original system inexactly in some sense.
It’s worth pointing out that Norton’s two definitions will, at least sometimes, exist on a continuum with one another: in some cases, approximations can be smoothly transformed into idealizations.
This interconversion is possible, for instance, in cases where the limits used in constructing idealized parameterizations are “well-behaved” in the sense that the exclusive use of limit quantities in the construction of the idealized system still results in a physically realizable system. This will not always be the case. For example, consider some system whose complete state at a time is described by an equation of the form
In this case, both and can be taken as parameterizations of . There are a number of approximations we might consider. For instance, we might wonder what happens to as and both approach 0. This yields a prediction that is perfectly mathematically consistent; approaches a real value as both those parameters approach 0. By Norton’s definition this is an approximation of , since we’re examining the system’s behavior in a particular limit case.
However, consider the difference between this approximation and the idealization of in which = 0 and = 0. Despite the fact that the approximation yielded by considering the system’s behavior as and both approach 0 is perfectly comprehensible (and hopefully informative as well), actually setting those two values to 0 yields a function value that’s undefined. The limits involved in the creation of the approximation are not “well behaved” in
- Norton (2012), p. 212