appeared as a chapter in his famous work Philosophiae Naturalis. Principia Mathematica . ... 0∆ where the acceleration A0 and velocity V0 is assumed constant in a small time interval ∆t t t. = － 0 . Further R0 gives the position at time t0 . Finally. R and V ... division by zero when trying to calculate the second term in this case.
The N-body Problem.
Jørgen Franck, Danish Amateur Rocket Club
During the great plague in 1666 the English universities closed. Isaac
Newton was then an undergraduate in his early twenties at Cambridge. He
returned to his mum’s farm in Woolsthorpe, Lincolnshire to remain out of
danger. Here he made an outstanding contribution to the gravitational
interaction between two bodies, either planets or small particles. Using
Kepler’s laws, he derived the law of universal gravitation. The law may be
stated as follows.
Every body in the universe attracts every other body by a force
proportional to the product of their masses and inversely proportional to
the square of the distance between them.
Mathematically the law of universal gravitation may be expressed in the
following manner. If m1
and m2 are the masses of two bodies separated by
a distance r between the bodies centers, the force of attraction F is
F G
m m
r
=
1 2
2
The constant G is called the constant of gravitation and its value is
G
Nm
kg = ´
-
6 67 10 11
2
2
.
Actually Newton postponed the publication of the discovery in nearly
twenty years, because he had difficulties in solving a particular problem in
integral calculus, which was conclusive for the whole theory. Strictly
Newton’s law of universal gravitation only apply for point masses, but
planets have finite size which may introduce some geometrical factor. This
problem teased Newton, but finally he could prove that a solid
homogeneous sphere produces a gravitational field identical to those of a
particle of the same mass located in the center of the sphere. Actually the
result still holds true when the sphere has its mass distributed with
spherical symmetry. Finally the discovery were published in 1687, when it
appeared as a chapter in his famous work Philosophiae Naturalis
Principia Mathematica .
Jørgen Franck The N-body Problem page 2 of 4
Newton’s law of universal gravitation is the cornerstone in the n-body
problem, which may be stated in the following manner.
Consider n bodies distributed in space. Assume that all masses, positions
and velocities are known and all the bodies experiences a gravitational
attraction according to Newton’s law of universal gravitation. What are
the position and velocity of each body at any time?
Newton solved the problem completely for n = 2. The complexity of the
problem grows considerably for n = 3 and is the famous three-body
problem. Since the eighteenth century the three-body problem is
considered as one of the most difficult to solve in mathematics. For the
solution of the three-body problem King Oscar II of Sweden offered a
prize and a gold medal. Poincaré got this price in 1889 for his discussion
of the problem, but up to date nobody have solved the n-body problem for
n equal or bigger than three.
Now we will use a computer and an algorithm to solve the n-body problem
numerical. This illustrates how a complex mathematical problem can be
expressed by an algorithm in a computer. Further the numerical solution
can be obtained, due to the computers capability to perform repeated
execution of an algorithm with very small time intervals. We use the
constant acceleration approximation expressed in vector form as
R = R +V t + A t
0 0 0
2 D ½ D
V =V + A t
0 0D
where the acceleration A0
and velocity V0
is assumed constant in a small
time interval Dt = t - t
0
. Further R0
gives the position at time t
0
. Finally
R and V are respectively the position and velocity at time t . In the
following we will use the notation
A(J,I) : Acceleration of the J’th planet in the I’th dimension.
V( J,I) : Velocity of the J’th planet in the I’th dimension.
R( J,I) : Position of the J’th planet in the I’th dimension.
