UC Riverside

Span categories provide an abstract framework for formalizing mathematical models of certain physical systems. The categories appearing in classical mechanics do not have pullbacks, limiting the utility of span categories in describing such systems. We introduce the notion of a generalized span category and an augmentation of a generalized span category. As an application of augmented generalized span categories, we introduce the categories $\LagSy$ and $\HamSy$ that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of certain classical mechanical systems. The morphisms of $\LagSy$ and $\HamSy$ contain all kinematical and dynamical information about these systems and composition of morphisms models the construction of systems from subsystems. A functor from $\LagSy$ to $\HamSy$ translates from the Lagrangian to the Hamiltonian perspective and is a categorical analog of the Legendre transformation.