New Newton Mechanics Taking Law of Conservation of Energy as Unique Source Law (Revised Version 3)

Transcription

1 New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law eised Vesion 3 u Yuhua CNOOC eseach Institute fuyh945@sina.com Abstact: 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. 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 and the oiginal law of gaity by using 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 u sing 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 pesen ted. 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. o 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. o 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: Uniqueness of tuth, law of conseation of enegy, unique souce law, New Newton Mechanics NNM Intoduction One of the deelopment tends of natual science is using fewe laws to sole inceasing poblems. In this pocess, some laws will play the inceasingly geat oles; while othes will play the smalle oles, o een disappea fom the anks of laws. 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.

2 Because the law of conseation of enegy is the most impotant one in natual sciences, it should play an inceasingly geat ole. o 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. 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 wellknown, 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. Taking Law of Conseation of Enegy as Unique Souce Law. eiing Oiginal Newton's Second Law and Oiginal Law of Gaity.. eiing Oiginal Newton's Second Law by Using Law of Conseation of Enegy In this section, only Newton's second law can be deied, but we hae to apply the law of gaity at the same time, so we pesent the geneal foms of Newton's second law and the law of gaity with undetemined constants fistly. Assuming that fo the law of gaity, the elated eponent is unknown, and we only know the fom of this fomula is as follows whee: is an undetemined constant, in the net section we will deie that its alue is equal to. Similaly, assuming that fo Newton's second law, the elated eponent is also unknown, and we only know the fom of this fomula is as follows ' ma

3 whee: is an undetemined constant, in this section we will deie that its alue is equal to. As shown in igue, supposing that cicle O denotes the Eath, M denotes its mass; m denotes the mass of the small ball teated as a mass point, A O is a plumb line, and coodinate y is paallel to AO. The length of AC is equal to, and O C equals the adius of the Eath. We also assume that it does not take into account the motion of the Eath and only consideing the fee falling of the small ball in the gaitational field of the Eath fom point A to point C. igue A small ball fee falls in the gaitational field of the Eath o this eample, the alue of which is the squae of the elocity fo the small ball located at point will be inestigated. To distinguish the quantities calculated by diffeent methods, we denote the alue gien by the law of gaity and Newton s second law as the alue gien by the law of conseation of enegy.,while denotes ' Now we calculate the elated quantities accoding to the law of conseation of enegy. om the law of gaity contained undetemined constant, the potential enegy of the small ball located at point is as follows V O ' Accoding to the law of conseation of enegy, we can get And theefoe O' A m' O ' GM [ ' O' ] Now we calculate the elated quantities accoding to the law of gaity and Newton s second law. o the small ball located at any point, we hae

4 We also hae Theefoe d / dt a dy dt d ady Accoding to the law of gaity contained undetemined constant, along the plumb diection, the foce acted on the small ball is as follows a O' om the Newton's second law contained undetemined constant, it gies a a m Then we hae / ' GM O' GM d { y / ' } / ' dy o the two sides of this epession, we un the integal opeation fom A to, it gies y p / ' GM y / ' dy GM / ' { [ y / ' / ' ] y p } / ' GM [ / ' / ' O' / ' ] Let, then we should hae: / ', and / ' ' ; these two equations all gie: ', this means that fo fee falling poblem, by using the law of conseation of enegy, we stictly deie the oiginal Newton's second law ma. ee, 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... eiing Oiginal Law of Gaity by Using Law of Conseation of Enegy In ode to eally deie the oiginal law of gaity fo the eample of fee falling poblem, we should conside the case that a small ball fee falls fom point A to point point is also shown

5 in igue though a ey shot distance the two endpoints of the inteal ae point A and point. As deiing the oiginal Newton's second law, we aleady eac h ] [ ' ' GM whee: ' O' o the eason that the distance of is ey shot, and in this inteal the gaity can be consideed as a linea function, theefoe the wok W of gaity in this inteal can be witten as follows W a whee, a is the aeage alue of gaity in this inteal, namely the alue of gaity fo the midpoint of inteal. Omitting the s econd ode tem of 4, it gies / W As the small ball fee falls fom point A to point, its kinetic enegy is as follows ] [ ' ' m Accoding to the law of conseation of enegy, we hae ' ' m W Substituting the elated quantities into the aboe epession, it gies ] [ / To compae the elated tems, we can each the following thee equations / All of these thee equations will gie the following esult

6 Thus, we aleady deie the oiginal law of gaity by using the law of conseation of enegy.. New Thee Laws of Motion and New Law of Gaity omula Ceated By Law of Conseation Of Enegy fo New Newton Mechanics The oiginal Newton's thee laws of motion ae as follows. Newton's ist 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. o 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 is = ma. The diection of the foce is the same as the diection of the acceleation. Newton's Thid Law of Motion: o eey action thee is an equal and opposite eaction. The oiginal Newton s law of gaity: The attactie foce betwee n two objects is as follows While 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 ist 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. o 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 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 factal: ma, whee: is a constant o a aiable. o 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 oi ginal Newton s thid law AB BA is as follows: AB BA, whee: is a constant o a aiable. o diffeent poblems, the foms of thid law may be diffeent.

