

In this paper we start developing the so-called Klein's formalism on dual Lie algebroids. The nonlinear connection associated to a regular section is naturally obtained. Particularly, this connection is found for the Hamiltonian case.

In this paper the problem of compatibility between a nonlinear connection and some other geometric structures on the cotangent bundle of a manifold is studied. We prove that the notions of semi-Hamiltonian vector field on cotangent bundle and the metric nonlinear connection on tangent bundle are dual structures, via Legendre transformation.