# $n$-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

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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

## 2 Answers

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

81Uxx Scattering theory [See also 34A55, 34L25, 34L40, 35P25, 47A40] 8728

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":

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.).