Mathematical Model of Gravitational and Electrostatic Forces

Transcription

1 Mahemaical Model of Graviaional and lecrosaic Forces Aleei Krouglov mail:

2 ABSTRACT Auhor presens mahemaical model for acing-on-a-disance aracive and repulsive forces based on propagaion of energy waves ha produces Newon epression for graviaional and Coulomb epression for elecrosaic forces. Model uses mahemaical observaion ha difference beween wo inverse eponenial funcions of he disance asympoically converges o funcion proporional o reciprocal of disance squared. Keywords: Graviaional Force; lecrosaic Force; Field Theory

3 . Inroducion Auhor described his Dual Time-Space Model of Wave Propagaion in work []. He ries o epress physical phenomena based only on concep of energy. Model assers ha when a some poin in space and in ime local energy value is differen from global energy level in surrounding area, energy disurbance is creaed a his poin. nergy disurbance propagaes boh in ime and in space as energy wave, and oscillaes. Auhor used he model o describe boh propagaion of energy waves in physics [] [5] and flucuaions of demand and supply in economics [6], [7]. Auhor presened in [] mahemaical model for aracive and repulsive forces ha demonsraed how graviaional and elecrosaic forces are creaed on a disance based on insrumen of Dual Time-Space Model of Wave Propagaion. However paper [] didn eplain one imporan poin. I described forces acing on a disance as being proporional o inverse eponenial funcion of disance. Laely auhor realized ha difference beween wo inverse eponenial funcions of disance is asympoically converging o funcion ha is proporional o reciprocal of disance squared. Thus resul provided by he model is in agreemen wih eperimenally obained formulas for forces acing on a disance such as graviaional and elecrosaic forces. In curren paper auhor presens quaniaive mahemaical model for acing-on-adisance saic forces.

4 . Model Mehodology As i was described in [], each poin in space and in ime is characerized by wo energy values local energy value in poin and global energy level in neighboring area Φ,. The difference beween quaniies U (, and Φ(, creaes energy disurbance, which propagaes boh in ime and in space according o ordinary differenial equaions presened below. U (, ( Differenial equaions in ime domain are, (, dp d d U (, Φ(, = λ ( U ( (, dp(, d = µ ( d dφ d (, dp(, = ν d ( where R, λ, µ, ν are consans, and P (, is dual variable evaluaed in poin a ime. Briefly speaking, meaning of equaions ( ( is as follows. Second derivaive of energy value U (, ( wih respec o ime is inversely proporional o energy disurbance U, Φ(,. Firs derivaive of energy level Φ(, wih respec o ime ( is direcly proporional o energy disurbance U, Φ(,. We can rewrie equaions ( ( o describe dynamics of changes for energy disurbance ( = U (, (,, Φ in ime, d (, d(, d (, = + λ + µ λ ν (4 d 4

5 quaion (4 is an ordinary differenial equaion of he second order, and can be resolved by regular mehods (for eample, see []. Noe ha value + for all. Differenial equaions in space domain are, (, when (, dp d d U (, Φ(, = λ ( U (5 (, dp(, d = µ (6 d dφ d (, dp(, = ν d (7 + where R, λ, µ, ν are consans. Similarly as above, meaning of equaions (5 (7 is as follows. Second derivaive of energy value U (, wih respec o he disance in space is inversely proporional o energy disurbance (,. Firs derivaive of energy level Φ(, wih respec o he disance in space is direcly proporional o energy disurbance (,. We can rewrie equaions (5 (7 o describe dynamics of changes for energy disurbance (, in space, d (, d(, d (, = + λ + µ λ ν (8 d quaion (8 is also an ordinary differenial equaion of he second order. Here ( again value, when + for all >. One of primary characerisics of curren model is ha i uses hree variables o describe dynamics of wave propagaion. Noneheless presen paper is devoed o saic 5

