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 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 $$S$$ whose complete state at a time $$t$$ is described by an equation of the form

In this case, both $$\alpha$$ and $$n$$ can be taken as parameterizations of $$S(t)$$. There are a number of approximations we might consider. For instance, we might wonder what happens to $$S(t)$$ as $$\alpha$$ and $$n$$ both approach 0. This yields a prediction that is perfectly mathematically consistent; $$S(t)$$ approaches a real value as both those parameters approach 0. By Norton’s definition this is an approximation of $$S(t)$$, since we’re examining the system’s behavior in a particular limit case.

However, consider the difference between this approximation and the idealization of $$S$$ in which $$\alpha$$ = 0 and $$n$$ = 0. Despite the fact that the approximation yielded by considering the system’s behavior as $$\alpha$$ and $$n$$ 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

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