7 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. o 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: 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. istly, 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 W = W 4 Accoding to LSM, fo the inteal [ t,t ],we can wite the following aiational pinciple: t W dt min t 5 Whee: min denotes the minimum alue of functional Π and it should be equal to zeo.

8 It should be noted that, in many cases W t is appoimate, and 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. o eample, fo inteal [, ], the following aiational pinciple can be gien accoding to the law of conseation of enegy: 6 W d 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. o 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. o 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 W can be edefined as follows: Q W = Q' 7 Substituting Eq.7into Eqs.5and6, 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.5o Eq.6, the equations deied by the condition of an etemum can be witten as follows: a i k i 8 Afte soling these equations, the impoed law of gaity, and Newton s second law can be

9 eached at once. 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 ig., 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. igue. A small ball olls fom A to B Supposing that cicle O ' denotes the Eath, M denotes 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, and O C equals the adius 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 conseation of enegy,then Eq.6can be witten as ' denotes the alue gien by the law of ' 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 ma whee: and ae constants.

10 Now we calculate the elated quantities accoding to the law of conseation of enegy. om 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 m' O ' 3 And theefoe GM ' [ ] 4 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 5 o the ball located at point, d/ dt a 6 Because dt ds d Theefoe d a d 7 Accoding to the impoed law of gaity, the foce along to the tangent is

11 a 8 O' Accoding to the impoed Newton s second law, fo point, the acceleation along to the tangent is a GM a 9 m / / O ' om Eq.7, it gies GM / d { } d / [ y ] Substituting Eq.5 into Eq., and fo the two sides, we un the integal opeation fom A to, it gies GM / / { } d / [ ] Then the alue can be calculated by a method of numeical integal. The gien data ae assumed to be: fo Eath, GM= m 3 /s ; the adius of the Eath = m, =/, ty to sole the poblem shown in ig., find the solution fo the alue of B,and deie the impoed law of gaity and the impoed Newton s second law. istly, 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 ee, 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 %. o the eason that the alue of Π is not equal to zeo, then the alues

12 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. istly, 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 ee 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 B =.73 7, the diffeence is about.7 % only. The esults suitable fo this eample with the constant dimension factal fom ae as follows The impoed law of gaity eads The impoed Newton s second law eads.458 ma 3 The aboe mentioned esults hae been published on efeence []. Accoding to the aboe esults, 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. o the eample shown in ig. 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

13 of aiable dimension factal can be witten as follows: k ma, u k u ; whee: u is the hoizon distance that the small ball olls u ; /,. 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. 4 o eample, the final Π 5.866, 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.93 % only. B =.777 7, the diffeence is about The esults suitable fo this eample with the aiable dimension factal fom ae as follows The impoed law of gaity eads u The impoed Newton s second law eads u ma 5 whee: u is the hoizon distance that the small ball olls u. 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 the hoizon distance that the small ball olls u. m Ed, u k 3 ;whee: u is

14 With the simila seaching method, we can get: k enegy fomula with aiable dimension factal fom eads 3, then the impoed kinetic u E d m Because the effect of impoement is ey small the alue of Π is only impoed fom into , theefoe these esults should be fo efeence only. 3 With the elp of Geneal elatiity and Accuate Epeimental ata to eie the Impoed Newton's omula of Uniesal Gaitation of. u Ning deied an equation accoding to geneal elatiity, with the help of u's equation and Binet s fomula, we get the following impoed Newton's fomula of uniesal gaitation [] 3G M c 4 mp 6 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 alu e 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 o 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 esul ts as gien by geneal el atiity fo the pobl em of planetay adance of peihelion and the pobl em of gaitational defection of a photon obit aound the Sun.