6 phenomena, paricularly o formaion of saic forces acing on a disance beween solid bodies. Dynamics of wave propagaion in space will involve a special sudy.. Transformaion of Global nergy Level near Solid Body Auhor considers in curren paper he saic siuaion. I means ha energy disurbance is no propagaed in ime. Here we deal wih solid bodies i means from he model s poin of view ha energy disurbance is no propagaed in space wih regard o local energy values U (,. Thus we only consider ransformaion of global energy level Φ(,. Mahemaically speaking, saic siuaion means ha equaions ( ( have null coefficiens λ, µ, ν =. Solid body means null coefficien = in equaions (5 (7 (oher coefficiens in equaions (5 (7 are nonnegaive λ, ν. Therefore equaions (5 and (7 can be combined ino he following one, µ dφ d (, = ν λ ( Φ(, U (, (9 Since we are dealing wih solid body, local energy value behind body s boundaries R. U (, can propagae The siuaion is illusraed on Figure, which shows ransformaion of energy level in space near boundary of solid body. Figure a shows siuaion when no energy disurbance is presen and local energy values in space mach global energy level in surrounding area. Figure b displays siuaion on he boundary of solid body where local energy value abruply changes is 6

7 magniude. Tha change creaes energy disurbance, which ransforms global energy level behind he boundaries of solid body. To calculae ransformaion of global energy level one has o solve equaion (9 for >. We have following iniial condiions a he body s boundary R for ( (, >, U, Φ U. We assume ha U, U is consan for > and >, and equaion (9 for global energy level (, = U Φ(,, ( Φ (, can produce following equaion for energy disurbance (, d d = ν λ (, ( where >, >, and ( U (, Φ( = U U = (, = λ ( } ( = ( U U λ ( } =,,. Thus soluion of equaion ( is as follows for >, where =. ep{ ν ( Therefore soluion of equaion (9 is for >, Φ ep{ +, ν U ( since = = U. U Figure c illusraes how global energy level is ransformed near he boundary of solid body according o equaion (. 7

8 4. Forces near Solid Body a Res As we know, concep of physical forces is relaed wih ideas of work and disance, and paricularly muliplicaion of average force applied o body by disance equals o he work, where work equals o change of body s energy. Following o discussion above, when a solid body has energy disurbance, i creaes energy disbalance in surrounding area. If energy disbalance occurs on he boundary of solid body, he body is rying o move o anoher posiion in order o reduce his energy disbalance. Thus energy disurbance creaes physical force. The magniude of ha force is proporional o changes of absolue value of energy disurbance wih respec o disance in space. Force poins o direcion of decrease of absolue value of energy disurbance. Ne definiion of force is used in he model. Definiion. Insananeous change of absolue value of energy disurbance a he boundary of solid body ( d, d in poin wih respec o disance in space is called force F (, applied o solid body in poin and direced o he decrease of absolue value of energy disurbance, F (, (, d U (, Φ(, d k = k ( d d where k > is coefficien of proporionaliy. Le s look how his definiion of force works for solid body ha is placed a res. The siuaion wih solid body a res is illusraed on Figure. Figure a displays local energy values U (, and global energy level Φ(, near solid body. Figure b presens disribuion of energy disurbance (, near solid 8

9 body. Figure c eamines adjused disribuion of energy disurbance near solid body, which akes ino accoun ransformaion of global energy level on each side of he solid body s border (his correcion of global energy level is no furher used in he paper. (, Thus if poins in space R and R, where >, are respecively righ and lef boundaries of solid body, ransformaion of global energy level Φ(, behind righ and lef boundaries of solid body is described as, Φ (, ep = ep { ν λ ( } { ν λ ( } + U + U if if > < (4 Then forces F (, and F (, acing on solid body a is boundaries are as follows (funcion (, U Φ( =, is decreasing o he righ of body s boundary >, and is increasing o he lef of body s boundary <, F F (, (, = lim k > = lim k < (, (, Φ(, Φ(, dφ(, = lim k > = k (, (, Φ(, Φ(, dφ(, = lim k < = k d d = ν = ν λ k (5 λ k (6 Since we assumed above ha solid body is a res (i.e. forces a body s boundaries are in balance wih each oher, i follows, (, + F(, F (7 = 9

10 5. Forces around Two Solid Bodies In his secion we sudy impac of energy disurbance on he boundaries of one solid body ha affecs energy disurbance on he boundaries of anoher solid body (as displayed on Figure. We consider hree poins in space R, R, and where >. Here is righ boundary of firs solid body, and are lef and > righ boundaries respecively of second solid body (see Figure a. R If we denoe as local energy value for second body and as = U U jump of energy value on he boundaries of second body, hen reverse forces, which hold second body a res, are as before, F F (, = ν λ k U (8 (, = ν λ k (9 (, + F(, F ( = When we consider resulan energy disurbance near second body, we have o consider no only ransformaion of global energy level o he lef of poin lef boundary of second body and o he righ of poin righ boundary of second body bu ransformaion of global energy level o he righ of poin righ boundary of firs body as well. Dual effec of ransformaions of global energy level near second body is refleced by subracing from global energy level ransformed by boundaries of second body and he global energy level ransformed by righ boundary of firs body. We may say ha we adjus global energy level near solid body by subracing from i he impac induced by oher solid bodies in surrounding area.