The above quantities are measured relative to a well-defined frame of
reference. The form of Newton’s law of universal gravitation is not very
usefull when the planets motion are described relative to a frame of
Jørgen Franck The N-body Problem page 3 of 4
reference, since the force of attraction is expressed as a function of
distance rather than coordinates. We will rewrite Newton’s law of
universal gravitation to express the acceleration acting on the J’th planet in
the I’th direction, we get
A J I A J I
G M K R K I R J I
K J D K J
N
0
0 0
( , ) ( , )
( ) ( ( , ) ( , ))
( , )
= +
* * -
¹
å
where
D K J D J K R K I R J I
I
( , ) = ( , ) = å ( ( , ) - ( , )) 0 0
3
The first term A(J,I) is the acceleration the planet got from the previous
calculation and the second term is the acceleration increment due to the
other planets. We can of course not include the term K = J in the sum,
because the actual planet does not influence on itself. Further we get
division by zero when trying to calculate the second term in this case.
Below is the algorithm implemented in the BASIC programming language.
The input and output section is omitted. Further the algorithm only solve
the n-body problem in two dimensions.
10 REM N-BODY PROBLEM
20 REM INPUT
.
.
.
230 REM CALCULATION
240 FOR J = 1 TO N
250 FOR K = 1 TO J - 1
260 D = (R(K,1) - R(J,1)) ^ 2 + (R(K,2) - R(J,2)) ^ 2
270 D(J,K) = D * SQR(D)
280 D(K,J) = D(J,K)
290 NEXT K
300 NEXT J
310 FOR J = 1 TO N
320 FOR I = 1 TO 2
330 FOR K = 1 TO N
340 IF K = J THEN GOTO 360
350 A(J,I) = A(J,I) + G * M(K) * (R(K,I) - R(J,I)) / D(K,J)
360 NEXT K
Jørgen Franck The N-body Problem page 4 of 4
370 R(J,I) = R(J,I) + V(J,I) * T0 + 0.5 * A(J,I) * T0 * T0
380 V(J,I) = V(J,I) + A(J,I) * T0
390 NETX I
400 NEXT J
410 T = T + T0
420 REM OUTPUT
.
.
.
530 GOTO 240
The input section of the program first read the time interval T0 and the
number of planets N. Then the individual planets masses, coordinates and
velocities are read. Last the entered planet positions are plotted on the
computer screen. Remember to define the constant of gravitation G and set
the duration T = 0.
The calculation section first determine the D matrix for all the planets (line
no. 240-300). When this is done, the acceleration, coordinates and
velocities are determined for the N planets (line no. 310-400). The
duration T is calculated by adding the time interval Dt (line no. 410).
The output section plot the planets new positions on the computer screen.
Finally the program jump back (line no. 530) and calculates the D matrix
again based on the new coordinates. Hereby the program run in an infinite
loop, calculating new accelerations, velocities and coordinates for all the
planets for a given time interval Dt .
The program solve the n-body problem for two dimensions, because it then
is easy to plot the planet positions on the computer screen. But it is easy to
expand the calculation to three dimensions. Add (R(K,3) - R(J,3)) ^ 2 in
line no. 260 and finally change 2 to 3 in line no. 320.
If the distance between the calculated planet position increase on the
computer screen the velocity of the planet increase too, because the time
interval Dt between the positions are constant.
