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Natural motion and gravitation
PERGAMON
Applied Mathematics Letters 13 (2000) 65-66
Applied Mathematics Letters www.elsevier.nl/locate/aml
Natural Motion and Gravitation Y. Y. NEYMAN 3553 Bantry Way Olney, MD 20832-2255, U.S.A. (Received September 1999; accepted November 1999)
A b s t r a c t - - I t is shown that for systems consisting of an arbitrary but finite set of celestial bodies, natural motion proves to be gravitation. ~) 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - N a t u r a l motion, Gravitation, Celestial body, Generalized sum.
1. I N T R O D U C T I O N With the help of elementary calculations, for the simplest system consisting of three celestial bodies, we have discovered t h a t these bodies possess a property which proves to be gravitation. Calculations carried out within the framework of Newton's theory of gravity for systems consisting of four, five, and six celestial bodies confirmed this. In this article, we want to confirm the same for systems consisting of an arbitrary but finite set of celestial bodies. We have published some fundamental elements of the mechanics of an infinite universe [1]. In the axiomatics, the central place belongs to the axiom asserting that motion is an inherent property of celestial bodies. Following Aristotle, we called such motion natural motion. In agreement with the axiomatics, natural motions of celestial bodies propagate between them and are transmitted from one to another. As a result of the superposition of such motions, every celestial b o d y executes violence motion, which is observable to the accessible part of the universe. Since action at a distance is accepted in Newton's theory, then violence motion of the considered system of celestial bodies is described by the infinite series My = D' + A B D ' + ( A B ) 2 D ' + ( A B ) 3 D ' + . . . ,
(A)
where D ' is a column matrix describing linear accelerations transmittable to celestial bodies of a system of the infinite universe, A is the matrix taking into account the configuration of the celestial bodies, B is the matrix taking into account the relative direction of the unit vectors issuing from the celestial bodies, taken as geometric points. Matrices A and B are square, nondegenerate, and of the same order, and matrix D has the same order.
2. S U M
OF SERIES
(A)
Necessary and sufficient conditions are known for convergence of (A) [2]. A sufficient condition is also known for convergence of infinite series of segments of (A) [3]. However, neither (A), nor a 0893-9659/00/$ - see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(00)00078-1
Typeset by fl,2VfS-TEX
66
Y . Y . NEYMAN
series of its segments satisfies these conditions. Therefore, we conclude that (A) diverges. There exists a detailed investigation of divergent scalar series [4]. In order to find the "generalized sum" of divergent matrix series (A), we proceed as follows. We assume that matrix D ~ in (A) is a variable matrix and denote it by X. Choose an arbitrary variable column matrix with the same order as D t and such t h a t the infinite series Co, C j , . . . , C ~ , . . . of matrices converges to a finite limn--,oo Cn = A. Since there exists a unique solution [5] of the equation
Ef i ( A B ) i X n = C,~, i=O
namely,
Xn =
(AB) i
]_1
C,,,
i=0
it follows t h a t each matrix Cn corresponds to a unique matrix X n and this means that matrix X proves to be a function of matrix argument C x
=
We extrapolate the result obtained for functions of a scalar argument [6] to a function of a matrix argument: if a series of matrices Cn has a limit, then a series of matrices Xn also has a limit. If limn~oo Cn = A, then limn--,oc X n = D, i.e.,
(AB) i
lim Xn =
n--~oo
i=O
lim Cn.
n ---*o(3
Thus, we get
My=
AB) n D=A,
(B)
where A is the "generalized sum" [7] of series (A). The meaning of (B) consists in the following: •for an arbitrary set of linear accelerations of celestial bodies of a finite system, arising under the action of gravity and described by matrix A, there exists a unique set of accelerations transmitted by the celestial bodies of the same system of the infinite universe and described by matrix D. Acceleration is one of the elements describing motion, therefore, one can say t h a t natural motion proves to be gravitation.
REFERENCES 1. Y.Y. Neiman, Celestial mechanics, An International Journal of Space Dynamics 5 (1), 55-66, (1972). 2. F.R. Gantmakher, Theory of Matrices, (in Russian), p. 108, Nauka, Moscow, (1988). 3. L.S. Pontryagin, Continuous groups, (in Russian), Izdatel'stvo Teoreticheskoy Literatury, Moscow, 160-E,D,
(1954). 4. G.H. Hardy, Divergent Series, Clarendon Press, Oxford, (1940). 5. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, (in Russian), p. 373, Nauka, (1968)• 6. G.M. Fikhtengolts, A Course in Di~erential and Integral Calculus, Volume I, (in Russian), p. 142, Izdatelstwo • Teoretickeskoy Literatury, Moscow-Leningrad, (1947). 7. G.M. Fikhtengolts, A Course in Di]:~erential and Integral Calculus, Volume III, (in Russian), p, 723, Izdatelstwo Teoretickeskoy Literatury, Moscow-Leningrad, (1949).
