Historic Examples of Multiscale Modeling

Newton and the Spherical Earth. One of the first quantitative effective theories that I can think of is that associated with Newton and the invention of the integral calculus. In particular, the key question that had to be faced was whether or not it was possible to pretend as though the entire mass of the earth is concentrated at its center, rendering it for gravitational purposes, nothing more than a point mass. Newton s argument was schematized pictorially as in fig. 12.1 and asserted as Proposition LXXVI, Theorem XXXVI of the famed Principia Mathematica (Motte 1934). 

If spheres be however dissimilar (as to density of matter and attractive force) in the same ratio onwards from the centre to the circumference but everywhere similar, at every given distance from the centre, on all sides round about and the attractive force of every point decreases as the square of the distance of the body attracted I say, that the whole force with which one of these spheres attracts the other will be inversely proportional to the square of the distance of the centres. 

The multiscale challenge arises from the recognition that the total gravitational potential associated with an object at a position r relative to the center of the earth 

For the purposes of the present discussion, this is the key result. What we have learned is that in the context of problems in which we interest ourselves in the gravitational influence of the earth at a distance r from the earth s center, the earth may be replaced conceptually by a point mass at the earth s center. This example satisfies one of the key criteria we have given as exemplifying multiscale modeling there is a massive reduction in the number of degrees of freedom, with the further clarification of physical insight. 

Bernoulli and the Pressure in a Gas. One of the elementary calculations undertaken in trying to effect the linkage between observed macroscopic behavior (such 

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Historic Examples of Multiscale Modeling The treatment of discrete and continuous representations of the vibrating string in Mathematical Thought from Ancient to Modern Times by Morris Kline, Oxford University Press, New York New York 1972, is enlightening. Here it is evident that both the continuous and discrete representations had features that recommended them as the basis for further study. 

Nuclei and Below. We conclude our series of historical examples of effective theories and multiscale modeling with a brief mention of the hierarchy of theories which present themselves in considering the subatomic world. Our first remark applies to most of the examples we have presented and it is to note that one man s microscopies is another s phenomenology. Nowhere is this more true than with the history of our exploration of the subatomic world and, in particular, the series of theories that have been set forth to explain what has been observed. The key point is that at each level of structure, the subscale structures can be ignored. For example. 

Hydrodynamics. Our historical interlude has made reference to effective theories of both gases (kinetic theory) and solids (elasticity), and now we take up yet a third example of enormous importance to modeling the natural world in general, and which serves as an example of the type of multiscale efforts of interest here, namely, the study of fluids. 

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