The elliptic integral in the sky
Vladimir Bondarenko
"Rayleigh's explanation [1, 2] of the polarization of clear-sky
daylight,
in terms of dipole radiation singly scattered by air molecules, would
predict that light is unpolarized in the direction of the Sun or anti-
Sun
(anti-solar point), whichever is above the horizon. But observations
had
already demonstrated the existence of three unpolarized directions:
above the Sun (the Babinet point), below the Sun (the Brewster point
[3]) and above the anti-Sun (the Arago point), and hinted at a fourth
point below the anti-Sun (the second Brewster point [4])"
"The global pattern of polarization lines can be generated as
contours
of an elliptic integral"
http://www.sciencedaily.com/releases/2004/11/041123112414.htm
"This is beautiful mathematics in the sky. Using elliptic integrals,
we’ve been able to replace pages and pages of formulae with one
very simple solution that predicts the pattern extremely well"
clicl...@freenet.de
>From <http://www.sciencedaily.com/releases/2004/11/041123112414.htm> Vladimir must have liked this part in particular:
"Many scientists and mathematicians believe that simple, concise
explanations of natural phenomena are better or closer to some
underlying truth than more complex ones. Professor Marcus du Sautoy,
from the Mathematical Institute at the University of Oxford, said:
Having a sense of beauty and aesthetics is an important part of being
a scientist. Nature seems to be a believer in Occam's Razor: given a
choice between something messy or a beautiful solution, Nature
invariably goes for beauty. This is why those scientists with an eye
for aesthetics are often better equipped for discovering the way
Nature works. We might find a complicated ugly solution but that is
probably a sign that we haven't yet found the best explanation. The
fact that there is so much beauty at the heart of Nature is what gives
scientists a constant sense of wonder and excitement about their
subject."
-
Here's a down-to-earth elliptic integral from electrical engineering,
INT((SINH(u)*COSH(u)-SINH(u0)*SIN(v0)*COSH(u)*SIN(v)-COSH(u0)*CO~
S(v0)*SINH(u)*COS(v))/((SINH(u0)*SIN(v0)-SINH(u)*SIN(v))^2+(COSH~
(u0)*COS(v0)-COSH(u)*COS(v))^2)^(3/2),v,0,2*pi)
Can 21st-century Computer Algebra Systems reduce this to a formula of
a few hundred ASCII characters?
The science dates from the 19th century, compare: J. Viriamu Jones, On
the magnetic field due to an elliptical current at a point in its
plane within it, Proc. Phys. Soc. London 14 (1895), 250-254. In the
integral, u0 and v0 are the elliptical coordinates of the "point", and
u and v those of the "current". (I fear this title would have been
rejected by present-day referees.)
Martin.
Vladimir Bondarenko
M> Can 21st-century Computer Algebra Systems reduce this to
M> a formula of a few hundred ASCII characters?
Here you are.
Maple 12> int((sinh(c)*cosh(c)-sinh(a)*sin(b)*cosh(c)*sin(z)-
cosh(a)*cos(b)*sinh(c)*cos(z))/((sinh(a)*sin(b)-sinh(c)*sin(z)
)^2+(cosh(a)*cos(b)-cosh(c)*cos(z))^2)^(3/2),z=0..2*Pi);
Error, (in int/ellalg/mapf1) square free base has wrong form
clicl...@freenet.de
Vladimir Bondarenko schrieb:
... while Mathmatica has obediently swallowed and is now trying to
digest it ... ?
Martin.
Dave Rusin
<clicl...@freenet.de> wrote:
>INT((SINH(u)*COSH(u)-SINH(u0)*SIN(v0)*COSH(u)*SIN(v)-COSH(u0)*CO~
>S(v0)*SINH(u)*COS(v))/((SINH(u0)*SIN(v0)-SINH(u)*SIN(v))^2+(COSH~
>(u0)*COS(v0)-COSH(u)*COS(v))^2)^(3/2),v,0,2*pi)
>
>Can 21st-century Computer Algebra Systems reduce this to a formula of
>a few hundred ASCII characters?
Well, not quite. But I can avoid all the hyperbolic and trig functions
and rewrite it as the integral (over the whole real line) of
q(t)*(1+t^2)/( Q(t) )^(3/2)
where q(t) is a quadratic polynomial and Q(t) a quartic polynomial,
where (reflecting the three parameters u,u0,v0 in the original problem)
the coefficients of q and Q are written as polynomials in three variables.
So it's a completely algebraic problem, asking for an integral of a 1-form
q(t)(1+t^2) dt/u
around one whole loop of a Riemann surface. I know in principle how to
reduce this to a linear combination of a few basic integrals on the surface
(i.e. it's a linear combination of a few entries of the "period matrix"),
but I don't know how to refer to those in any computer-algebra language.
