Expanding Newton Mechanics with Neutrosophy and Quadstage Method New Newton Mechanics Taking Law of. Conservation of Energy as Unique Source Law

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1 Neutosophic Sets and Systems, Vol. 3, 4 3 Epanding Newton Mechanics with Neutosophy and Quadstage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law Fu Yuhua CNOOC Reseach Institute, No.6, ongzhimenwaiiaojie Steet, Beijing, 7, China. fuyh945@sina.com Abstact. Neutosophy is a new banch of philosophy, and "Quad-stage" (Fou stages) is the epansion of Hegel s tiad thesis, antithesis, synthesis of deelopment. Applying Neutosophy and "Quad-stage" method, the puposes of this pape ae epanding Newton Mechanics and making it become New Newton Mechanics (NNW) taking law of conseation of enegy as unique souce law. In this pape the eamples show that in some cases othe laws may be contadicted with the law of conseation of enegy. The oiginal Newton's thee laws and the law of gaity, in pinciple can be deied by the law of conseation of enegy. Though the eample of fee falling body, this pape deies the oiginal Newton's second law by using the law of conseation of enegy, and poes that thee is not the contadiction between the oiginal law of gaity and the law of conseation of enegy; and though the eample of a small ball olls along the inclined plane (belonging to the poblem cannot be soled by geneal elatiity that a body is foced to moe in flat space), deies impoed Newton's second law and impoed law of gaity by using law of conseation of enegy. Whethe o not othe conseation laws (such as the law of conseation of momentum and the law of conseation of angula momentum) can be utilized, should be tested by law of conseation of enegy. When the oiginal Newton's second law is not coect, then the laws of conseation of momentum and angula momentum ae no longe coect; theefoe the geneal foms of impoed law of conseation of momentum and impoed law of conseation of angula momentum ae pesented. In the cases that law of conseation of enegy cannot be used effectiely, New Newton Mechanics will not eclude that accoding to othe theoies o accuate epeiments to deie the laws o fomulas to sole some specific poblems. Fo eample, with the help of the esult of geneal elatiity, the impoed Newton's fomula of uniesal gaitation can be deied, which can be used to sole the poblem of adance of planetay peihelion and the poblem of deflection of photon aound the Sun. Again, accoding to accuate epeimental esult, the synthesized gaitational fomula (including the effects of othe celestial bodies and sunlight pessue) fo the poblem of deflection of photon aound the Sun is pesented. Unlike the oiginal Newton Mechanics, in New Newton Mechanics, fo diffeent poblems, may hae diffeent laws of motion, diffeent fomulas of gaity, as well as diffeent epessions of enegy. Fo eample, fo the poblem of a small ball olls along the inclined plane, and the poblem of adance of planetay peihelion, the two fomulas of gaity ae completely diffeent. Keywods: Neutosophy, "Quad-stage" (Fou stages), law of conseation of enegy, unique souce law, New Newton Mechanics Intoduction As a new banch of philosophy, Neutosophy studies the oigin, natue, and scope of neutalities, as well as thei inteactions with diffeent ideational specta. Accoding to Neutosophy that thee is a 3 Neutosophic Space, whee each dimension of the space epesents espectiely the tuth (T), the falsehood (F), and the indeteminacy (I) of the statement unde consideation. Moe infomation about Neutosophy may be found in efeences [,]. Quad-stage is intoduced in efeence [3], it is the epansion of Hegel s tiad-stage (tiad thesis, antithesis, synthesis of deelopment). The fou stages ae geneal theses, geneal antitheses, the most impotant and the most Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

2 4 Neutosophic Sets and Systems, Vol. 3, 4 complicated uniesal elations, and geneal syntheses. In quad-stage method, geneal theses may be consideed as the notion o idea <A> in neutosophy; geneal antitheses may be consideed as the notion o idea <Anti- A> in neutosophy; the most impotant and the most complicated uniesal elations may be consideed as the notion o idea <Neut-A> in neutosophy; and geneal syntheses ae the final esults. The diffeent kinds of esults in the aboe mentioned fou stages can also be classified and induced with the iewpoints of neutosophy. Thus, the theoy and achieement of neutosophy can be applied as many as possible, and the method of quad-stage will be moe effectie. The combination of