A GAUGE-INVARIANT RELATIVISTIC THEORY OF THE RUTHERFORD-SANTILLI NEUTRON

Transcription

1 aonic Jounal GUGE-INRINT RELTIISTIC TEORY OF TE RUTERFORD-SNTILLI NEUTRON.O.E. nialu Institut fo asic Rsach Lipopo Stt F... Maitaa buja. & Dpt. of hysics & stonoy Univsity of Nigia Nsukka Nigia -ass: nascinc@aol.co ib@aol.co Rciv ugust 6 Rvis Spt. bstact W psnt a gaug-invaiant lativistic constuction of ou ali non- lativistic thoy of th Ruthfo-Santilli nuton fin as th stuctu of th nuton as a boun stat of on poton an on lcton bas on haonic chanics M n p in th fo M but foulat as a supflui phas of a copss -ato n p of th lcton which chaactizs a Coop-lik paiing M an a nw assiv nutal spin- paticl in s configuation aoun th poton p. ft a viw of th non-lativistic thoy its gaug invaianc is foulat in such a way that th tansition fo p Coulob intaction potntial C to p ulthn intaction potntial M /[ ] is scib by a gnal nonlina fist-o iffntial Riccati s quation fo whos xact solution tins th atio of th two asss M fining. Finally a gaug-invaiant lativistic thoy is constuct fo th of th usual Diac quation fo p n p M syst consisting p lativ otion an an iso-iac quation fo lativ otion iv fo consvation of an iso-cunt nsity whos xact solution fo th ass of is us without ajustabl constants to pict th ± asss of th nuton an th oth bs of th S bayon octt Λ Σ Ξ. Goo agnt btwn th pict ass atios an xpintal ata is obtain. CS Nos..65.z 7.65.n 67.5 Fi Copyight aonic ss Inc. al abo FL

2 . INTRODUCTION In a cnt pap[] w hav psnt a non-lativistic thoy of th Ruthfo-Santilli nuton fin by Santilli[] as th stuctu of th nuton as a boun stat of on poton an on lcton bas on haonic chanics M n p M [s Fig. a]. owv w chaactiz th stuctu in a nw fo n p [s Fig. b] as a supflui phas of a M copss -ato fo by Coop-lik paiing lcton an a nw assiv nutal spin- paticl of th call spinion in supconuctivity thoy[] in a clos s shll configuation aoun th poton p tigg. f p tigg. R.. p a c b Fig.: a Stuctu of th nuton[] n p M whn th su of th classical lctic chag aii of an p is qual to f un copssion; an b th nw stuctu[] n p showing utual ovlapping of th M wavpackts of th pai aoun th poton p.

3 Th non-lativistic thoy of th n p M syst in Fig.b will b viw in Sc. in o to unsco th fatus that ak it quivalnt to th n p syst in Fig. a an also tin th signatu of th nw paticl u to th xpct isotopic lifting of unitay syty SU S U qui by haonic chanics. This will involv an xplicit constuction of th isounit of haonic chanics lating factionally-chag SU quaks to intgally-chag iso-paticls in gnal lptons in paticula. W shall s that sinc th isotopic lifting SU S U fo factionally chag quaks to intgally chag iso-paticls is not uniqu th al pobl is to istinguish btwn physical an athatical constitunts in a oh-lik ol of stuctu which woul flct th wll-stablish M paalllis btwn olcula focs ivabl fo noal p syst an nucla focs iv fo th copss syst as inicat in Fig.. Moov th justification fo th intouction of th nw n p syst lis in its siilaity to ou ali nonlativistic thoy[] of Coop paiing of lctons C M M z aoun a copp ion Cu tigg in th cupat supconuctos in z th fo C Cu M. This appoach l in f.[] to a piction of th citical tpatus fo th supconucting phas tansition in goo agnt with xpintal ata. Subsquntly th basic wav quations of th thoy w vifi by nialu an Santilli[5] an foun to b consistnt with th axios of haonic chanics. owv sinc a nub of ajustabl paats a intouc in th non-lativistic foulation an in o to liinat ths ajustabl paats so as to ak th thoy o pictiv w intn in this pap to tak th nxt stp of constucting a gaug-invaiant non-lativistic thoy of th Ruthfo-Santilli nuton in Sc. an thn gnalizing it to a gauginvaiant lativistic thoy in Sc.. s th pobl of istinguishing btwn physical an athatical haonic constitunts within a oh-lik ol of stuctu lis in fining igoous analytical tools fo non-ptubativ tatnt of th tansition fo th static Coulob lctoagntic p bining potntial ngy C / in th noal -ato to a stong

