  We present a new general method to construct an action functional for a
non-potential field theory. The key idea relies on representing the governing
equations of the theory relative to a diffeomorphic flow of curvilinear
coordinates which is assumed to be functionally dependent on the solution
field. Such flow, which will be called the conjugate flow of the theory,
evolves in space and time similarly to a physical fluid flow of classical
mechanics and it can be selected in order to symmetrize the Gateaux derivative
of the field equations with respect to suitable local bilinear forms. This is
equivalent to requiring that the governing equations of the field theory can be
derived from a principle of stationary action on a Lie group manifold. By using
a general operator framework, we obtain the determining equations of such
manifold and the corresponding conjugate flow action functional. In particular,
we study scalar and vector field theories governed by second-order nonlinear
partial differential equations. The identification of transformation groups
leaving the conjugate flow action functional invariant could lead to the
discovery of new conservation laws in fluid dynamics and other disciplines.