Applied Mathematics Letters 13 (2000) 65-66
Applied Mathematics Letters www.elsevier.nl/locate/aml
Natural Motion and Gravitation Y. Y. NEYMAN 3553 Bantry Way Olney, MD 20832-2255, U.S.A. (Received September 1999; accepted November 1999)
A b s t r a c t - - I t is shown that for systems consisting of an arbitrary but finite set of celestial bodies, natural motion proves to be gravitation. ~) 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - N a t u r a l motion, Gravitation, Celestial body, Generalized sum.
1. I N T R O D U C T I O N With the help of elementary calculations, for the simplest system consisting of three celestial bodies, we have discovered t h a t these bodies possess a property which proves to be gravitation. Calculations carried out within the framework of Newton's theory of gravity for systems consisting of four, five, and six celestial bodies confirmed this. In this article, we want to confirm the same for systems consisting of an arbitrary but finite set of celestial bodies. We have published some fundamental elements of the mechanics of an infinite universe [1]. In the axiomatics, the central place belongs to the axiom asserting that motion is an inherent property of celestial bodies. Following Aristotle, we called such motion natural motion. In agreement with the axiomatics, natural motions of celestial bodies propagate between them and are transmitted from one to another. As a result of the superposition of such motions, every celestial b o d y executes violence motion, which is observable to the accessible part of the universe. Since action at a distance is accepted in Newton's theory, then violence motion of the considered system of celestial bodies is described by the infinite series My = D' + A B D ' + ( A B ) 2 D ' + ( A B ) 3 D ' + . . . ,
(A)
where D ' is a column matrix describing linear accelerations transmittable to celestial bodies of a system of the infinite universe, A is the matrix taking into account the configuration of the celestial bodies, B is the matrix taking into account the relative direction of the unit vectors issuing from the celestial bodies, taken as geometric points. Matrices A and B are square, nondegenerate, and of the same order, and matrix D has the same order.
2. S U M
OF SERIES
(A)
Necessary and sufficient conditions are known for convergence of (A) [2]. A sufficient condition is also known for convergence of infinite series of segments of (A) [3]. However, neither (A), nor a 0893-9659/00/$ - see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(00)00078-1
Typeset by fl,2VfS-TEX
66
Y . Y . NEYMAN
series of its segments satisfies these conditions. Therefore, we conclude that (A) diverges. There exists a detailed investigation of divergent scalar series [4]. In order to find the "generalized sum" of divergent matrix series (A), we proceed as follows. We assume that matrix D ~ in (A) is a variable matrix and denote it by X. Choose an arbitrary variable column matrix with the same order as D t and such t h a t the infinite series Co, C j , . . . , C ~ , . . . of matrices converges to a finite limn--,oo Cn = A. Since there exists a unique solution [5] of the equation
Ef i ( A B ) i X n = C,~, i=O
namely,
Xn =
(AB) i
]_1
C,,,
i=0
it follows t h a t each matrix Cn corresponds to a unique matrix X n and this means that matrix X proves to be a function of matrix argument C x
=
We extrapolate the result obtained for functions of a scalar argument [6] to a function of a matrix argument: if a series of matrices Cn has a limit, then a series of matrices Xn also has a limit. If limn~oo Cn = A, then limn--,oc X n = D, i.e.,
(AB) i
lim Xn =
n--~oo
i=O
lim Cn.
n ---*o(3
Thus, we get
My=
AB) n D=A,
(B)
where A is the "generalized sum" [7] of series (A). The meaning of (B) consists in the following: •for an arbitrary set of linear accelerations of celestial bodies of a finite system, arising under the action of gravity and described by matrix A, there exists a unique set of accelerations transmitted by the celestial bodies of the same system of the infinite universe and described by matrix D. Acceleration is one of the elements describing motion, therefore, one can say t h a t natural motion proves to be gravitation.
REFERENCES 1. Y.Y. Neiman, Celestial mechanics, An International Journal of Space Dynamics 5 (1), 55-66, (1972). 2. F.R. Gantmakher, Theory of Matrices, (in Russian), p. 108, Nauka, Moscow, (1988). 3. L.S. Pontryagin, Continuous groups, (in Russian), Izdatel'stvo Teoreticheskoy Literatury, Moscow, 160-E,D,
(1954). 4. G.H. Hardy, Divergent Series, Clarendon Press, Oxford, (1940). 5. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, (in Russian), p. 373, Nauka, (1968)• 6. G.M. Fikhtengolts, A Course in Di~erential and Integral Calculus, Volume I, (in Russian), p. 142, Izdatelstwo • Teoretickeskoy Literatury, Moscow-Leningrad, (1947). 7. G.M. Fikhtengolts, A Course in Di]:~erential and Integral Calculus, Volume III, (in Russian), p, 723, Izdatelstwo Teoretickeskoy Literatury, Moscow-Leningrad, (1949).