11 (, Figure b shows ransformaions of global energy level Φ (,, Φ (,, and Φ caused by righ boundary of firs body, lef boundary of second body, and righ boundary of second body respecively. (, Figure c displays disribuions of energy disurbances,, (,, and ( caused by righ boundary of firs body, lef boundary and righ boundary of second body respecively if he energy disurbances were separae. Now we wan o move from separae energy disurbances,, (,, and (, o resulan energy disurbance ˆ(, a he boundaries of second body and,. We assume ha energy disurbances (, and ( in poins and are decreased by energy disurbance (, propagaed from poin of firs body. Thus energy disurbances (, and (, in poins and are decreased by values (, and (, respecively. I gives us resulan energy disurbances ˆ(, and ˆ(, F ˆ (, F ˆ, in poins and, and creaes forces and ( acing on boundaries of second body. Discussion above can be summarized by following asserion. ( Assumpion. Quaniy of energy disurbance (, in poin boundary of solid body is decreased by sum of energy disurbances i i (, propagaed o poin from oher solid bodies in viciniy, (, (, ( ˆ i, ( i

12 We may look a equaion ( from a new poin of view energy disurbances a he boundaries of solid body are summed wih energy disurbances propagaed o hese boundaries from refleced images of oher solid bodies in viciniy (refleced image of solid body has energy disurbance of opposie sign. Figure d presens disribuion of resulan energy disurbance ˆ (, near he boundaries of second body in poins and, and shows forces F ˆ (, acing on second body in poins and. To calculae forces F ˆ (, and ( in poins and we denoe as F ˆ, and as he absolue values of resulan energy disurbance ˆ(, in poins and U. Then forces acing on he second body in hese poins are, (, = λ k U ˆ, and U ˆ (, ν ( (, = λ k where ( = ν (. = Consequenly resulan force acing on second body is as follows, (, + (, = λ k ( ν (4 On he oher hand, absolue values of resulan energy disurbances and a he boundaries of second body in poins and are as follows, (, (, = Φ (, Φ (, = ' { ν λ ( } = ep (5 (, (, = Φ (, Φ (, = ' { ν λ ( } = ep (6

13 Le us assume for simpliciy furher ha ' ep ν ' ep { ν λ ( } = ' ep = ' ep and i follows from equaion (4, { λ ( }. Then i follows from equaions (5 and (6, { λ ( } (, + (, = ν λ k ( ep{ ν λ ( } { ν λ ( } (, ( and ν (7 { λ ( } ν (8 ep ( ( Therefore resulan force ˆ, + F, > when > and > >. I means ha resulan force F ˆ, is direced o he righ of poin of second + body, i.e. second body is repulsed from firs body in his siuaion. (9 Le us esimae resulan force on second body ( = (, ( F ˆ, + presened by equaion (9. Since eponen may be represened hrough series epansion, τ τ τ ep{} τ = ( 6 hen i follows for τ > τ, > ep { τ } ep{ τ } δ τ + + τ τ = τ δ ( τ + δ τ ( τ + δ τ ( τ + δ δ τ δ δ + δ τ δ τ δ + τ ( where δ = τ τ and τ >> δ.

14 As a resul, by combining equaions (9 and ( we obain ne approimaion for resulan force ( ( acing on second body, r k ( R where R =, r =, and R >> r. Thus we have esablished Inverse Square Law for resulan force second body and given by equaion (. ( acing on Le us eend formula for resulan force on wo- and hree-dimensional spaces. A firs we refer o observaion ha resulan force acing on solid body is obained by summarizing all forces acing on ha body. Assumpion. Resulan force acing on a solid body is equal o sum of all forces acing on he body, ( F i ( ( i Secondly, if we move from one-dimensional space ha was considered so far o muli-dimensional spaces, we imagine pluraliy of lines saring perpendicularly o he surface of firs solid body, and peneraing he surface of second solid body. One line produces forces on he boundaries of second body. ach resulan force is proporional o inerval of he line ha is conained beween he boundaries of second body (as saed by equaion (. To obain oal resulan force acing on second body we have o sum ogeher all resulan forces acing a boundaries of he body. Toal resulan force acing on second body is proporional o he sum of all such inervals ha are locaed beween he boundaries of second body. ( 4

