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Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing as $1/d^p$ for distance separation $d$ and some power $p$. Two questions:

  1. Presumably the 2-body problem still factors into two independent 1-body problems, results in planar motion, and can be solved. Have the orbits (the equivalents of elliptical and parabolic orbits for $p=2$) been worked out for other (perhaps specific) values of $p$?

  2. In some sense, the 3-body problem for $p=2$ cannot be solved. Most systems are choatic; see this interesting collection of Eugene Butikov. Only a few periodic solutions are known; see the nice article by Bill Casselman on the discovery of "choreographies." Is the situation simpler for other values of $p$? Perhaps $p=1$?

References and pointers would be appreciated. Thanks!

Edit. Thanks to Agol, Ken, and José. I've now looked at Arnolʹd's Huygens and Needham (but not yet Arnolʹd's Classical Mechanics). Indeed, as the commenters say, there is a remarkable 2-body result for $p=1$: the orbits for a linearly attractive force are ellipses, for a linearly repulsive force, hyperbolae. This depends on the Kasner-Arnolʹd theorem stating that for each power law, there is a dual power law that maps orbits of one to orbits of the other. Newton proved in Principia that elliptical orbits result if and only if the force is inverse-linear or inverse-square. The Kasner-Arnolʹd theorem explains why.

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@Agol: Thanks for the reference! I am not familiar with Arnolʹd's Huygens and Barrow, Newton and Hooke, but I see that my college library has it, so I can retrieve it later. –  Joseph O'Rourke Aug 18 '10 at 17:09
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There's more on the 2-body case in Needhams' Visual Complex Analysis Book, see pp. 241-247. There, he refers to Arnol'd's work, but also mentions that Arnol'd rediscovered the general result of E. Kasner in Kasner's 1913 work Differential-Geometric Aspects of Dynamics. –  Ken Fan Aug 18 '10 at 17:20
    
@Ken: Great! I'll check out Needham as well. Thanks! –  Joseph O'Rourke Aug 18 '10 at 17:36
    
If I remember correctly, for the 2-body problem, Arnold showed that the orbits are also ellipses (maybe for p=-1?) in the appendix of his book: books.google.com/… I don't have access to the book now, so I can't quite remember what he proved. (I reposted this because the link was wrong). –  Ian Agol Aug 18 '10 at 18:06
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5 Answers

The answers to question (1) for the 2 body problem are fine, and complete enough.

Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler than with $p \ne 3$. The added simplicity is due to the occurenc of an additional integral which comes out of the Lagrange Jacobi identity for the evolution of the total moment of inertia $I$. This identity asserts that $d^2 I/ dt^2 = 4 H + (4 - 2(p-1)) U$ where $H = K - U$ is the total energy, with $K$ the kinetic energy and $U$ the NEGATIVE of the total potential energy, a function which is homogeneous of degree $p-1$. When $p =3$ we get $d^2 I/ dt^2 = 4H = const.$!. (The total moment of inertia is a the squared norm relative to the ``mass inner product' and as such measures the total size of the system. )

For details on this Lagrange-Jacobi identity and its use see the first sections of my paper `Hyperbolic Pants fit a three-body problem', Ergodic Theory and Dynamical Systems, Volume 25, - June 2005, 921-947, which you can also find on my web site http://count.ucsc.edu/~rmont/papers/list.html or on the arXivs. Also see the references there.

For a study of choreographies with various $p$- potentials see the paper by Fujiwara et al. `Choreographic Three Bodies on the Lemniscate':J. Phys. A: Math. Gen. 36 (21 March 2003) 2791-2800, available on his web site ( or the ArXivs). http://www.clas.kitasato-u.ac.jp/~fujiwara/nBody/nbody.html

The discoverer of the figure eight, Cris Moore, in his beautiful 2 page paper `Braids and Classical Gravity' (which Casselman should have a ref. to) found numerically, and argues convincingly that as one increases $p$ more and more ``braid types'' (and hence choreographies) appear. It is known that all possible braid types (and so choreography types) occur as soon as $p =2$.

The cases $p \ge 2$ are often called ``strong-force potentials'' and from the variational perspective are much simpler than $p < 2$ for the reason that with the strong force potentials all collision paths have infinite action. This fact regarding action is surprising, since with the strong force condition in force it seems that almost all bounded solutions end in collision. This "seems" is a theorem for the 2-body problem, and for the negative energy three body problem when $p=2$.

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Very interesting (and counterintuitive to naïve me) that strong force potentials are simpler. Thanks for this rich collection of information! –  Joseph O'Rourke Sep 1 '10 at 20:08
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Not a complete answer, but there is a theorem (proved, for instance, in Arnold's Mathematical Methods in Classical Mechanics) that says that the only radial potentials for which all bound orbits are closed are the Kepler potential $\propto 1/r$ and the harmonic oscillator $\propto r^2$.

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Not really an answer but too long for a comment.

In addition to the cases already mentioned, there is a third one where the orbits can be described simply in analytical terms. This is where the force is proportional to $\frac 1{r^3}$---the curves are then Cotes spirals. These three cases were known to Newton.

Maclaurin then discovered a two parameter family of orbits for each power law. They are obtained from the Maclaurin spirals (as they are now known), i.e., the curves $r^n=\cos (n \theta)$---by dilation and rotation. If $n$ is a whole number, then these orbits are closed---this is not in contradition to Bertrand's theorem, of course.

The reason for this is that the functions $(\cos (d \theta))^{\frac 1 d}$ are such that $f+f''$ is proportional to a power of $f$.

This fact lies behind further remarkable properties of these curves (for example, their curvatures are proportional to a power of the distance from the origin and they are catenaries for suitable power laws (the index of the power law depends in a simple fashion on $d$).

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Regarding question 1):

a) In addition to the potentials for which all bound orbits are closed there are further central force potentials with analytic solutions, please cf.
http://en.wikipedia.org/wiki/Classical_central-force_problem#Central_forces_with_exact_solutions (Sorry, not quite what you were asking for [I read too fast], but perhaps interesting nevertheless.)

b) Perhaps this paper on "Multiparticle systems with a particle-interaction potential homogeneous of degree α = −2" by Borisov et al. is interesting for you: http://ics.org.ru/doc?pdf=1359&dir=e. ("Some new integrable and superintegrable systems generalizing the classical ones are also described.").

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"A central-force problem is said to be integrable if this integration [of the Binet equation] can be solved in terms of known functions." Great, thanks Andreas! –  Joseph O'Rourke Aug 18 '10 at 18:54
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A couple of Don Saari's papers from the 1970's era look at p = 2 + epsilon and show that there is no dramatic change in dynamics for a small perturbation of the exponent.

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