Quantization of theories with non-Lagrangian equations of motion

We present an approach to the canonical quantization of systems with non-Lagrangian equations of motion. We first construct an action principle for an equivalent first-order equations of motion. A hamiltonization and canonical quantization of the constructed Lagrangian theory is a non-trivial problem, since this theory involves time-dependent constraints. We adopt the general approach of hamiltonization and canonical quantization for such theories (Gitman, Tyutin, 1990) to the case under consideration. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with the given set of equations. We give a complete description of this ambiguity. It is remarkable that the quantization scheme developed in the case under consideration provides arguments in favor of fixing this ambiguity. Finally, as an example, we consider the canonical quantization of a general quadratic theory.