Invariance Equation in Differential Form

Abstract

Definition of invariance in terms of motions and trajectories assumes, at least, existence and uniqueness theorems for solutions of the original dynamical system. This prerequisite causes difficulties when one studies equations relevant to physical and chemical kinetics, such as, for example, equations of hydrodynamics. Nevertheless, there exists a necessary differential condition of invariance: The vector field of the original dynamic system touches the manifold at every point. Let us write down this condition in order to set up the notation.