Action principle for so-called non-Lagrangian systems

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Eu ler-Lagrange equations for an action principle. The existence of an action principle for a given p hysical system, or what is the same, the existence of a Lagrange function for such a system, allows one to proceed with canonical quantization schemes. This, in particular, emphasizes the importance of formulating an action principle for any physical system. From simple consideration, we find n ecessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order equations with the structure of Euler-Lagrange equations. An explicit form of the Lagrangian is constructed for a system which admits the existence of such a multiplier. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange functi on with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.