# $n$-body problem = many-body problem? [closed]

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 ZachosAug 14 at 14:17

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• It depends on the context, I would say. – Norbert Schuch Aug 6 at 18:43
• Whenever you hear a term like "n-body problem", it must come with a definition of the included variables. If "n" is not defined along with the term, this term essentially is meaningless. – Steeven Aug 6 at 19:11
• @Steeven: Maybe semantics, but I tend to (respectfully) disagree: In astrophysics you use the term $N$-body codes as a generic term for particle simulations that do not involve hydrodynamics. A given $N$-body simulation will of course contain a fixed (or variable) number of particles, which in general will vary from run to run. You could thus talk about "solving a problem of structure formation as an $N$-body problem", without specifying any number. – pela Aug 6 at 19:39
• In the cold atoms community, people also study the few-body problem (e.g. Efimov physics, 3-body bound states, etc). – Adam Aug 6 at 20:25
• @my2cts What's the [terminology] tag for, then? – Geremia Aug 6 at 20:50

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.

• Not sure where this terminology is defined, but I would agree based on my experience. Astronomers that I know always referred to n-body problems of celestial mechanics. Colleagues that were solid state and condensed matter physicists worked on many-body problems. – amateurAstro Aug 6 at 22:08
• What about the many-body problem in quantum physics? – Norbert Schuch Aug 7 at 9:17

The Mathematics Subject Classification (MSC2010) classifies "many-body" under quantum mechanics:

Whereas "$$n$$-body" appears to have more extension:

70-XX Mechanics of particles and systems [For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-XX] 62456

APS's Physics Subject Headings doesn't appear to have "$$n$$-body," nor does AAS's Astronomical Subject Keywords even have "many-body".
The OED doesn't have "$$n$$-body", but it says this for "many-body":