Neutosophy and quad-stage will be a poweful method to ealize many innoations in aeas of science, technology, liteatue and at. Theefoe, this pape epands Newton Mechanics with Neutosophy and Quad-stage Method and ceates New Newton Mechanics (NNW) taking law of conseation of enegy as unique souce law. One of the deelopment tends of natual science is using fewe laws to sole inceasing poblems. In this pocess, accoding to the iewpoint of neutosophy, some laws will play the inceasingly geat oles; some laws will play the smalle oles, o een disappea fom the anks of laws; and the middle ones will be impoed and epanded to play the geate oles. As epanding Newton mechanics with neutosophy and quad-stage, the whole pocess can be diided into the following fou stages. The fist stage (stage of geneal theses ), fo the beginning of deelopment, the thesis (namely Newton mechanics) should be widely, deeply, caefully and epeatedly contacted, eploed, analyzed, pefected and so on. Regading the adantages of Newton mechanics, that will not be epeated hee, while we should stess the deficiencies of Newton mechanics. As well-known, Newton mechanics cannot be used to sole the poblem of adance of planetay peihelion and the poblem of deflection of photon aound the Sun. Fo othe pespecties on Newton mechanics, we will discuss in detail below, in ode to aoid duplication. The second stage, fo the appeaance of opposite (antithesis), the antithesis should be also widely, deeply, caefully and epeatedly contacted, eploed, analyzed, pefected and so on. Thee ae many opposites (antitheses) to Newton mechanics. Fo eample: special and geneal theoy of elatiity, "theoy of eeything", law of conseation of enegy, and so on, this pape focuses on the poblems elated to law of conseation of enegy. The thid stage is the one that the most impotant and the most complicated uniesal elations. The pupose of this poision stage is to establish the uniesal elations in the widest scope. To link and combine Newton mechanics with law of conseation of enegy, as well as the billiant achieements of moden science and technology, then Newton mechanics can be epanded and deeloped effectiely and successfully in the maimum aea. The fouth stage, to cay on the unification and synthesis egading aious opposites and the suitable pieces of infomation, factos, and so on; and each one o moe esults to epand Newton mechanics which ae the best o ageed with some conditions; this is the stage of geneal syntheses. Now we discuss the law of conseation of enegy. Its main contents ae as follows: In a closed system, the total enegy of this system emains unchanged. Because the law of conseation of enegy is the most impotant one in natual sciences, it should play an inceasingly geat ole. Fo this eason and accoding to the pinciple of the uniqueness of tuth, this pape pesents the New Newton Mechanics (NNM) taking law of conseation of enegy as unique souce law with Neutosophy and Quad-stage Method. In the aea of Newton Mechanics, thee should be one tuth only. Othe so-called tuth, eithe it can be deied by the unique tuth, o we can poe that in cetain cases it is not tue. As well-known, when Newton founded the classical mechanics, fou laws wee poposed, they wee Newton's thee laws and the law of gaity. If the law of conseation of enegy is choosing as the unique souce law, that in pinciple, all the Newton's fou laws can be deied accoding to the law of conseation of enegy; afte studying caefully we found that this may indeed be the eal case. In addition, in the aeas such as physics, mechanics, engineeing and so on, thee ae thee ey impotant laws: the law of conseation of enegy, the law of conseation of momentum and the law of conseation of angula momentum. If we beliee that the law of conseation of enegy is the tuth, then fo the law of conseation of momentum and the law of conseation of angula momentum, eithe they can be deied by the law of conseation of enegy, o we can poe that in cetain cases they ae not tue. We beliee that the tue situation is the latte, namely, the law of conseation of momentum and the law of conseation of angula momentum ae not tue in some cases (o thei esults ae contadicted to the law of conseation of enegy). Of couse, we can also find that in some cases, these two laws still can be used. Taking the eample that a man walks along the ca located on the hoizontal smooth ail, we can see that at pesent in the aea of Newton mechanics, some people do not notice the case of the contadiction between the law of conseation of enegy and the law of conseation of momentum. New thee laws of motion and new law of gaity (fomula) ceated by law of conseation of enegy fo New Newton Mechanics Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

