A geometrical description of the Lagrangian dynamics in quasi-coordinates on the tangent bundle, using the Lie algebroid framework, is given. Linear nonholonomic systems on Lie algebroids are solved in local coordinates adapted to the constraints, through Lagrangian multipliers and Gibbs-Appell generalized methods.

In this work we establish the relation between the Jacobi last multiplier, which is a geometrical tool in the solution of problems in mechanics and that provides Lagrangian descriptions and constants of motion for second-order ordinary differential equations, and nonholonomic Lagrangian mechanics where the dynamics is determined by Hamel’s equations.

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [20]. We also provide several examples to illustrate our construction.

The virial theorem is formulated both intrinsically and in local coordinates for a Lagrangian system of mechanical type on a Riemann manifold. An import case studied in this paper is that of an affine virial function associated to a vector field on the configuration manifold. The special cases of a virial function associated to a Killing, a homothetic and a conformal Killing vector field are considered and the corresponding virial theorems are established for this type of functions.

Based on the ideas of Marsden–Ratiu, a reduction method for Lie algebroids is developed in such a way that the canonical projection onto the reduced Lie algebroid is a homomorphism of Lie algebroids. A relation between Poisson reduction and Lie algebroid reduction is established. Reduction of Lie algebroids with symmetry is also studied using this method.