ANALYTICAL SOLUTION OF THEN PARTICLE GRAVITATIONAL PROBLEM:
STOKES' THEOREM AND ORBITAL CIRCULATION.
by
M. W. Evans and H. Eckardt,
Civil List and AlAS,
(wv1w.aias. us. www. webarchive.org. uk, www.atomicprecision.com, www. upitec.org,
www.et3m.net)
ABSTRACT
The N particle gravitational problem is solved analytically in the pairwise
additive approximation. The solution is illustrated for three interacting particles, and Stokes'
Theorem used to introduce the concept of orbital circulation. It is shown numerically that the
orbital circulation is non-zero for all conical sections and precessing conical sections with the
exception of circular orbits. The orbital circulation may therefore be used to characterize all
orbits known in cosmology.
Keywords: Classical limit ofECE theory, solution ofthe N particle gravitational problem,
orbital circulation.
-I
~
1. INTRODUCTION
Recently in this series of papers { 1 - 10} on the applications of ECE theory the
Einsteinian general relativity (EGR) has been refuted in many ways and replaced by a
relativity based on the constrained Minkowski metric. In the classical limit of this new
relativity it has been shown that all the features of planetary precession can be explained
straightforwardly with the equation of the precessing conical section, and a myriad of new
properties discovered in terms of the precession constant x. All known orbits in cosmology
have been shown to be explicable in terms ofx and the equation ofthe precessing conical
section. In immediately preceding papers the investigation of the classical limit ofthe new
relativity has been extended to the multi particle gravitational problem, thought to have no
known analytical solution, and in the preceding paper a new form of Kepler's third law
in!Crred for a precessing orbit.
In Section 2 of this paper it is shown that the N particle gravitational problem
can be solved analytically in a relatively straightforward way given the usual form ofthe
st~In ing
lagrangian for the problem. This is a four hundred year old problem in cosmology,
and up to now it has been thought to have only specialized solutions discovered by Euler,
/
Lagrange, Poincare and others. Stokes' Theorem is used to develop the solution for three
intnacting particles. and a new concept developed of orbital swirl or circulation. In Section 3
some of the results are illustrated graphically.
;
SOLUTION AND CONCEPT OF ORBITAL CIRCULATION
Consider three masses interacting simultaneously with the following lagrangian
This is the standard format of the lagrangian for what is known as the three particle
~~r<t\
iwtional problem in Newtonian dynamics. Herem are the masses, G is Newton's
constant, and the coordinates are defined in Fig. ( 1).
The centre of mass of the three particle system is defined to be at the origin of the coordinate
S\
stem. So:
fh,i,.\ 1\-;in Fig. (1 ):
~")
(2,
(2_)
-
!.t
-
fl-> Therefore the lagrangian is:
-()
(,
{)
-
+
V)\.li.J,.
-'j
(~
-
-
'3
-::.
-0 - (~)
- ~)
(~)
{s)
From Eqs. ( ')_ ) and (
l'h.,
~
£,
0
-h)
-\-1'\q. ( ~- ~) +11-.> ~
):
-
~ ~-t, ~-~
1.<..'.
+-
t'h l
.- (~)
th..).
Si m i I arl -v .·
I. C.:
Therefore:
-\- n-..~
(?
t
( ~, -\-~")))
,;, · ing [q:;. (
\\
-•-
;) Y)., ,__!. (2,
) and ( ·\)) cives
0 "
the reduced lagrangian:
(_!,
where the reduced mass is:
.
1!' •Line poL1r coordinates:
•
\~\ \
~~LR-(8\
'1
,, \
)
-y--
Nmv consider the Euler Lagrange equations:
"'
~
--tAt
--
}1
--.--d~\
_(n)
\3 ). Ths solution of Eqs. ( \t )and ( \1 ) is well known { 11} to
be the elliptical orbit:
''here
J\
is the semi right latitude and
f- \ is the eccentricity. Similarly:
- (t~)
-I
and
{<.3
~
- r:L~
,-
'-\ E-3 c~
"here:
R)
....