3 Neutosophic Sets and Systems, Vol. 3, 4 5 The oiginal Newton's thee laws of motion (patial theses) ae as follows. Newton's Fist Law of Motion: Eey object in a state of unifom motion (o at est) tends to emain in that state of motion (o at est) unless an etenal foce is applied to it. Fo shot: est emains est, and moing emains moing. Newton's Second Law of Motion: The elationship between an object's mass m, its acceleation a, and the applied foce F is F = ma. The diection of the foce is the same as the diection of the acceleation. Newton's Thid Law of Motion: Fo eey action thee is an equal and opposite eaction. The oiginal Newton s law of gaity (patial theses): The attactie foce between two objects is as follows F () While though the stage of geneal antitheses and the stage of the most impotant and the most complicated uniesal elations, fo NNM, taking law of conseation of enegy as unique souce law, then we hae the following NNM s thee laws of motion and law of gaity. NNM's Fist Law of Motion: Eey object in a state of unifom motion (o in a state of unifom otation, o at est) tends to emain in that state of motion (o in a state of unifom otation, o at est) unless an etenal foce is applied to it; othewise the law of conseation of enegy will be destoyed. Fo shot: est emains est, moing emains moing, and otating emains otating. NNM's Second Law of Motion: The elationship between an object's mass m, its acceleation a, and the applied foce F is a function that should be deied by law of conseation of enegy. The diection of the foce is the same as the diection of the acceleation. In geneal, the function can be witten as the fom of aiable dimension F ma factal:, whee: is a constant o a aiable. Fo diffeent poblems, the foms of second law may be diffeent. NNM's Thid Law of Motion: In geneal, fo eey action thee is an equal and opposite eaction. In special case, the function elationship between action and eaction should be deied by law of conseation of enegy. The impoed fom of the oiginal Newton s thid law ( F AB F BA ) is as follows: F, AB F BA whee: is a constant o a aiable. Fo diffeent poblems, the foms of thid law may be diffeent. NNM s law (fomula) of gaity: The attactie foce between two objects is a function that should be deied by law of conseation of enegy, o epeimental data; o deied with the help of othe theoies. Fo diffeent poblems, the foms of law (fomula) of gaity may be diffeent. The esults of oiginal Newton s law of gaity ae only accuate in the cases that two objects ae elatie static o unning the staight line between one cente and anothe cente, and the like; fo othe cases its esults ae all appoimate. In geneal, NNM s law (fomula) of gaity may be taken as the fom that adding the amending tem to oiginal Newton s law of gaity, o the following fom of aiable dimension factal: F () whee: is a constant o a aiable. Now fo an eample, a NNM s law (fomula) of gaity (an impoed Newton s law of gaity) and a NNM's second law of motion (an impoed Newton s second law of motion), they ae suitable fo this eample only, ae deied simultaneously by law of conseation of enegy. Fistly, though uniesal elations, the aiational pinciples established by the law of conseation of enegy can be gien with least squaes method (LSM). Supposing that the initial total enegy of a closed system equals W (), and fo time t the total enegy equals W (t), then accoding to the law of conseation of enegy: W () = (t) W (3) This can be witten as: W ( t) R W = W () (4) Accoding to LSM, fo the inteal [ t,t ],we can wite the following aiational pinciple: Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

4 6 Neutosophic Sets and Systems, Vol. 3, 4 t dt W min t R (5) whee: min denotes the minimum alue of functional Π and it should be equal to zeo. It should be noted that, in many cases W (t) is appoimate, and R W is not identically equal to zeo, theefoe Eq.