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

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1 New Newton Mechanics Taking Law of Conseation of Enegy as Unique Souce Law (Reised Vesion) Fu Yuhua CNOOC Reseach Institute 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. Examples 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 example 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 example 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 exclude that accoding to othe theoies o accuate expeiments to deie the laws o fomulas to sole some specific poblems. Fo example, 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 expeimental 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 expessions of enegy. Fo example, 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. 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. 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 example 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 The oiginal Newton's thee laws of motion 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 extenal 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: The attactie foce between two objects is as follows F ()

3 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 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 extenal 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 factal: F ma, 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 expeimental 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 appoximate. 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 example, 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 example only, ae deied simultaneously by law of conseation of enegy. Fistly, the aiational pinciples established by the law of conseation of enegy can be gien with least squaes method (LSM).

4 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 () = W (t) (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: t R W dt min (5) t 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 appoximate, 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 example, fo inteal [ x, x ], the following aiational pinciple can be gien accoding to the law of conseation of enegy: x x R W dx min (6) 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 example, 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

5 conseation of enegy as Q ',then the alue of R W can be edefined as follows: Q R W = Q' (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 extent 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 extemum 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 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 example. 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. Figue. 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 x is othogonal to O A, coodinate y

6 is othogonal to coodinate x (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 example, 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 conseation of enegy,then Eq.(6)can be witten as,while ' denotes the alue gien by the law of H ( ' ) dx 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 F () D F ma () whee: D 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 () ( D ) D O ' Accoding to the law of conseation of enegy, we can get ( D ) D O' A m' ( D ) D O ' (3) And theefoe GM ' [ ] (4) D D D ( R H) O'

7 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 x H (5) Fo the ball located at point, d/ dt a (6) Because dt ds dx Theefoe d a dx (7) Accoding to the impoed law of gaity, the foce along to the tangent is Fa (8) D O' Accoding to the impoed Newton s second law, fo point, the acceleation along to the tangent is Fa GM a ( ) ( ) (9) m / / D O ' Fom Eq.(7), it gies GM / d { } dx () D/ [( H x) ( R H y) ] Substituting Eq.(5) into Eq.(), and fo the two sides, we un the integal opeation fom A to, it gies

8 x GM / / { } ( ) dx D / [( H x) ( R x) ] () H 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 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 D = 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 D 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. Fistly, the alue of D is fixed so let D =,then seach the alue of,as =.46, the alue of Π eaches the minimum ;then the alue of is fixed,and seach the alue of D,as D =.99989, the alue of Π eaches the minimum ;then the alue of D is fixed,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 D=.99989,ε=.458, =37.33 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 B =.73 7, the diffeence is about.7 % only.

9 The esults suitable fo this example with the constant dimension factal fom ae as follows The impoed law of gaity eads 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 example shown in Fig., it can be said that we could not ely on any expeimental 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 appoximately fo this example. 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 the oiginal law of gaity by using the law of conseation of enegy. Assuming that fo the oiginal law of gaity and Newton's second law, the elated exponents ae unknown, only know the foms of these two fomulas ae as follows: F, D whee: D and D ae undetemined constants. D' F ma ; 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: GM [ D D ( R H) ' D O' ]

10 y p / D' ( GM ) ( R H y) D / D ' dy ( GM ) / D ' { [( R H y) D / D' D / D' ] y p } / D' ( GM ) [ ( D / D' ) ( D / D') O' ( R H) ( D / D') ] ' Let, then we should hae: / D', and D ( D / D') ; these two equations all gie: D ', 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 D may be any constant, cetainly including the case that D=), 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. In ode to eally deie the oiginal law of gaity fo the example of fee falling body, we should conside the case that a small ball fee falls fom point A to point (point is not shown in Fig.) though a ey shot distance Z (the two endpoints of the inteal Z ae point A and point ). As deiing the oiginal Newton's second law, we aleady each GM [ D ( R H Z) ( R H ) ' D D ] whee: R H Z O ' Fo the eason that the distance of Z 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 F whee, a Z R H Z) ( D Z F a is the aeage alue of gaity in this inteal Z midpoint of inteal Z. Omitting the second ode tem of Z, it gies, namely the alue of gaity fo the

11 W ( R H Z RH RZ HZ ) D/ As the small ball fee falls fom point A to point, its kinetic enegy is as follows m ' D D ( R H ) ( R H Z) [ D ( R H RH RZ HZ) D ] Accoding to the law of conseation of enegy, we hae W m' To compae the elated expessions, we can each the following thee equations D D / D Z ( R H ) D ( R H Z) D All of these thee equations will gie the following esult D Thus, we aleady deie the oiginal law of gaity by using the law of conseation of enegy. Fo the example 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: ; k F ma, u F /, k u ; whee: u is the hoizon distance that the small ball olls ( u x H ). 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 Fo example, 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 B =.777 7, the diffeence is about