The algorithm does not solve the n-body problem very accurate, due to the
simple numerical method used.http://dark.dk/documents/technical_notes/Nbody1.pdf
再生核研究所声明297(2016.05.19) 豊かなゼロ、空の世界、隠れた未知の世界
ゼロ除算の研究を進めているが、微分方程式のある項を落とした場合の解と落とす前の解を結び付ける具体的な方法として、ゼロ除算の解析の具体的な応用がある事が分かった。この事実は、広く世の現象として、面白い視点に気づかせたので、普遍的な現象として、生きた形で表現したい。
ある項を落とした微分方程式とは、逆に言えば、与えられた微分方程式はさらに 複雑な微分方程式において、沢山の項を落として考えられた簡略の微分方程式であると考えられる。どのくらいの項を落としたかと考えれば、限りない項が存在して、殆どがゼロとして消された微分方程式であると見なせる。この意味で、ゼロの世界は限りなく広がっていると考えられる。
消された見えない世界は ゼロの世界、空、ある隠された世界として、無限に存在していると考えられる。たまたま、現れた項が 表現する物理現象を記述していると言える。
これは、地球に繁茂する動植物が、大海や大地から、生まれては、それらに回帰する現象と同様と言える。大量に発生した卵の極一部がそれぞれの生物に成長して、やがて元の世界に戻り、豊かな大海や大地は生命の存在の元、隠れた存在の大いなる世界であると考えられる。無数の生命の発生と回帰した世界の様は 生物、生体の様の変化は捉えられても、人間の精神活用や生命の生命活動の様の精しい様などは 殆ど何も分からない存在であると言える。我々の認知した世界と発生して来た世界と消えて行った認知できない世界である。
このような視点で、人間にとって最も大事なことは 何だろうか。それは、個々の人間も、人類も 大きな存在の中の小さな存在であることを先ず自覚して、背後に存在する大いなる基礎、環境に畏敬の念を抱き、謙虚さを保つことではないだろうか。この視点では日本古来の神道の精神こそ、宗教の原点として大事では ないだろうか。未知なる自然に対する畏敬の念である。実際、日本でも、世界各地でも人工物を建設するとき、神事を行い、神の許しを求めてきたものである。その心は大いなる存在と人間の調和を志向する意味で人間存在の原理ではないだろうか。それはそもそも 原罪の概念そのものであると言える。
しかしながら、人類が好きなように生きたいも道理であり、巨大都市を建設して、環境を汚染して生存を享受したいも道理であるから、それらの一面も否定できず、それは結局全体的な有り様の中でのバランスの問題ではないだろうか。人類の進化の面には必然的に人類絶滅の要素が内在していると考えられる：
再生核研究所声明 144（2013.12.12） 人類滅亡の概念 － 進化とは 滅亡への過程である
そこで、結局は全体的な調和、バランスの問題である：
再生核研究所声明 56： アースデイ の理念
発想における最も大事なことに触れたが、表現したかった元を回想したい。― それは存在と非存在の間の微妙な有り様と非存在の認知できない限りない世界に想いを致す心情そのものであった。無数とも言える人間の想いはどこに消えて行ったのだろうか。先も分からず、由来も分からない。世の中は雲のような存在であると言える。
以 上
The N-body Problem.
Jørgen Franck, Danish Amateur Rocket Club
During the great plague in 1666 the English universities closed. Isaac
Newton was then an undergraduate in his early twenties at Cambridge. He
returned to his mum’s farm in Woolsthorpe, Lincolnshire to remain out of
danger. Here he made an outstanding contribution to the gravitational
interaction between two bodies, either planets or small particles. Using
Kepler’s laws, he derived the law of universal gravitation. The law may be
stated as follows.
Every body in the universe attracts every other body by a force
proportional to the product of their masses and inversely proportional to
the square of the distance between them.
Mathematically the law of universal gravitation may be expressed in the
following manner. If m1
and m2 are the masses of two bodies separated by
a distance r between the bodies centers, the force of attraction F is
F G
m m
r
=
1 2
2
The constant G is called the constant of gravitation and its value is
G
Nm
kg = ´
-
6 67 10 11
2
2
.
Actually Newton postponed the publication of the discovery in nearly
twenty years, because he had difficulties in solving a particular problem in
integral calculus, which was conclusive for the whole theory. Strictly
Newton’s law of universal gravitation only apply for point masses, but
planets have finite size which may introduce some geometrical factor. This
problem teased Newton, but finally he could prove that a solid
homogeneous sphere produces a gravitational field identical to those of a
particle of the same mass located in the center of the sphere. Actually the
result still holds true when the sphere has its mass distributed with
spherical symmetry. Finally the discovery were published in 1687, when it
appeared as a chapter in his famous work Philosophiae Naturalis
Principia Mathematica .
Jørgen Franck The N-body Problem page 2 of 4
Newton’s law of universal gravitation is the cornerstone in the n-body
problem, which may be stated in the following manner.