4 Fig. : aalllis btwn olcula an nucla focs appopiatly scal in ngy. c / h bining ulthn potntial ngy Mc /[ ] in th copss -ato th objctiv of th gaug invaianc pincipl in Sc. is to fin an xpssion fo th pogssiv gnalization  of th usual ti-coponnt of th lctoagntic potntial.this will b achiv by xploiting on on han th analogy in f.[] btwn th ikhoffian function q& p& of haonic

5 chanics an th Gibbs function of classical thoynaics an on th oth han th lativistic gnalization of th Ginzbug-Lanau quation[6] by iggs[7] in th fawok of Yang-Mills-iggs gaug thoy[7] to xpss th ffctiv foc p & q as a functional of th ffctiv potntial ngy. This will la in Sc. to a non-lina fist-o iffntial Riccati squation[8] fo th ffctiv ulthn potntial ngy whos xact solution will pscib a pcis lation btwn th asss M fining. noth pobl which will ais in th gaug-invaiant lativistic thoy in Sc. is how stating fo th usual Diac quation fo th lativ p otion chaactiz by a spin fil ψ x of ass M to constuct an xactly solubl iso-diac quation fo lativ p otion. This will b solv by iving th qui iso-iac quation fo consvation of an iso-cunt nsity J x ψ x γ i t ψ x that inclus a convctiv cunt [9] gnat via a gaug pincipl that lats th ulthn potntial to a iggs scala fil of ass. Th solution of th iso-iac quation in Sc. fo th ass will b us to pict th asss of th nuton an oth bs of th SU bayon octt. Finally in Sc. 5 w shall iscuss th sults an aw th attnant conclusions.. REIEW OF TE NON-RELTIISTIC TEORY ND RERESENTTION OF ISO-URKS S LETONS. Rviw of th Non-Rlativistic thoy Lt us now poc to viw th non-lativistic thoy of th Ruthfo-Santilli nuton psnt in f.[] in o to systatiz notations as wll as ak this pap slf-contain. Th non-lativistic thoy of th p M syst is bas on a pai of non-lativistic wav quations in Nabu psntation : p Ψ STΨ Ê Ψ * τ ψ Ψ *. ψ 5

6 wh τ an S scal tansfoation of spac-ti ST cooinats: ct bct b such that isolativistic tansfoation law hols in th fo: c b t' b x' y' z' c b t b x y z b b bing paats psnting th ffcts of xtnal pssu an tpatu an c th sp of light in vacuu an * b is an ffctiv uc ass p of th lcton. Explicitly in Eq.. * T ψ ψ T T. is th isotopic lifting opato of aonic Mchanics intouc in f.[] wh ψ * but ψ * ψ Z so that ψ * ψ T Z whil ψ * T an hnc th chag on th ψ paticl ψ ψ is plt by an aount Z whil th chag on th paticl ψ appas to vanish altogth. Fo this ason ψ was intifi in f.[] with a own-spin lcton whil ψ was intifi with an up-spin nutal assiv paticl call spinion in supconuctivity thoy[]. ccoingly. las to th pai of s- wav quations h ψ E C ψ ψ. fo th lativ p otion an * b h * * STψ E. ψ ψ. fo th lativ p otion. C / is th usual p Coulob intaction potntial ngy in th noal -ato an is th ffctiv p ulthn intaction potntial ngy fin as a paatization of th noaliz Coulob potntial an * th non-potntial t u to th ovlapping of ψ an ψ : 6

