This article discusses the side condition approach as set down in the work by P. Olver and P. Rosenau [Phys. Lett., A 114, No. 3, 107–112 (1986; Zbl 0937.35501)] for the class of ordinary differential equations. Certain important notions are discussed for clarity and necessary and sufficient criteria for admissibility of the side conditions are derived. These are useful when employing a trial-and-error approach, however a trial-and-error side condition ansatz will not in general yield any invariant sets. Thus, to make the general side condition heuristics more useable appropriate restrictions would need to be imposed. This then, is the focus of the article: to further specify a priori restrictions or specializations.
An example considered provides the motivation for these restrictions: the first restriction is motivated by a theorem of LaSalle on limit sets of dynamical systems – {LaSalle type} side condition. In this instance, the invariant sets these define are characterized in a more concise manner. Furthermore, the classical side conditions induced by local transformation groups are reviewed and generalised; two-dimensional systems are considered.
In the final section, it is shown that side conditions appear naturally in the context of some applied problems: quasi-steady state (QSS) in chemistry and biochemistry. Examples are considered which show that the side condition approach provides a conceptually straightforward and computationally feasible way to determine QSS conditions for prescribed variables. The applications considered are the Michaelis-Menten and Lindemann-Hinsley system.