15 o the poblem of pl anetay adance of peihelion, the impoed Newton s uniesal gaitation fomula eads 3G M ma e 4 c 7 o the poblem of gaitational defection of a photon obit aound the Sun, the impoed Newton s uniesal gaitation fomula eads 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? 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 includin g 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 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 c c

16 whee: w is a constant to be detemined. igue 3. eflection of photon aound the Sun Now We etemine The Value Of W Accoding To Accuate Epeimental ata. istly the poblem of deflection of photon aound the Sun as shown in ig.3 will be soled with Eq.9. The method to be used is the same as pesented in efeences [] and [3]. 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 y 3 / Whee: The alue of is gien by Eq.9. Because m dt dy y c dy 3 ence GM dy 6G M p dy c 3 3 y c y / 5/ 3 3 wg M p 5 c dy 7 y / 3

17 Because dy 3 y /, Theefoe dy 5/ 4 y 3 GM G M p wg M p c 4 c c 3 3, dy 8 7/ y 5 6 Because tg c By using the half nomal chod gien in efeence [], it gies p c GM Then the deflection angle is as follows 4GM c w 5 33 Whee: is the adius of Sun. Because 4 GM c 34 Then, it gies w 5 35 Thus the alue of w can be soled as follows

18 w 5 36 Now we can detemine the alue of w accoding to the epeimental data. 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.A.Samak.57±.8 97 J.M.ill.87± ± ± ±. Now we choose the epeimental data in 975, it gies.76 φ.8 Then, we hae.857 w.4857 Taking the aeage alue, it gies w=.574 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 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 cetain conditions. o 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.

19 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. owee, 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. ee 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 ig.4, a man walks along the ca located on the hoizontal smooth ail, the len gth 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]. igue 4 A Man Walks along the Ca Located On the oizontal Smooth ail

20 As soling this poblem by using the oiginal classical mechanics, the law of conseation of momentum will be used, it gies m m owee, 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. o 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 t Const and the law of conseation of 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: is a constant o a aiable. L L t efeences. u Yuhua, eiing Impoed Newton s Second Law and the Law of Gaity at One Time with om of actal omula, Engineeing Science. 3,Vol.5,No.6, u Yuhua, Impoed Newton s fomula of uniesal gaitation, ianzazhi Natue Jounal,, C. Kittel et al, Tanslated into Chinese by Chen Bingqian et al, Mechanics, Beijing: Science ess, 979, Liu Liao, Geneal elatiity, Beijing: ighe education pess, 987, 5. Xu eing, Mechanics eised edition, Shanghai: East China Nomal Uniesity ess, 998, Appendi: Soling oblems of Adance of Mecuy s eihelion and eflection of hoton Aound the Sun with New Newton s omula of Gaity u Yuhua CNOOC eseach Institute, fuyh945@sina.com Abstact: Accoding to the new Newton's fomula of gaity the oiginal law of gaity plus a coection tem, i.e., impoed Newton's fomula of gaity, applying the methods of classical mechanics to sole the poblem of adance of Mecuy's peihelion and the poblem of deflection

21 of photon aound the Sun espectiely, and the esults ae the same as gien by geneal elatiity. ointing out that the futhe topic is based on deiing oiginal law of gaity and oiginal Newton s second law with law of conseation of enegy to deie the new Newton's fomula of gaity the impoed Newton's fomula of gaity with law of conseation of enegy. To ealize the pupose that patially eplacing elatiity and soling some poblems that cannot be soled by elatiity with the methods of classical mechanics. Key wods: Law of gaity, new Newton's fomula of gaity impoed Newton's fomula of gaity, adance of Mecuy's peihelion, deflection of photon aound the Sun, law of conseation of enegy Intoduction In efeences [, ], the new Newton s fomula of gaity the oiginal law of gaity plus a coection tem, i.e., the impoed Newton's fomula of gaity, was pesented; but it did not pesent the detailed pocess to sole the poblem of adance of Mecuy's peihelion and the poblem of deflection of photon aound the Sun espectiely with the methods of classical mechanics. While, in this pape the detailed pocess will be gien the esults ae the same as gien by geneal elatiity. The impoed Newton's fomula of gaity is as follows 3G M mp 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 with mass m moing aound the object with mass M along a cue, and the alue of p is gien by: p a -e fo ellipse, p a e - fo hypebola, p = y / fo paabola. Soling the poblem of adance of Mecuy's peihelion with new Newton s fomula of gaity In classical mechanics, acted by the cental foce, the obit diffeential equation Binet s fomula eads h u u" u m whee: u. As deiing Eq., it aleady gies h GMp 3 Substituting Eq. and Eq.3 into Eq., we hae the following equation of planet s moement aound the Sun 3GMu u" u 4 p c o ellipse, p a e -, thus the appoimate solution fo Eq.4 is as follows