15 Thirdly, when we consider solid bodies ha are relaively depared from each oher (i.e. R >> r as saed above, he lines connecing wo bodies are drawn in parallel o each oher. Therefore for wo-dimensional space sum of all parallel inervals ha lie beween boundaries of wo-dimensional body equals o he body s inernal area. Respecively for hree-dimensional space sum of all parallel inervals ha lie beween boundaries of hree-dimensional body equals o he body s inernal volume. The discussion above is illusraed on Figure 4. Figure 4a shows forces, whose sum makes resulan force acing on wodimensional second body. Similarly Figure 4b demonsraes forces, whose sum creaes resulan force ( acing on hree-dimensional second body. Therefore in hree-dimensional space resulan force body is epressed by following equaion, ( ( ( acing on second solid V k (4 R where R is disance beween firs and second bodies, V is second body s inernal volume, is jump of energy value on he boundaries of firs body, and k > is coefficien of proporionaliy. Since indeing on firs and second bodies can be inerchanged, we obain pair of epressions for absolue values of and ˆ resulan forces acing on firs and second bodies respecively, V ( U ˆ ( ( F F ~ (5 R 5

16 V ( U ~ (6 R where and are jumps of energy values on boundaries of firs and second bodies respecively, and V are inernal volumes of firs and second bodies V respecively, and R is disance beween firs and second bodies. ˆ ( ( A he same ime, from Newon s hird law resulan forces and ˆ acing on firs and second bodies respecively should have he same absolue values, and poin o opposie direcions. Therefore i akes place for ˆ ( ( = ( F =, F F ( ~ ( V ( V R (7 or equivalenly, ( ( V ( V K (8 R where K > is coefficien of proporionaliy. 6. lecrosaic Forces To ge Coulomb s formula for elecrosaic forces we consider wo separae cases elecrosaic forces for repulsion and aracion. a Repulsive lecrosaic Forces A firs, I consider for diversiy one-dimensional space wih wo solid bodies, each wih negaive jump of energy values = U U and = U U, where < jumps of energy values have magniudes of common order. < 6

17 Le s eamine absolue values of resulan energy disurbances and a he boundaries of second body in poins and obained in viciniy of firs body, and presened by equaions (5 and (6. Since values and are negaive and of common order, i akes place ' ep{ ν λ ( } and ' ep λ ( from equaions (5 and (6, = ' + ep = ' + ep { λ ( } { ν }. Then i follows ν (9 (, + (, = ν λ k ( ep{ ν λ ( } { ν λ ( } { λ ( } and from equaion (4 for resulan force, ν (4 ep (4 Since < and > > resulan force ( ˆ, + F(, > i.e. force is direced o righ way of second body, and second body is repulsed from firs body. Secondly, according o equaion (8 we have an esimae of he absolue value of resulan force beween wo hree-dimensional bodies. To ge formula for elecrosaic forces we look a facor ( V from equaion (8. I represens elecrosaic energy of solid body bu ha energy is radiionally epressed via elecrosaic charge Q. Therefore i follows he Coulomb s formula from equaion (8 for repulsive force beween wo solid bodies ha boh have negaive elecrosaic charges Q = V (noice ha elecrosaic charge Q is concenraed on he body s surface, ˆ Q ( K F (4 R Q 7

18 where and Q are elecrosaic charges of firs and second bodies respecively, R is Q disance beween firs and second bodies, and K > is Coulomb s consan. Figure 5 illusraes repulsive forces beween wo solid bodies where each body has a negaive elecrosaic charge. b Aracive lecrosaic Forces A firs, I consider here one-dimensional space wih wo solid bodies where firs body has negaive jump of energy value = U U < and second body has posiive jump of energy value = U U >, and jumps of energy values have magniudes of common order. Le s look again a absolue values of resulan energy disurbances and a he boundaries of second body in poins and obained in viciniy of firs body, and presened by equaions (5 and (6. Since value is negaive and value is posiive and boh are of common order, i akes place ' ep λ and ' ep ν λ. Then i follows from equaions (5 and (6, { ν ( } { ( } = ' ep { ν λ ( } (4 = ' ep { λ ( } and from equaion (4 for resulan force, ν (44 (, + (, = ν λ k ( ep{ ν λ ( } { ν λ ( } ep (45 8

