A179177 Integral exponents n for radial potentials V=a*r^(n+1) that lead to motions in terms of circular functions and Legendre Elliptic Integrals.

-7, -5, -4, 0, 3, 5

Offset

1,1

Comments

n = {-2, -3} correspond to inverse square and inverse cube force force laws. n = 1 also gives a solution in simple functions. n=-1, corresponding to a constant potential is excluded, as being no force; it is an equally anomalous case if the exponent -1 is used in the power law directly, as force varying as r^-1 corresponds to a logarithmic potential, not a power law, and typical of a line source rather than a point source. Thus a solution in terms of simple functions is obtained for n = {1, -2, -3}. This does not mean that other powers are not integrable, merely that they lead to functions not as well known. n = {-7,-5,-4,0,3,5} can be reduced to forms involving circular functions and Legendre elliptic integrals of the first, second, and third kind. See Goldstein. Although this exhausts the possibilities for integral exponents, with suitable transformations some fractional exponents can also be shown to lead to elliptic integrals.

References

Herbert Goldstein, Classical Mechanics, 2nd edn., Addison-Wesley, 1980, Chapter 3-5: "The Differential Equation for the Orbit, and Integrable Power-law Potentials", pp.85-90.

Links

Table of n, a(n) for n=1..6.

Crossrefs

Sequence in context: A160670 A145985 A109863 * A021574 A021061 A334399

Adjacent sequences: A179174 A179175 A179176 * A179178 A179179 A179180

Keyword

fini,full,sign,bref

Author

Jonathan Vos Post, Jan 05 2011

Status

approved