22 GM 3GM u [ ecos ] 5 a e c a e c ence, the alue of fo adance of planetay peihelion fo one cicuit is as follows 3 4 a 6 T c e whee: T, a, and e ae obital peiod, semi-majo ais and eccenticity espectiely. Obiously, this esult is the same as gien by geneal elatiity. In addition, accoding to Eq., fo poblem of planetay motion aound the Sun, the impoed Newton's fomula of gaity eads 3G M ma e 4 c Soling the poblem of deflection of photon aound the Sun with new Newton s fomula of gaity As soling this poblem by using the impoed fomula of gaity, the method to be used is the same as pesented in efeences [3], in which the oiginal law of gaity was used. ig. eflection of photon aound the Sun Supposing that m epesents the mass of photon, fo the eason that it will be eliminated, so it is not necessay to gie its alue. As shown in ig., epesents the neaest distance between the photon and the cente of the Sun, fo the eason that the deflection is ey small, so actually the alue of is the same as the photon is not deflected. When the photon is located at, ythe alue of y is measued fom point in ig., the foce acted on photon is as follows 7 / y 3G M mp whee: y c y

23 Because m dt dy y c dy Theefoe GM dy 6G M p dy c y c y 3/ 3 5/ Afte calculating, it gies GM 4G M p 8 c c 3 3 ence, the deflection angle is as follows GM 4G M p tg 9 c c c 4 3 While, the alue of should be detemined by iteation method. Befoe detemining the alue of, fistly we will alidate that the alue of the second tem in Eq.9 is equal to the alue of the fist tem, that means that the deflection gien by the second tem in Eq.9 is equal to that gien by the fist tem. As soling poblem of deflection of photon aound the Sun with geneal elatiity, the photon s obit is a hypebola, and its equation is as follows GMu sin u u cos c whee: u ence, the ecipocal of the half nomal chod p is as follows u p / GM c Substituting this alue of the half nomal chod p into Eq.9, it gies 4GM 4GM c c s whee: s is the adius of the Sun. Thus, the alue of the second tem in Eq.9 is eally equal to the alue of the fist tem. Now we detemine the alue of in Eq.9 by iteation method.

24 Suppose KGM 3 c In ode to apply iteation method, the elationship between and p should be gien. Consideing two staight lines, the fist one is, the second one is passing though the oigin O and the fist quadant, and it makes an angle of / with the positie diection of Y ais; the alue of the half nomal chod p is equal to the distance between the oigin O and the intesection of the two staight lines. Supposing that the intesection of the two staight lines is the point it is not shown in ig., then its coodinates ae, p. om the tiangle fomed by the thee points of oigin O,, and, it gies sin p p 4 Now, consideing the esult of the alue of gien by the oiginal law of gaity, it gies K ee, the deflection angle is as follows GM c om Eq.4, its coesponding half nomal chod p is as follows p c GM Substituting the alue of p into Eq.9, it gies 6GM c Namely K 6

25 Similaly, the alues of K, K 3, and the like ae as follows: , 4.4, 3.88, 4.95, , 4.34, , 4.59, 3.997, 4.5, , 4.4, , 4., 4., 4.; finally it gies K 4 This esult is also the same as gien by geneal elatiity. Accoding to Eq., fo poblem of deflection of photon aound the Sun, the impoed Newton's fomula of gaity eads.5 4 whee: is the shotest distance between the photon and the Sun, if the light and the Sun is tangent, it is equal to the adius of the Sun. The inteesting fact is that, fo this poblem, the maimum gaitational foce gien by the impoed Newton s fomula of gaity is.5 times of that gien by the oiginal Newton s law of gaity. 3 uthe topic In efeences [4-6], the oiginal law of gaity and the oiginal Newton s second law hae been deied with law of conseation of enegy, based on this, the futhe topic is to deie the new Newton's fomula of gaity the impoed Newton's fomula of gaity, i.e., Eq., with law of conseation of enegy. And, finally, to ealize the pupose that patially eplacing elatiity and soling some poblems that cannot be soled by elatiity with the methods of classical mechanics. efeences u Yuhua, Impoed Newton s fomula of uniesal gaitation, ianzazhi Natue Jounal,, loentin Smaandache, V. Chistianto, u Yuhua,. Khapko, J. utchison. Unfolding the Labyinth: Open oblems in hysics, Mathematics, Astophysics, and Othe Aeas of Science. eis-hoeni, 6, Kittel C., Knight W.. and udeman M. A., Mechanics. Bekeley hysics Couse Vol., McGaw-ill, u Yuhua, eiing Impoed Newton s Second Law and the Law of Gaity at One Time with om of actal omula, Engineeing Science. 3,Vol.5,No.6, u Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law, Neutosophic Sets and Systems, Vol.3, 4 6 u Yuhua, New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law, Science Jounal of hysics, Volume 5, Aticle I sjp-3, ages, 5, doi:.737/sjp/3