(5) can be used to sole the poblem. Besides the time coodinate, anothe one can also be used. Fo eample, fo inteal [, ], the following aiational pinciple can be gien accoding to the law of conseation of enegy: Accoding to the alue of Π, the effect of the solution can be judged. The neae the alue of Π is to zeo, the bette the effect of the solution. It should be noted that besides of soling equations, optimum-seeking methods could also be used fo finding the minimum and the constants to be detemined. In fact, the optimum seeking method will be used in this pape. Now we sole an eample. As shown in Fig., supposing that the small ball olls along a long incline fom A to B. Its initial elocity is zeo and the fiction and the otational enegy of small ball ae neglected. R (6) d W min The aboe-mentioned pinciples ae established by using the law of conseation of enegy diectly. Sometimes, a cetain pinciple should be established by using the law of conseation of enegy indiectly. Fo eample, a special physical quantity Q may be inteested,not only it can be calculated by using the law of conseation of enegy, but also can be calculated by using othe laws (fo this pape they ae the law of gaity, and Newton s second law). Fo distinguishing the alues, let s denote the alue gien by othe laws as Q,while denote the alue gien by the law of conseation of enegy as Q ',then the alue of R can be edefined as follows: Q R W = Q' W (7) Substituting Eq.(7)into Eqs.(5)and(6),as Q ' is the esult calculated with the law of conseation of enegy, it gies the aiational pinciple established by using the law of conseation of enegy indiectly. Othewise, it is clea that the etent of the alue of Q accods with Q '. Substituting the elated quantities into Eq.(5)o Eq.(6),the equations deied by the condition of an etemum can be witten as follows: Fig. A small ball olls fom A to B O ' denotes the Eath, M denotes Supposing that cicle its mass; m denotes the mass of the small ball (teated as a mass point ), O A is a plumb line, coodinate is othogonal to O A, coodinate y is othogonal to coodinate (paallel to O A), BC is othogonal to O A. The lengths of OA, OB, BC, and AC ae all equal to H, and O C equals the adius R of the Eath. In this eample, the alue of which is the squae of the elocity fo the ball located at point is inestigated. To distinguish the quantities, denote the alue gien by the impoed law of gaity and impoed Newton s second law as,while denotes the alue gien ' by the law of conseation of enegy,then Eq.(6)can be witten as H ( ' ) d min (9) Supposing that the impoed law of gaity and impoed Newton s second law can be witten as the following constant dimension factal foms a i k i (8) F () Afte soling these equations, the impoed law of gaity, and Newton s second law can be eached at once. F ma () Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

5 Neutosophic Sets and Systems, Vol. 3, 4 7 whee: and ae constants. Now we calculate the elated quantities accoding to the law of conseation of enegy. Fom Eq.(), the potential enegy of the small ball located at point is V () ( ) O ' Accoding to the law of conseation of enegy, we can get ( ) O' A And theefoe m' ( ) O ' (3) GM ' [ ] (4) ( R H) O' Now we calculate the elated quantities accoding to the impoed law of gaity and impoed Newton s second law. Supposing that the equation of olling line is y H (5) Fo the ball located at point, because d/ dt a (6) dt Theefoe ds d d a d (7) Accoding to the impoed law of gaity, the foce along to the tangent is Fa (8) O' Accoding to the impoed Newton s second law, fo point, the acceleation along to the tangent is Fa GM a ( ) ( ) (9) m / / O ' Fom Eq.(7), it gies d { [( H ) () GM ( R H y) ] / } / Substituting Eq.(5) into Eq.(), and fo the two sides, we un the integal opeation fom A to, it gies () H { [( H ) GM ( R ) ] / then the alue can be calculated by a method of numeical integal. } ( ) / / d The gien data ae assumed to be: fo Eath, GM= m 3 /s ; the adius of the Eath R= m, H=R/, ty to sole the poblem shown in Fig., find the solution fo the alue of B,and deie the impoed law of gaity and the impoed Newton s second law. Fistly, accoding to the oiginal law of gaity, the oiginal Newton s second law (i.e., let = in Eq.(), = in Eq.()) and the law of conseation of enegy, all the elated quantities can be calculated, then substitute them into Eq.(9), it gies =57.45 Hee, accoding to the law of conseation of enegy, it gies ' B =.767 7,while accoding to the oiginal law of gaity, and the oiginal Newton s second law, it gies B =.35 7,the diffeence is about 5.4 %. Fo the eason that the alue of Π is not equal to zeo, then the alues of and can be decided by the optimum seeking method. At pesent all the optimum seeking methods can be diided into two types, one type may not depend on the initial alues which pogam may be complicated, and anothe type equies the bette initial alues which pogam is simple. One method of the second type, namely the seaching method will be used in this pape. d Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