19 Since < and > > resulan force ( ˆ, + F(, < i.e. force is direced o lef way of second body, and second body is araced o firs body. According o equaion (8 we have an esimae of he absolue value of resulan force beween wo hree-dimensional bodies. I is represened by he same formula (4. Figure 6 illusraes aracive forces beween wo solid bodies where firs body has a negaive elecrosaic charge, and second body has a posiive elecrosaic charge. 7. Graviaional Forces This secion is devoed o Newon s formula for graviaional forces. We consider wo solid bodies where firs solid body produces energy disurbance ha is much bigger han energy disurbance produced by anoher solid body. We look a wo differen cases siuaion wih roaion of small body around is ais and siuaion wih idally locked small body (i.e. wihou roaion of small body around ais. a Graviaional Force wih Tidally Locked Aracion A firs, I consider wo solid bodies in one-dimensional space wih posiive jumps of energy values = U U and = U U, when jump of energy value > > for firs body is much bigger han jump of energy value for second body i.e. >>. Le s eamine he absolue values of resulan energy disurbances and a he boundaries of second body in poins and obained in viciniy of firs body, and presened by equaions (5 and (6. 9

20 Since values and are boh posiive and >>, i akes place ' ep{ ν λ ( } and ' ep λ ( from equaions (5 and (6, = ' + ep = ' + ep { λ ( } { ν }. Then i follows ν (46 { λ ( } and from equaion (4 for resulan force, ν (47 (, + (, = ν λ k ( ep{ ν λ ( } { ν λ ( } ep (48 Since > and > > resulan force ( ˆ, + F(, < i.e. force is poined o he lef way of second body, and second small body is araced o firs massive body. If we eamine resuls of equaions (46 and (47 we see ha absolue values, and are in realiy he propagaed energy disurbances (, and ( of U refleced image of massive solid body ( revealed a he boundaries of small body, and summed wih energy disurbances of small body (, and (. Thus we, may say ha graviaional force of aracion beween massive body and small body is resul of ineracion of massive body wih is refleced image (I suppose from symmery ha force, which aracs massive body o small body, is resul of ineracion of small body wih is refleced image as well. Ulimaely according o equaion (8 we have an esimae of he absolue value of resulan force beween wo hree-dimensional bodies. To ge formula for graviaional

21 forces we eamine facor ( U V from equaion (8. I represens graviaional energy of solid body. I assume (his assumpion is in agreemen wih general asserion ha body s energy is proporional o is mass ha energy value proporional o is densiy ρ (where γ >, of solid body is = γ ρ (49 Merger of equaions (8, (49 gives formula for resulan graviaional force (, ˆ ( and ulimaely, ( ρ V ( ρ V F ~ = R m R m (5 ˆ m ( G F (5 R m where and m are masses of firs and second bodies respecively, R is disance m beween firs and second bodies, and G is graviaional consan. Figure 7 illusraes idally locked graviaional forces beween wo solid bodies where one body has much bigger jump of energy value han anoher body does. b Graviaional Force wih Roaion of Small Body Le us move small body away from massive body in righ direcion. Then a some momen we have ne resulan energy disurbances a he boundaries of small body in poins and (where > >, ˆ (, = ' + ( ep{ ν λ } > (5 ˆ (, = ' + ( } ep{ ν λ (5 =

22 A his posiion force a righ boundary of small body in poin (, = k λ < becomes zero, ν (54 (, = k ν (55 λ = where U ˆ (, and U ˆ (, disance =. = If we again move small body in righ direcion from poins and on minor δ > ˆ ˆ, hen resulan energy disurbances are, ( + δ, = ' + ν λ ( + δ ep{ } ε > (56 ( + δ, = ' + ν λ ( + δ ep{ } ε < (57 where ε > and ε > are small. + δ and Therefore resulan energy disurbances a he boundaries of small body in poins + δ have opposie signs. Conrary boundaries of small body are araced o each oher, and i generaes slow roaion of small body. Since ε > and ε > are small, graviaional force sill can be approimaed by formula (5. When we move small body furher away from massive body, conrary boundaries of small body become sronger araced o each oher and roaion of small body becomes faser. A he same ime magniude of graviaional force acing beween small and massive bodies approaches and finally reaches zero. A ha momen absolue values of resulan energy disurbances a he boundaries of small body in poins and become equal, ˆ (, = ' + ( ep{ ν λ } (58 >

