  The paper presents necessary and sufficient conditions for the order
reduction of optimal control systems. Exploring the corresponding Hamiltonian
system allows to solve the order reduction problem in terms of dynamical
systems, observability and invariant differential forms. The approach is
applicable to non-degenerate optimal control systems with smooth integral cost
function. The cost function is defined on the trajectories of a smooth
dynamical control system with unconstrained controls and fixed boundary
conditions. Such systems form a category of Lagrangian systems with morphisms
defined as mappings preserving extremality of the trajectories. Order reduction
is defined as a factorization in the category of Lagrangian systems.