12 .93 % only. The esults suitable fo this example with the aiable dimension factal fom ae as follows The impoed law of gaity eads (4).7 u F 3 The impoed Newton s second law eads u F ma (5) whee: u is the hoizon distance that the small ball olls ( u x 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 x 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 alue of Π is only impoed fom into ), theefoe these esults should be fo efeence only. 3 With the Help of Geneal Relatiity and Accuate Expeimental Data to Deie the Impoed Newton's Fomula of Uniesal Gaitation 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 [] F 3G M c 4 mp (6)

13 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 /x (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 D 4 c It gies GMp D 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: D. 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. Fo the poblem of planetay adance of peihelion, the impoed Newton s uniesal gaitation fomula eads F 3G M ma( e 4 c (7) ) Fo the poblem of gaitational defection of a photon obit aound the Sun, the impoed Newton s uniesal gaitation fomula eads F.5 4 (8) 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 maximum gaitational foce gien by the impoed

14 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 exactly 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 (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 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 whee: w is a constant to be detemined. Figue. Deflection of photon aound the Sun Now We Detemine The Value Of W Accoding To Accuate Expeimental 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].

15 Supposing that m epesents the mass of photon. Because the deflection angle is ey small, we can assume that x= ; thus on point (x, y), its coodinate can be witten as (,y), then the foce acted on photon eads F x F (3) / ( y ) Whee: The alue of F is gien by Eq.(9). Because m x F dt x F x dy y c F dy x (3) Hence GM dy 6G M p dy x c 3 3 ( y ) c ( y ) / 5/ 3 3 wg M p 5 c dy 7 ( y ) / (3) Because dy 3 ( y ) /, Theefoe dy ( 5/ 4 y ) 3, dy 8 ( 7/ y ) 5 6 GM 4G M p 6wG M p x c c 5c 3 3 Because tg x c By using the half nomal chod gien in efeence [], it gies

16 c p 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 w 5 ( ) (36) Now we can detemine the alue of w accoding to the expeimental data. Table shows the expeimental data of adio astonomy fo the deflection angle of photon aound the Sun (taken fom efeence [4]). Table. The expeimental 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 D.O.Muhleman et al I.I.Shapio.8±. 97 R.A.Samak.57±.8 97 J.M.Hill.87± ± ± ±.

17 Now we choose the expeimental 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 expeimental 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. Fo example, 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

18 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 example is taken fom efeences [5]. Figue 3 A Man Walks along the Ca Located On the Hoizontal Smooth Rail 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 paadox, 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: constant o a aiable), and the impoed law of conseation of angula momentum: t ( is a L L t

19 ( is a constant o a aiable). Refeences. Fu Yuhua, Deiing Impoed Newton s Second Law and the Law of Gaity at One Time with Fom of Factal Fomula, Engineeing Science. 3,Vol.5,No.6, Fu Yuhua, Impoed Newton s fomula of uniesal gaitation, Zianzazhi (Natue Jounal), (), C. Kittel et al, Tanslated into Chinese by Chen Bingqian et al, Mechanics, Beijing: Science ess, 979, Liu Liao, Geneal elatiity, Beijing: Highe education pess, 987, 5. Xu Hexing, Mechanics (eised edition), Shanghai: East China Nomal Uniesity ess, 998, 为了对自由落体问题真正导出原有的万有引力定律, 需要考虑小球从 A 点自由下落一段极短距离 Z, 到达端点 时 ( 图中未画出 ) 的情况 在导出原有的牛顿第二定律时, 我们已经得到 GM [ D ( R H Z) ( R H ) ' D D 式中 : R H Z O ' 由于 Z 极短, 在此区间引力可视为线性变化, 所以在此区间引力所做功 W 为 W F a Z R H Z) ( D Z 式中, F 为区间 Z 的引力平均值, 亦即区间中点的引力值 a 略去 Z 的二次项可得 W ( R H Z RH RZ HZ ) D/ 当小球下落至区间 Z 的端点 时, 其动能为 m ' D D ( R H ) ( R H Z) [ D ( R H RH RZ HZ) 根据能量守恒定律, 应有 W m' ] D ]

20 对比有关的公式, 可以得到下面的三个等式 D D / D Z ( R H ) D ( R H Z) D 从这三个等式都可以得到 D 于是我们就根据能量守恒定律导出了原有的万有引力定律