7 7 ] / xp[ * * * c Mc E b h ψ ψ ψ ψ..5 It was also stat in f.[] that th total ngy of th stuctu ol p n givn in f.[]: c b b E E c b b E p kin p S T ].6 woul b qual to th total ngy of M p n givn in f.[]: ] [ ] [ ] [ ] [ c b b c E E c b b E E c E p o kin p kin o T.7 wh bing th st asss of an un two conitions. Th fist conition aiss fo th fotuitous cancllation of th uc st-ass of th lcton by its non-lativistic bining ngy / R E in th configuation shown in Fig. a at p spaation of o / / c c f R π h. Th scon is ith that th kintic ngy kin E of th lcton shoul vanish upon th contact of its classical lctic chag sph with that of th poton at st o ls that its lag valu / Mc c E kin π shoul b absob in th ulthn potntial in Eq..5. In this pap w choos to absob kin E in th ulthn potntial so that Eq.. ay now b wittn as a non-lativistic wav quation fo otion of th nutal assiv spin- paticl lativ to p in th o xplicit fo: * * ] ] / xp[ / Mc c E c E E b kin kin π ψ ψ h h.8

8 Finally with this valu of M th ol ts th quint of having a iniu siz of th ulthn potntial hol fo a boun stat to b fo. Moov th st ass of th poton is givn quit accuatly by th quation: / / α p π α / hc /7.9 which is analogous to th xpssion iv in f.[] fo th citical tpatu of th supconucting phas tansition bas on a siila stuctu ol fo z Coop paiing C Cu in th cupat supconuctos. M owv vn with th quality of th total ngis of n p an n p M th pnnc of both ols on a ath lag nub of piical paats M b b aks it is ncssay to viw how th signatus of an coul b iv fo thos of th SU quaks in any chaactization of th tansition fo th noal to th copss syst bas on haonic chanics... Rpsntation of Iso-uaks as Lptons gnalization of th unitay syty goup SU to th isounitay syty goup S U is qui by haonic chanics. This quint will now b viw fo th pupos of slcting fo sval possibilitis an isotopic lnt fo th lifting SU S U incopoating th pojction opato popty T T of th non-local isotopic lifting opato fin in Eq... W bgin by calling that th convntional Gll-Mann-Zwig[] quaks us a fin by th convntional Li-algbaic stuctu of th SU gnatos if i j k...8.a i j j i ijk k with factional bayon nub an lctic chags q q 8 q /..b qustion aiss thfo as to whth on can slct an isotopic lnt G so that th isotopic lifting SU S U fin by a Liisotopic gnalization of Eq..a in th following way q 8

9 9 k k G G G if G G k ijk i j j i.a will chaactiz iso-quaks cosponing to th physical lpton tiplt ν with intgal fion nub l an lctic chags l i.. / 8 l l..b n affiativ answ to this qustion follows if w us ou xpinc with th pocu fo isotopic lifting of th SU algba of Coop pai cation an annihilation opatos in supconuctos s f.[] to slct an isotopic lnt G incopoating th pojction opato T in th following way: T G T G.. W vify ictly that this las to th si sult. / / 8 8 l q T T T T G G povi that w slct th ignvalu T of th pojction opato T cosponing to th noal nutino stat. ll oth iso-quak stats chaactiz utations of th nutino. In paticula whn T w obtain th intgal iso-quak chags iv by Santilli [] bas on a iffnt slction of a sytic atix isotopic lnt