23 ˆ (, = ' + ( ep{ ν λ } < (59 (, = ˆ ( ˆ, (6 A his posiion (when small body is in poins and he resulan force ( beween small and massive bodies becomes zero. When we move small body furher away from poins and, roaion of small body becomes slow. Graviaional force acing beween small and massive bodies changes is direcion, and small and massive bodies sar being repulsed from each oher. Figure 8 illusraes siuaion wih graviaional force beween small and massive bodies a posiion when he force equals zero. 8. Conclusion Unforunaely model sill doesn provide answers for all quesions. For eample, I am rying o undersand process behind Newon s hird law for forces acing on a disance beween wo solid bodies.

24 References. V. I. Arnol d, Ordinary Differenial quaions, rd ediion, Springer Verlag, Berlin; New York, 99.. A. Krouglov, Dual Time-Space Model of Wave Propagaion, physics/9994, available a hp://ariv.org.. A. Krouglov, Mahemaical Model of Aracion and Repulsion Forces, physics/99, available a hp://ariv.org. 4. A. Krouglov, planaion of Faraday s perimen by he Time-Space Model of Wave Propagaion, physics/757, available a hp://ariv.org. 5. A. Krouglov, Mahemaical Model of Shock Waves, physics/999, available a hp://ariv.org. 6. A. Krouglov, Dynamical lemens of he Moneary Theory, in Trends in Macroeconomics Research, L.Z. Pelzer, d., Nova Science Publishers, New York, A. Krouglov, Mahemaical Dynamics of conomic Markes, Nova Science Publishers, New York, 6 (forhcoming. 4

25 Figure. Transformaion of Global nergy Level U Φ Figure a. nergy Local Values and Global Level a Res U U( U( Figure b. Change of Local nergy Values Φ Φ( Φ( Figure c. Transformaion of Global nergy Level 5

26 Figure. Disribuion of nergy Disurbance near Solid Body Φ(=U( Φ( Φ( U( U( Figure a. nergy Local Values and Global Level near Solid Body (= ( ( Figure b. Disribuion of nergy Disurbance near Solid Body ( (= ( F( < F( > Figure c. Adjused Disribuion of nergy Disurbance 6

27 Figure. Disribuion of nergy Disurbance near Two Bodies ' Figure a. Two Solid Bodies in Space and Local nergy Values Φ Φ Φ Figure b. Transformaion of Global nergy Level Figure c. Disribuion of nergy Disurbances F( < F( > Figure d. Disribuion of Resulan nergy Disurbance near Second Body 7

28 Figure 4. Forces in Two- and Three-Dimensional Spaces Firs Body Second Body Σ F i Figure 4a. Forces Acing on Second Body in Two-Dimensional Space Firs Body Second Body Σ F i Figure 4b. Forces Acing on Second Body in Three-Dimensional Space 8

29 Figure 5. Repulsive lecrosaic Forces beween Two Bodies ' Figure 5a. Two Solid Bodies wih Negaive nergy Values in Space Φ Φ Φ Figure 5b. Transformaion of Global nergy Level F> Figure 5c. nergy Disurbance near Second Body and Repulsive Force 9

30 Figure 6. Aracive lecrosaic Forces beween Two Bodies ' Figure 6a. Two Solid Bodies wih Negaive and Posiive nergy Values in Space Φ Φ Φ Figure 6b. Transformaion of Global nergy Level F< Figure 6c. nergy Disurbance near Second Body and Aracive Force

31 Figure 7. Graviaional Force wih Tidally Locked Aracion ' Figure 7a. Massive and Small Solid Bodies wih Posiive nergy Values in Space Φ Φ Φ Figure 7b. Transformaion of Global nergy Level F< Figure 7c. nergy Disurbance near Second Body and Graviaional Force

32 Figure 8. Graviaional Force wih Null Magniude ' Figure 8a. Massive and Small Solid Bodies wih Posiive nergy Values in Space Φ Φ Φ Figure 8b. Transformaion of Global nergy Level F= Figure 8c. nergy Disurbance near Second Body and Null Graviaional Force