6 8 Neutosophic Sets and Systems, Vol. 3, 4 Fistly, the alue of is fied so let =,then seach the alue of,as =.46, the alue of Π eaches the minimum ;then the alue of is fied,and seach the alue of,as =.99989, the alue of Π eaches the minimum ;then the alue of is fied,and seach the alue of,as =.458, the alue of Π eaches minimum Because the last two esults ae highly close, the seaching can be stopped, and the final esults ae as follows =.99989,ε=.458,=37.33 these two fomulas ae as follows: F ma ' F, ; whee: and ae undetemined constants. As shown in Fig., supposing that a small ball fee falls fom point A to point C. Simila to the aboe deiation, when the small ball falls to point (point is not shown in Fig.), the alue of calculated by the undetemined Newton's second law and the law of gaity, as well as the alue of ' calculated by the law of conseation of enegy ae as follows: Hee the alue of Π is only 4% of Π. While accoding to the law of conseation of enegy, it gies ' B =.785 7,accoding to the impoed law of gaity and the impoed Newton s second law, it gies =.73 7, the diffeence is about.7 % only. B The esults suitable fo this eample with the constant dimension factal fom ae as follows The impoed law of gaity eads GM [ ( R H) ' O' / ' ( GM) ( GM) / y p ( R H y) ] / ' dy ' { [( R H y) / ' / ' ] y p } F () The impoed Newton s second law eads.458 F ma (3) The aboe mentioned esults hae been published on efeence []. Accoding to the esults fo the eample shown in Fig., it can be said that we could not ely on any epeimental data, only apply the law of conseation of enegy to deie the impoed law of gaity, and impoed Newton's second law; and demonstate that the oiginal Newton s law of gaity and Newton's second law ae all tenable appoimately fo this eample. So, can only apply the law of conseation of enegy to deie that these two oiginal laws o demonstate they ae tenable accuately in some cases? The answe is that in some cases we can indeed deie the oiginal Newton's second law and poe the oiginal Newton s law of gaity is tenable accuately. Now, in the case that a small ball fee falls (equialent to fee fall fom A to C in Fig. ), we deie the oiginal Newton's second law and poe the oiginal Newton s law of gaity is tenable accuately. Assuming that fo the oiginal law of gaity and Newton's second law, the elated eponents ae unknown, only know the foms of / ' ( GM) [ ( / ' ) ( / ') O' ( R H ) ( / ') ' Let, then we should hae: / ', and ( / ' ) ' ; these two equations all gie:, this means that fo fee fall poblem, by using the law of conseation of enegy, we stictly deie the oiginal Newton's second law F ma. Hee, although the oiginal law of gaity cannot be deied (the alue of may be any constant, cetainly including the case that =), we aleady poe that the oiginal law of gaity is not contadicted to the law of conseation of enegy, o the oiginal law of gaity is tenable accuately. Fo the eample shown in Fig. that a small ball olls along the inclined plane, in ode to obtain the bette esults, we discuss the aiable dimension factal solution with Eq.(4) that is established by the law of conseation of enegy diectly. Supposing that the impoed Newton s second law and the impoed law of gaity with the fom of aiable dimension factal can be witten as follows: F ma, k u ; F /, k u ; whee: u is the hoizon distance that the small ball olls ( u H ). ] Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

7 Neutosophic Sets and Systems, Vol. 3, 4 9 With the simila seaching method, the alues of k, k can be detemined, and the esults ae as follows u,.7 u The esults of aiable dimension factal ae much bette than that of constant dimension factal. Fo eample, the Π final, it is only.9% of Π (3.7). While accoding to the law of conseation of enegy, it gies ' B =.767 7,accoding to the impoed law of gaity and the impoed Newton s second law, it gies B =.777 7, the diffeence is about.93 % only. The esults suitable fo this eample with the aiable dimension factal fom ae as follows The impoed law of gaity eads F 3.7 u (4) The impoed Newton s second law eads F u ma (5) whee: u is the hoizon distance that the small ball olls ( u H ) Thee is anothe poblem should also be discussed. That is the impoed kinetic enegy fomula. As well-known, the kinetic enegy fomula has been modified in the theoy of elatiity, now we impoe the kinetic enegy fomula with the law of conseation of enegy. Supposing that the impoed kinetic enegy fomula is Ed m, k 3 u ;whee: u is the hoizon distance that the small ball olls ( u H ). With the simila seaching method, we