This paper improves a new inclusion concept, called ’extension’ to mechanical systems described by matrix second-order equations. Recent work on specialization of the inclusion principle has been generalized to the expansion and contraction of input, state and output spaces, but without the possibility of constructing completely general contractible controllers. In this paper, necessary and sufficient conditions for extensions and contractible feedback controllers are proved. They enable the construction of general contractible controllers without any constraint on the structure of controllers for mechanical systems.