(The process is straightforward, replacing say ( COS(v0), SIN(v0) ) with
( (1-t^2)/(1+t^2), (2t)/(1+t^2) ) for suitable t, and using similar
formulae for the hyperbolic terms.)
I then took a sample integral of this family and asked Maple to do something
with it. It had no suggestions except to return the integral unevaluated.
dave
OwlHoot
> In article <82ff274c-4d16-4e19-8020-7db6fc10e...@p2g2000prn.googlegroups.com>,
I'm sure I've seen ancient books that reduce the integral of any
rational
function of t and SQRT(Q(t)) in your notation (where Q has degree at
most 4
and no repeated roots) to standard forms such as Legendre forms. There
must
be some standard algorithm for this.
Cheers
John Ramsden
clicl...@freenet.de
cliclic...@freenet.de schrieb:
> Vladimir Bondarenko schrieb:
> > On Dec 12, 9:58?pm, cliclic...@freenet.de wrote:
> > >
> > > Here's a down-to-earth elliptic integral from electrical engineering,
> > >
> > > INT((SINH(u)*COSH(u)-SINH(u0)*SIN(v0)*COSH(u)*SIN(v)-COSH(u0)*CO~
> > > S(v0)*SINH(u)*COS(v))/((SINH(u0)*SIN(v0)-SINH(u)*SIN(v))^2+(COSH~
> > > (u0)*COS(v0)-COSH(u)*COS(v))^2)^(3/2),v,0,2*pi)
> > >
> > > Can 21st-century Computer Algebra Systems reduce this to a formula of
> > > a few hundred ASCII characters?
> > >
> > > The science dates from the 19th century, compare: J. Viriamu Jones, On
> > > the magnetic field due to an elliptical current at a point in its
> > > plane within it, Proc. Phys. Soc. London 14 (1895), 250-254. In the
> > > integral, u0 and v0 are the elliptical coordinates of the "point", and
> > > u and v those of the "current". (I fear this title would have been
> > > rejected by present-day referees.)
> > >
> >
> > M> a formula of a few hundred ASCII characters?
> >
> > Here you are.
> >
> > Maple 12> int((sinh(c)*cosh(c)-sinh(a)*sin(b)*cosh(c)*sin(z)-
> > cosh(a)*cos(b)*sinh(c)*cos(z))/((sinh(a)*sin(b)-sinh(c)*sin(z)
> > )^2+(cosh(a)*cos(b)-cosh(c)*cos(z))^2)^(3/2),z=0..2*Pi);
> >
> > Error, (in int/ellalg/mapf1) square free base has wrong form
>
> ... while Mathmatica has obediently swallowed and is now trying to
> digest it ... ?
For the record (and perhaps to help Maple and Mathematica developers),
here is the formula of a few hundred ASCII characters:
2*SINH(u)*COSH(u)/(SINH(u)^2+SIN(v0)^2)*(CEL_K(2*((SINH(u0)^2+SI~
N(v0)^2)*(SINH(u)^2+SIN(v0)^2))^(1/4)/(SQRT(SINH(u0)^2+SIN(v0)^2~
)+SQRT(SINH(u)^2+SIN(v0)^2)))/(SQRT(SINH(u0)^2+SIN(v0)^2)+SQRT(S~
INH(u)^2+SIN(v0)^2))-CEL_E(2*((SINH(u0)^2+SIN(v0)^2)*(SINH(u)^2+~
SIN(v0)^2))^(1/4)/(SQRT(SINH(u0)^2+SIN(v0)^2)+SQRT(SINH(u)^2+SIN~
(v0)^2)))/(SQRT(SINH(u0)^2+SIN(v0)^2)-SQRT(SINH(u)^2+SIN(v0)^2)))
where
CEL_K(k):=INT(1/SQRT(1-k^2*SIN(t)^2),t,0,pi/2)
CEL_E(k):=INT(SQRT(1-k^2*SIN(t)^2),t,0,pi/2)
This formula holds for points inside (u0<u) as well as outside (u0>u)
the elliptical current. It can be simplified considerably if only
inside points are considered.
Martin.
Vladimir Bondarenko
> cliclic...@freenet.de schrieb:
>
>
>
> > Vladimir Bondarenko schrieb:
> > > On Dec 12, 9:58?pm, cliclic...@freenet.de wrote:
>
Elliptic integrals appear in many problems, in geometry,
mechanics, potential theory, ferromarnetism, cosmology...
Field theory? Here you are.
Vladimir Bondarenko
http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.2055v1.pdf