can get: k , then the impoed kinetic enegy fomula with aiable dimension factal fom eads u E d m Because the effect of impoement is ey small (the 4 alue of Π is only impoed fom into ), theefoe these esults should be fo efeence only. 3 With the help of geneal elatiity and accuate epeimental data to deie the impoed Newton's fomula of uniesal gaitation We aleady point out that, accoding to Neutosophy and Quad-stage Method, aious esults can be eached. of. Hu Ning deied an equation accoding to geneal elatiity, with the help of Hu's equation and Binet s fomula, we get the following impoed Newton's fomula of uniesal gaitation [] 3G M mp F (6) 4 c whee: G is gaitational constant, M and m ae the masses of the two objects, is the distance between the two objects, c is the speed of light, p is the half nomal chod fo the object m moing aound the object M along with a cue, and the alue of p is gien by: p = a(-e ) (fo ellipse), p = a (e -) (fo hypebola), p = y / (fo paabola). It should be noted that, this impoed Newton's fomula of uniesal gaitation can also be witten as the fom of aiable dimension factal. Suppose 3G M mp 4 c It gies GMp 3 ln( ) / ln 4 c Fo the poblem of gaitational defection of a photon obit aound the Sun, M=.99 3 kg, = m, c= m/s, then we hae: The impoed Newton s uniesal gaitation fomula (Eq.(6)) can gie the same esults as gien by geneal elatiity fo the poblem of planetay adance of peihelion and the poblem of gaitational defection of a photon obit aound the Sun. Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

8 Neutosophic Sets and Systems, Vol. 3, 4 Fo the poblem of planetay adance of peihelion, the impoed Newton s uniesal gaitation fomula eads 3G M ma( e ) F (7) 4 c Fo the poblem of gaitational defection of a photon obit aound the Sun, the impoed Newton s uniesal gaitation fomula eads whee: w is a constant to be detemined. Now we detemine the alue of w accoding to accuate epeimental data. Fistly the poblem of deflection of photon aound the Sun as shown in Fig. will be soled with Eq.(9). The method to be used is the same as pesented in efeences [] and [3]..5 F (8) 4 whee: is the shotest distance between the light and the Sun, if the light and the Sun is tangent, it is equal to the adius of the Sun. The funny thing is that, fo this poblem, the maimum gaitational foce gien by the impoed Newton s uniesal gaitation fomula is.5 times of that gien by the oiginal Newton s law of gaity. Although the deflection angles gien by Eq.(6) and Eq.(8) ae all eactly the same as gien by geneal elatiity, they hae still slight deiations with the pecise astonomical obseations. What ae the easons? Accoding to uniesal elations, the answe is that the deflection angle not only is depended on the gaitational effect of the Sun, but also depended on the gaitational effects of othe celestial bodies, as well as the influences of sunlight pessue and so on. If all factos ae taken into account, not only geneal elatiity can do nothing fo this poblem, but also fo a long time it could not be soled by theoetical method. Theefoe, at pesent the only way to sole this poblem is based on the pecise obseations to deie the synthesized gaitational fomula (including the effects of othe celestial bodies and sunlight pessue) fo the poblem of deflection of photon aound the Sun. As well-known, the deflection angle gien by geneal elatiity o the impoed Newton's fomula of uniesal gaitation is as follows =.75 Fig. eflection of photon aound the Sun Supposing that m epesents the mass of photon. Because the deflection angle is ey small, we can assume that = ; thus on point (, y), its coodinate can be witten as (,y), then the foce acted on photon eads F F (3) / ( y ) whe:the alue of F is gien by Eq.(9). Because Hence m F dt F dy y c F dy (3) GM dy 6G M p dy c 3 3 ( y ) c ( y ) / 5/ Adding an additional tem to Eq.(8), it gies the synthesized gaitational fomula between the photon and the Sun as follows GMp wg M p F ( ) (9) c c Because 3 3 wg M p 5 c dy 7 ( y ) / (3) Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