10 g iag g g g such that k fo k 8 an hnc th lctic chag hav th fos: gg g g 8 g g g g gg g 8. whn g 6 g g. This spctu of intgal chags a copaabl with thos of th an-nabu quaks[]. t this junctu it is ptinnt to call that aut s constuction of haon ultiplts[] was bas on witing th quality of th quak an lpton fion nub an lctic chag pojction opatos ali obsv by nialu[5]..5 l l q q in th fo l l l l q q to fin that l l l q an l l q..6 Fo this aut conclu that quaks a obtain fo lptons by shifting / of th fionic chag to th lctic chag. It is now appant that such a shifting is in achiv though th isotopic lifting tansfoation of haonic chanics scib abov an las to intification of iso-quaks with th lptons. owv whas th quaks oby th Gll-Mann-Nishijia lation I Y with q I Y q.7 q

11 th iso-quaks i.. th lptons ust oby th appopiat lation qui by isotopic lifting. In paticula contay to aut s poposal to kp th sa I assignnts fo lptons w fin T I G IG.8 which inicats a utation of th usual isospin assignnt to th lpton tiplt; an in lik ann T Y q G YqG.9 is iffnt fo aut s assignnt but as xpct w cov q I Y q l.. Consquntly whil ou nuton stuctu ol n p M is copatibl with iso-quak thoy it is funantally iffnt fo but analogous to aut s ol of th nuton n ~ p ν as fa as intification of with th anti-nutino ν is concn. In this sns w a inclin to ag with Santilli[pivat counication] that h os not accpt th xistnc of nutinos bcaus h has shown that whn haons a psnt as xtn objcts all physical laws incluing that on th consvation of ngy an angula ontu can b vifi without any n fo conjctuing th xistnc of nutinos s It is ncssay thfo to pob o ply into th natu of th nw paticl by constucting th nonlativistic thoy in a gaug-invaiant fawok.. GUGE-INRINT RECONSTRUCTION OF TE NON-RELTIISTIC TEORY W hav stat in Sc. that th al pobl of istinguishing btwn physical an athatical haonic constitunts within a ohlik ol of stuctu lis in fining igoous analytical tools fo nonptubativ tatnt of th tansition fo lctoagntic Coulob

12 bining focs in noal atoic systs to stong haonic bining focs in copss atoic systs. To this n w poc in this sction to constuct th non-lativistic thoy of th n p M syst by highlighting its gaug invaianc in such a way that th pogssiv gnalization of th Coulob potntial to th ulthn potntial can b scib by a class of nonlina fist-o iffntial Riccati s quations[8] whos xact solutions will yil analytical xpssions of gnal intst an nabl us to lat so of th unknown paats in Eq..7 to ach oth. W bgin by witing th convntional Schoing quation fo th noal -ato in th fo: h / c E o ψ ψ..a This aks vint th fact that th static Coulob p intaction is tin by th ti-coponnt of th lctoagntic fouvcto potntial C / co /. Consquntly if w xpss th intpaticl Coulob foc C / / as a functional of th potntial ngy C by liinating xplicit -pnnc btwn C / an C th sult woul b a non-lina fist-o iffntial quation: C C o /..b This is a spcial cas of Riccati s quation[8] which suggsts that an obvious stp to tak in o to achiv a pogssiv gnalization of th Coulob potntial to th ulthn potntial in th isoschoing quation fo th copss syst b h * * E ψ ψ.a is to plac Eq..b by th ost gnal fo of th Riccati s quation R o / ζ κ.b K R an ζ κ bing constants in gnal functions of. Not that th ivativ an nonlina pats { / } of Eq..b ay b

