Physics Asked on July 17, 2021
Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for certain situations, where many particles are present and where we are interested only in bulk motion. In situations like modeling a string, we see that the continuum limes leads to a very good approximation for the motion of pointlike masses connected by springs under certain conditions. This seems to point in the direction that continuum mechanics is an approximation to point particle mechanics.
But point particles should also be easy to accommodate as a special case in continuum mechanics by admitting delta distributions and the like. Taking this viewpoint, continuum mechanics seems to be a generalization of point particle mechanics.
Therefore my question: Is continuum mechanics a generalization or an approximation to point particle mechanics? Or can it be argued that both are equally valid starting points?
Part of my motivation to ask this question is that I sometimes have difficulties to connect concepts from one viewpoint to the other (see e.g. my question about the point of application of a force).
My guess is that point particles were initially introduced as approximations to avoid the effect of mass distribution in inertial and gravity phenomena description. Spring - mass systems usually only offer a very crude model of solids, although they can sometimes be used as a first approach (e g. phonons in crystals). You can always isolate a finite volume within a medium and apply the equations of mechanics, the difference is that you need to take into account internal efforts acting on its boundary from the surrounding part of the medium (see the central notion of stress). And no, continnuum mechanics is certainly not only concerned with bulk motion, think of the entire field of elasticity!
Answered by user8736288 on July 17, 2021
It's neither a generalization of, nor approximation to, the classical mechanics of particles, it's simply a different perspective. As described by Walter Noll in discussing the connection in The foundations of classical mechanics in the light of recent advances in continuum mechanics (1959):
It is true that the mechanics of systems of a finite number of mass points has been on a sufficiently rigorous basis since Newton. Many textbooks on theoretical mechanics dismiss continuous bodies with the remark that they can be regarded as the limiting case of a particle system with an increasing number of particles. They cannot. The erroneous belief that they can had the unfortunate effect that no serious attempt was made for a long period to put classical continuum mechanics on a rigorous axiomatic basis.
Worth a read if you have the time.
Answered by A rural reader on July 17, 2021
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