9 Neutosophic Sets and Systems, Vol. 3, 4 dy 3 ( y ) /, dy 8 ( 7/ y ) 5 Theefoe Because 6 dy ( 5/ 4 y ) 3, GM 4G M p 6wG M p c c 5c tg c 3 3 By using the half nomal chod gien in efeence [], it gies Table shows the epeimental data of adio astonomy fo the deflection angle of photon aound the Sun (taken fom efeence [4]). Table. The epeimental data of adio astonomy fo the deflection angle of photon aound the Sun Yea Obsee Obseed alue / 969 G.A.Seielstud et al.77±. 969.O.Muhleman et al I.I.Shapio.8±. 97 R.A.Samak.57±.8 97 J.M.Hill.87± ± ±.5 p c GM ±. Then the deflection angle is as follows Now we choose the epeimental data in 975, it gies 4GM c w 5 (33) Then, we hae.76 φ.8 whee: is the adius of Sun. Because 4 GM c (34).857 w.4857 Taking the aeage alue, it gies w=.574 Then, it gies w ( ) 5 Thus the alue of w can be soled as follows w 5 ( ) (35) (36) Now we can detemine the alue of w accoding to the epeimental data. Thus, accoding to the epeimental data, the synthesized gaitational fomula can be decided. 4 Contadiction between the law of conseation of enegy and the law of conseation of momentum as well as the law of conseation of angula momentum Accoding to Neutosophy, any law may be in thee states: coect, wong, and it is coect unde cetain conditions. As well-known, unlike the law of conseation of enegy, the law of conseation of momentum and the law of conseation of angula momentum ae only coect unde Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

10 Neutosophic Sets and Systems, Vol. 3, 4 cetain conditions. Fo eample, consideing fiction foce and the like, these two laws will not be coect. Now we point out futhe that fo NNM the law of conseation of momentum as well as the law of conseation of angula momentum will be not coect unde cetain conditions (o thei esults contadict with the law of conseation of enegy). As well-known, in ode to poe the law of conseation of momentum as well as the law of conseation of angula momentum, the oiginal Newton's second law should be applied. Howee, as we hae made clea, the oiginal Newton's second law will not be coect unde cetain conditions, fo such cases, these two laws also will not coect. Hee we find anothe poblem, if the oiginal thee conseation laws ae all coect, theefoe fo cetain issues, the law of conseation of enegy and the othe two conseation laws could be combined to apply. While fo NNM, if the othe two conseation laws cannot be applied, how to complement the new fomulas to eplace these two conseation laws? The solution is ey simple: accoding to the law of conseation of enegy, fo any time, the deiaties of total enegy W (t) should be all equal to zeo, then we hae n d W ( t) n dt n,,3, (37) In addition, unning the integal opeations to the both sides of Eq.(3), it gies W ( )t = W ( t) dt t (38) Now we illustate that, because thee is one tuth only, een within the scope of oiginal classical mechanics, the contadiction could also appea between the law of conseation of enegy and the law of conseation of momentum. As shown in Fig.3, a man walks along the ca located on the hoizontal smooth ail, the length of the ca equals L, the mass of the man is m and the ca is m. At beginning the man and the ca ae all at est, then the man walks fom one end to the othe end of the ca, ty to decide the moing distances of the man and the ca. This eample is taken fom efeences [5]. Fig.3 A man walks along the ca located on the hoizontal smooth ail. As soling this poblem by using the oiginal classical mechanics, the law of conseation of momentum will be used, it gies m m Howee, at beginning the man and the ca ae all at est, the total enegy of the system is equal to zeo; while once they ae moing, they will hae speeds, and the total enegy of the system is not equal to zeo; thus the law of conseation of enegy will be destoyed. Fo this paado, the oiginal classical mechanics looks without seeing. In fact, consideing the lost enegy of the man and applying the law of conseation of enegy, the completely diffeent esult will be eached. As the oiginal law of conseation of momentum ( Const ) and the law of conseation of t angula momentum ( L t L Const ) ae not coect, we can popose thei impoed foms of aiable dimension factal. The impoed law of conseation of momentum: t ( is a constant o a aiable), and the impoed law of conseation of angula momentum: Refeences L t ( is a constant o a aiable). L [] Fu Yuhua, eiing Impoed Newton s Second Law and the Law of Gaity at One Time with Fom of Factal Fomula, Engineeing Science. 3,Vol.5,No.66, Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law

11 Neutosophic Sets and Systems, Vol. 3, 4 3 [] Fu Yuhua, Impoed Newton s fomula of uniesal gaitation, Zianzazhi (Natue Jounal), (), [3] C. Kittel et al, Tanslated into Chinese by Chen Bingqian et al, Mechanics, Beijing: Science ess, 979, [4] Liu Liao, Geneal elatiity, Beijing: Highe education pess, 987, [5] Xu Heing, Mechanics (eised edition), Shanghai: East China Nomal Uniesity ess, 998, Receied: May 7 th, 4. Accepted: May th, 4. Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law