13 -intpt as appopiat coponnt of th SU Yang-Mills gaug fil[7] a a a ν abc b c Fν g ε ν ν. x x wh g is a coupling constant. Not also that th atix Riccati s quation has an unlying Li-aissibl algbaic stuctu[6] which aks.b an xplicit unquivocal alization of th iso-athatical stuctu of haonic chanics. Eqs..a an.b fin thfo th gnalization w a aft. n altnativ way of looking at th gnalization lis in th analogy btwn on on han th ikhoffian function p& q& of haonic chanics an Gibbs f ngy of classical thoynaics iscuss in f.[] an on th oth han iggs gnalization of th Ginzbug-Lanau quation fo th o paat in supconuctvity thoy[6] in th fawok of Yang-Mills-iggs gaug thoy[7]. W call as iscuss in f.[] that an xtnsion of th analogy btwn classical thoynaics an classical chanics las to a cosponnc btwn th Gibbs f ngy of classical thoynaics as wll as its thoynaic iffntial lations: G T U TS G ST;. G / S G / T an th ikhoffian function of Santilli s haonic chanics as wll as th associat canonical quations of otion: p& q& qp & qp& qp& pq& ;.5 q / p& p / q&. Thfo bsis th aition of q & p qp& as xtnal t to th ailtonian q p to gt th ikhoffian th ikhoffian function p& q& iffs futh fo q p in th fact that it is a function of th vlocity q& an of th at of chang of ontu p& quivalnt to th foc q. Consquntly fo th pupos of pscibing th natu of th scon-o phas tansition fo th noal -ato stat to its copss supflui stat via a pogssiv gnalization of th Coulob intpaticl foc psnt by p & / Eq..b is a statnt of a gnal hypothsis cf th Ginzbug-Lanau xpansion of th f ngy as a functional of th o paat[6] that th ffctiv

14 intpaticl foc is a functional of th potntial ngy. Consquntly a futh gnalization of Eq..b to inclu high o ts is concivabl: fo xapl w coul as in iggs gnalization of th Ginzbug-Lanau thoy in th fawok of Yang-Mills-iggs gaug thois of spontanous syty baking xpan th foc to quatic o : K R K T..6 This woul ak th attainnt of asyptotic fo anifst itslf in th sa way as spontanous syty baking as th sktchs of / vsus in Fig. suggst. K T a b Fig. : Gaphs of / vsus fo a Coulob foc in Eq..b an b th gnalization in Eq..6. Lt us now tun to analytically xain th physical contnt of th gnalization fo.b to.b by obsving that whn K R a constants inpnnt of th Riccati quation.b can b

15 5 intgat xactly as shown in ppnix to fin a gnal solution of th fo: R R R R R R xp.7 wh is a constant of intgation. This is quivalnt to th stana fo of th ulthn potntial ] / xp[ c Mc h.8 povi that R R giving < R ;.9 an hnc sinc in units such that c h R R R R M R. Th abov foal ivation of th ulthn potntial is by vitu of th fact that it is abl to pict th ipotant atio / M vintly o satisfactoy than th a hoc psciption of th ulthn potntial via th paatization in Eq..5 vn though th two

16 appoachs coplnt ach oth. Th significanc of this atio will g in th lativistic thoy to which w now tun.. CONSTRUCTION ND ERIFICTION OF GUGE INRINT RELTIISTIC TEORY. Constuction Tchnically as shown in 996 by Santilli[7] fo th n p M syst th constuction of an iso-iac quation fo a copss -ato woul qui th intouction of th isotopis of th Minkowski spacti of th oinca syty an of spcial lativity with th consquntial aitional piical paats fo an up-to-at suay s owv bcaus of th foulation of th ulthn potntial fo th non-lativistic thoy in th fo of a fist-o nonlina iffntial Riccati s quation so novl physical insight woul b gain in this sction by constucting an xactly solubl gaug-invaiant lativistic wav quation fo th n p M syst involving only th two paats M with M / in th ulthn potntial. This can b on ath siply by using a Lagangian foulation in which th lctoagntic intaction is intouc via an isotopic lifting tansfoation of th convntional gaug-invaiant intaction t x x x x lating th lativ p otion in J J an xtnal lctoagntic fil x to th lativ p otion in an xtnal ffctiv fil x. Th qui lativistic an isolativistic wav quations woul thn b thos that consv th spctiv cunt an iso-cunt nsitis J x ψ x γ ψ x an J x ψ x γ i t ψ x which w now poc to constuct. W stat with th convntional lativistic Diac quation fo th lativ p otion in an xtnal lctoagntic fil: 6