Consider n bodies distributed in space. Assume that all masses, positions
and velocities are known and all the bodies experiences a gravitational
attraction according to Newton’s law of universal gravitation. What are
the position and velocity of each body at any time?
Newton solved the problem completely for n = 2. The complexity of the
problem grows considerably for n = 3 and is the famous three-body
problem. Since the eighteenth century the three-body problem is
considered as one of the most difficult to solve in mathematics. For the
solution of the three-body problem King Oscar II of Sweden offered a
prize and a gold medal. Poincaré got this price in 1889 for his discussion
of the problem, but up to date nobody have solved the n-body problem for
n equal or bigger than three.
Now we will use a computer and an algorithm to solve the n-body problem
numerical. This illustrates how a complex mathematical problem can be
expressed by an algorithm in a computer. Further the numerical solution
can be obtained, due to the computers capability to perform repeated
execution of an algorithm with very small time intervals. We use the
constant acceleration approximation expressed in vector form as
R = R +V t + A t
0 0 0
2 D ½ D
V =V + A t
0 0D
where the acceleration A0
and velocity V0
is assumed constant in a small
time interval Dt = t - t
0
. Further R0
gives the position at time t
0
. Finally
R and V are respectively the position and velocity at time t . In the
following we will use the notation
A(J,I) : Acceleration of the J’th planet in the I’th dimension.
V( J,I) : Velocity of the J’th planet in the I’th dimension.
R( J,I) : Position of the J’th planet in the I’th dimension.
The above quantities are measured relative to a well-defined frame of
reference. The form of Newton’s law of universal gravitation is not very
usefull when the planets motion are described relative to a frame of
Jørgen Franck The N-body Problem page 3 of 4
reference, since the force of attraction is expressed as a function of
distance rather than coordinates. We will rewrite Newton’s law of
universal gravitation to express the acceleration acting on the J’th planet in
the I’th direction, we get
A J I A J I
G M K R K I R J I
K J D K J
N
0
0 0
( , ) ( , )
( ) ( ( , ) ( , ))
( , )
= +
* * -
¹
å
where
D K J D J K R K I R J I
I
( , ) = ( , ) = å ( ( , ) - ( , )) 0 0
3
The first term A(J,I) is the acceleration the planet got from the previous
calculation and the second term is the acceleration increment due to the
other planets. We can of course not include the term K = J in the sum,
because the actual planet does not influence on itself. Further we get
division by zero when trying to calculate the second term in this case.
Below is the algorithm implemented in the BASIC programming language.
The input and output section is omitted. Further the algorithm only solve
the n-body problem in two dimensions.
10 REM N-BODY PROBLEM
20 REM INPUT
.
.
.
230 REM CALCULATION
240 FOR J = 1 TO N
250 FOR K = 1 TO J - 1
260 D = (R(K,1) - R(J,1)) ^ 2 + (R(K,2) - R(J,2)) ^ 2
270 D(J,K) = D * SQR(D)
280 D(K,J) = D(J,K)
290 NEXT K
300 NEXT J
310 FOR J = 1 TO N
320 FOR I = 1 TO 2
330 FOR K = 1 TO N
340 IF K = J THEN GOTO 360
350 A(J,I) = A(J,I) + G * M(K) * (R(K,I) - R(J,I)) / D(K,J)
360 NEXT K
Jørgen Franck The N-body Problem page 4 of 4
370 R(J,I) = R(J,I) + V(J,I) * T0 + 0.5 * A(J,I) * T0 * T0
380 V(J,I) = V(J,I) + A(J,I) * T0
390 NETX I
400 NEXT J
410 T = T + T0
420 REM OUTPUT
.
.
.
530 GOTO 240
The input section of the program first read the time interval T0 and the
number of planets N. Then the individual planets masses, coordinates and
velocities are read. Last the entered planet positions are plotted on the
computer screen. Remember to define the constant of gravitation G and set
the duration T = 0.
The calculation section first determine the D matrix for all the planets (line
no. 240-300). When this is done, the acceleration, coordinates and
velocities are determined for the N planets (line no. 310-400). The
duration T is calculated by adding the time interval Dt (line no. 410).