--
(2,
~(~J
--
e.!
-
Rl
..\-
())
This is the same result as in previous work { 1 - 10} and obtained with a different method.
Therefore it follows that the general solution ofthe N particle problem is:
\
gi \en the lagrangian (
( ). \
:.
LS )extended to N particles. For N masses in a plane the constraint
) is extended toN masses. For precessing orbits [ 1 - 10 J the general solution is
)
''here x . is the precession factor for each orbit. Therefore theN particle gravitational
\
J'l.,lt~km
hds been solved for the first time.
Straightforward application of Stokes' Theorem gives:
•
''here S .
is the orbital circulation:
s.
r
-t
•
-
(.
-
From Eq. (
-
--
..->,
)')
) in the Newtonian limit:
del_ dei
-
su:
s~
-
f\
R~
_s,·h
,1rhit;1l ..:irculation is a
lle\\
- (Jt)
cJ_l
t-~ s;~ 8;
\t fl
.,
9i
-----9;
-
(~)
{o.s
concept in cosmology and some examples are graphed in
Section 3. For precessing orbits:
i1· ,,J:ich the total angular momentum is the constant:
L
In genera I for any curve in a plane { 11}:
"I
1\ssuming:
d~JL) J}} .-
For precessing orbits:
( l-\- f-.:c thrcL'
!',lrticlc problem:
-
(~ f-
(o
S (
- (
:>c,ff.)) ~
&) . 4f2:t_
-(~
19
~~)
.
~\
t!1c circuldtion vectors are additive as follows:
-
-
pm\ iding the constraint on the solution (
;}l ):
f., s<,_e,
\ +(:I
\\here the eccentricities are:
and \vhere
r( \
(t15
f} (
~-
'
are the ti mcs taken for each orbit to transcribe 2'('f' radians.
The orbital circulation seems to be a new concept of cosmology, and may be used to
chdracterize every orbit. It is non-zero for every orbit in general. with the exception of a
circular orbit. For a circular orbit:
-
f ~. J~
-I
In the case oftl1e eire
. Ie:
e ~·
Q{".1
~
1 . . Rs.-" e, - (~>)
-(~)
- (4()
- - Rs,.1.G OJ1
(2_
(Of
& t(j
1., '"( x.~x. *I d0~ ~ t•s8 {- Rs.~-e)w
therefore:
'l1(
l
T~1c
0
cont
-·
·
-
O
f2s;"O {R
c~)
r~~P.o
of a c·Ire Ie IS
. zero. The curl is:
om mtegrai around the circmnterence
._.
'5/_
"( 1
y._ (2
"
)!_ - JX ~J~
~o
Stoke-.;·
, .
.IS \erilied . In cv.
. Tl lt..:OICill
.
. lmdncai
pohr
coo I.c11nates:
'
.j'
(2
- (2 R(
:
-
t
~,·circle
is delined b'- th,c conical
. .
section eq uat'1011:
-\
(2_
J(2
dO
i .- {s:)l
/
~
\ + .f
--
R-
-I
~
(oJ e
- (s>)
\\ i th:
--
t-
0
--(s~
So:
.df2
dlh.~
0
-
-
(59
de
t hL'I"C fore
qxR- --
-
''
~-
----
0
l)
(]RAPHICAL ILLUST RAIIONS
r~
Section by 0 r. H orst Eckardt
_(~~
-1
I<
;\CKNOWLEDGMENTS
The British Government is thanked for a Civil List Pension and the staff of AlAS
Ill\
interesting discussions. Dave Burleigh is thanked for posting, and Alex Hill. Robert
Cheshire and Simon Clifford for translation and broadcasting. The AlAS was established in
Spring 2012 under the aegis of the New lands Family Trust.
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r 1:
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: .~: ;'v1.
Fi~..:ld
:-+:
SL'\
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: (J:
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_, 1 :
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~
': :! : \1. \\·. hans and 1\. A. 1-Jasanein, "The Photomagneton in Quantum Field Theory"
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