17 i i h γ c ψ.a hc wh th ti-coponnt of th lctoagntic fou-vcto potntial psnting th static p Coulob intaction satisfis as bfo th siplst fo of Riccati s quation: /..b Fo th lativ p otion w natually psu that th ticoponnt  of th ffctiv ulthn potntial  is tin as bfo by th coplt fo of th Riccati quation: / ζ κ. ζ κ bing constants in gnal functions of. owv w now convt th ulthn potntial  into a iggs scala fil by using th wll-known stana tansfoation that convts th nonlina fist-o Riccati Eq..b into a lina scon-o iffntial quation fo s p. of iaggio[8] : log -.a wh. Not that this tansfoation ay b wittn as a Wyl-lik gaug pincipl in th intgal fo: xp /..b Fo.a w fin. so that on substitution in Eq.. th ts in isappa an hnc on ultiplying th sulting quation though by / w obtain a lina scon-o iffntial quation: ζ κ /..5 7

18 In aition if w slct ζ / an put κ / this quation taks th stana fo κ.6 which is th static liit of th Klin-Goon quation fo a spin- scala fil of ass in units such that h c. s is wll-known a solution of Eq..6 has th fo of a Yukawa potntial: g / xp.7 which chaactizs a stong nucla foc. y copaing this potntial with an xpansion of th ulthn potntial fin by Eq..8 in th following way M M M /.8 O w fin an g M / which is in th ang of stong nucla coupling constants. Ou intst now lis in th fact that having tansfo Riccati s quation fo  into Klin-Goon quation fo iggs fil x w can us stana Lagangian fil thotic tchniqus to put togth th spin- fil x th spin- fil ψ x an th aining coponnts k k an oth bosons aking up th Yang-Millsfils x to unify lctoagntic wak an stong intactions in a a gaug-invaiant ol. Thn in th zo gaug-coupling liit a of intst fo th copss -ato pobl w hav th following consv cunt an fil quations h c : t J x x γ ψ x i * x x.9a ψ γ M ψ x Γψ x x i x ψ x Γψ x x.9b.9c wh M is th ass of th spino fil ψ x an th ass of th iggs scala fil x. Sinc th wavfunction ψ x of has spin 8

19 as ichotoic vaiabl consiabl siplification can b achiv by using spin labls ψ x an ψ x to appoxiat th spin- iggs fil though a lation involving ψ o ψ fo insional asons:.. This nabls us to abstact fo Eq..9a th iso-cunt nsity J x ψ x γ i t ψ x.a which is consv by th following lativistic wav quation: i γ M / ψ..b This is th final gaug-invaiant iso-lativistic wav quation fo that w a aft. Not that it contains only th two paats M fining.. iction of th Nuton Mass W obsv h that th contibution to th iso-cunt nsity in Eq..a fo th iggs fil call convctiv cunt is sponsibl fo th anoalous agntic ont of th lcton in th havy lcton thoy vlop by aut Coo an Ghiai[9]. Moov bcaus th convctiv cunt givs is to th scon-o ivativ non-noalizabl stuctu Eq..b las to two ass valus: j [ p j p j ] j.a givn by th two oots of th quaatic ass quation: M.b M. nc sinc / M th two asss ay b wittn with as ± M ± 7 o c M M This has th significanc that sinc M π / K bcaus π / α w ay siilaly xpss ± using th sa scal in th K fos: K an so that in plac of.c w hav /.65 /.65. 9

20 Thus fo th M p n syst w pict th ass.. with.5.65 p p p n. which is consistnt with th xpintal valu of th nuton st ass. Th sall piically tin bining ngy. is also consistnt with Eq..b bing inicativ of a syst that is asyptotically f. Th bauty of having a thoy with a iniu of f paats lis howv in its pictiv pow which in this cas aiss fo th physical significanc of th ngativ ass atio in Eq..c to which w now tun.. iction of th Masss of th SU ayon octt Following ou ali convctiv cunt appoach to th tination of th ass atio of quaks in coposit systs nialu[9] lt us plac th ass paats M in Eq..b by iagonal SUatics as follows: 8 M.5a so that / / / b

