In this talk we will present some recent work on control theory for mechanical systems whose Lagrangian consists of the kinetic energy with respect to a Riemannian metric.

In the first part of the talk we will give a review of affine connections. We also present an interpretation of a product, which we have dubbed the "symmetric product," on the set of vector fields on a manifold with an affine connection.

With an understanding of some basic properties of affine connections, we may concisely present a decomposition for control systems whose drift vector field is the Lagrangian vector field for kinetic energy Lagrangians. This decomposition is the natural one associated with a version of controllability for these systems. In particular, we are able to present computable conditions which determine whether a given system is controllable.

In the last part of the talk, we turn to systems with the same kinetic energy Lagrangians, but which are also subject to linear constraints in their velocities. Using a simple procedure, we may think of the solutions of the constrained system as being geodesics of an affine connection. This affine connection possesses certain properties and we able to give a complete characterisation of all affine connections with these properties (at least locally). If time permits, we will also give a short discussion of symmetry in problems with constraints.

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