Are the terms "$n$-body problem" and "many-body problem" synonymous? Or does one refer to a numerical problem an the other to an analytical problem?
closed as too broad by Norbert Schuch, stafusa, ZeroTheHero, Aaron Stevens, Cosmas Zachos Aug 14 at 14:17
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N-body problem refers to the problem of having any number objects with initial positions and velocity and predict their dynamics according to Newton's laws; for n = 2 there are analytical and practical solutions, and though it is proven that analytic solutions exists for n > 3 those are not practical and numerical methods are used instead.
Many-body problems refer to the family of problems pertaining a quantum system with an arbitrary number of particles.
The Mathematics Subject Classification (MSC2010) classifies "many-body" under quantum mechanics:
Whereas "$n$-body" appears to have more extension:
The OED doesn't have "$n$-body", but it says this for "many-body":
many-body adj. relating to or involving three or more bodies or particles; spec. with reference to the problem of predicting their positions and motions at any future time given their present values and the way the bodies interact (through collision, gravitational attraction, etc.).