``Simple'' mechanical systems are those which, in the Lagrangian framework, have kinetic energy minus potential energy Lagrangians. These systems can, even in the presence of nonholonomic constraints, be effectively modelled in an affine connection framework. When dealing with control systems, the resulting interplay with geometric control and affine differential geometry is quite striking. The main emphasis of this talk will be on the role of affine differential geometry in the controllability of mechanical systems. Simple examples illustrate the main ideas.
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