The output section plot the planets new positions on the computer screen.
Finally the program jump back (line no. 530) and calculates the D matrix
again based on the new coordinates. Hereby the program run in an infinite
loop, calculating new accelerations, velocities and coordinates for all the
planets for a given time interval Dt .
The program solve the n-body problem for two dimensions, because it then
is easy to plot the planet positions on the computer screen. But it is easy to
expand the calculation to three dimensions. Add (R(K,3) - R(J,3)) ^ 2 in
line no. 260 and finally change 2 to 3 in line no. 320.
If the distance between the calculated planet position increase on the
computer screen the velocity of the planet increase too, because the time
interval Dt between the positions are constant.
The algorithm does not solve the n-body problem very accurate, due to the
simple numerical method used.http://dark.dk/documents/technical_notes/Nbody1.pdf
再生核研究所声明297(2016.05.19) 豊かなゼロ、空の世界、隠れた未知の世界
ゼロ除算の研究を進めているが、微分方程式のある項を落とした場合の解と落とす前の解を結び付ける具体的な方法として、ゼロ除算の解析の具体的な応用がある事が分かった。この事実は、広く世の現象として、面白い視点に気づかせたので、普遍的な現象として、生きた形で表現したい。
ある項を落とした微分方程式とは、逆に言えば、与えられた微分方程式はさらに 複雑な微分方程式において、沢山の項を落として考えられた簡略の微分方程式であると考えられる。どのくらいの項を落としたかと考えれば、限りない項が存在して、殆どがゼロとして消された微分方程式であると見なせる。この意味で、ゼロの世界は限りなく広がっていると考えられる。
消された見えない世界は ゼロの世界、空、ある隠された世界として、無限に存在していると考えられる。たまたま、現れた項が 表現する物理現象を記述していると言える。
これは、地球に繁茂する動植物が、大海や大地から、生まれては、それらに回帰する現象と同様と言える。大量に発生した卵の極一部がそれぞれの生物に成長して、やがて元の世界に戻り、豊かな大海や大地は生命の存在の元、隠れた存在の大いなる世界であると考えられる。無数の生命の発生と回帰した世界の様は 生物、生体の様の変化は捉えられても、人間の精神活用や生命の生命活動の様の精しい様などは 殆ど何も分からない存在であると言える。我々の認知した世界と発生して来た世界と消えて行った認知できない世界である。
このような視点で、人間にとって最も大事なことは 何だろうか。それは、個々の人間も、人類も 大きな存在の中の小さな存在であることを先ず自覚して、背後に存在する大いなる基礎、環境に畏敬の念を抱き、謙虚さを保つことではないだろうか。この視点では日本古来の神道の精神こそ、宗教の原点として大事では ないだろうか。未知なる自然に対する畏敬の念である。実際、日本でも、世界各地でも人工物を建設するとき、神事を行い、神の許しを求めてきたものである。その心は大いなる存在と人間の調和を志向する意味で人間存在の原理ではないだろうか。それはそもそも 原罪の概念そのものであると言える。
しかしながら、人類が好きなように生きたいも道理であり、巨大都市を建設して、環境を汚染して生存を享受したいも道理であるから、それらの一面も否定できず、それは結局全体的な有り様の中でのバランスの問題ではないだろうか。人類の進化の面には必然的に人類絶滅の要素が内在していると考えられる：
再生核研究所声明 144（2013.12.12） 人類滅亡の概念 － 進化とは 滅亡への過程である
そこで、結局は全体的な調和、バランスの問題である：
再生核研究所声明 56： アースデイ の理念
発想における最も大事なことに触れたが、表現したかった元を回想したい。― それは存在と非存在の間の微妙な有り様と非存在の認知できない限りない世界に想いを致す心情そのものであった。無数とも言える人間の想いはどこに消えて行ったのだろうか。先も分からず、由来も分からない。世の中は雲のような存在であると言える。
以 上
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