21 This aks Eq..b to siulat th Gll-Mann Oaks an Rn[8] ol of chial SUxSU gaug syty baking own to th SU of stong intactions without ixing isotopic spin oublts so that it ucs to two istinct quations: i γ ψ.6a with two ass ignvalus o ; an γ ψ i.6b with two ass ignvalus tin by th oots of th quation which is th sa as Eq..b giving ± ± 7 i o This ans that in th aitiv constitunt quak ol of th spin bayon octt to which th nuton blongs with w tin fo th obsv asss of th nuton n an th oth bs of th octt Λ' s 6 9 Λ Σ th two ass atios: s Λ' which a in agnt with thos in Eq..7. In oth wos th pict ass atios of th bs of th SU bayon octt a in agnt with xpint in accoanc with th qual-spacing ul whn splitting u to lctoagntic intactions a igno. It is also notwothy that th scon ass atio in Eq..8 is lat to th ass of th stang paticl Λ as an xcit stat of th nuton scib by Eq..6b which can b wittn in ts of th nuton ass in th fo: n iγ ψ n n n..9 Th last t givs is to th agntic tansitions iscuss by aut t al[9]; an Eq..9 focs on us th conclusion that th nutal spin- u

22 paticl with wavfunction ψ n constuct in th gaug-invaiant ~ n. lativistic thoy has th sa signatu as th nuton itslf 5. DISCUSSION ND CONCLUSION In this pap w hav foulat th non-lativistic thoy of th Ruthfo-Santilli nuton psnt in f.[] as a gaug-invaiant lativistic thoy stablishing in th pocss a fuitful linkag btwn th paats M of th ulthn potntial an unifi gaug thois in a oh-lik ol of haonic stuctu. Ou cntal objctiv at ach stag laing to th gaug-invaiant lativistic thoy has bn to constuct an xactly solubl non-ptubativ physical ol of haonic stuctu that woul povi a igoous fawok involving a iniu nub of piical paats fo confonting thotical sults with xpintal ata. Th achivnt of th abov objctiv was a possibl by th psntation of th ffctiv intapaticl foc in th copss -ato as a functional of th potntial ngy so that ath than intouc an a hoc potntial ngy fo th copss syst th potntial ngy is tin by th solutions of a nonlina fist o Riccati s iffntial quation which can also b tansfo albit non-unitaily into a lina scon-o iffntial quation. This is a natual out fo haonic chanics to th so-call nonlina consvation laws s Stang[9] laing to solitons instantons tc. Moov as is wll-known [p. 9 of f.[] Schwazschil s clbat solution g [ / ] κ π ρ of Einstin s quation fo th gavitational fil is also govn by Riccati s quation g g 8 πκρ g 5. wh s g g θ sin θϕ an 6 πκρ is th scala cuvatu ρ bing th cuvatu ngy nsity. W conclu an con thfo that th stuy of th Ruthfo-Santilli nuton shoul b intnsifi fo both thotical an xpintal points of viw in o to ap th potntial bnfits of having an xactly solubl ol of th intnal stuctu of th stongly intacting paticls haons fo tsting vaious contning ols an fo vification of haonic chanics laws of physics.

23 cknowlgnt I wish to thank ofsso R.M. Santilli fo stiulating ias an suggstions an fo sustaining y intst in th pobl solv in this pap fo clos to two cas. I also thanks D. C. N. nialu fo fining th fo of th xact solution of th Riccati quation in pnix. ENDIX SOLUTION OF TE RICCTI EUTION In this appnix w wish to fin in a clos fo th solution of th Riccati s quation: R bing constants. Th quation ay b xpss an intgat in th fo: R wh R ; R Thus by using th patial faction xpansion w fin

24 ln ln ln i.. xp xp xp R R R R R R xp xp M. This it th gnal sult that w a aft.

25 REFERENCES..O.E. nialu aonic J. in pss. R. M. Santilli: aonic J ; Counication of th JINR Dubna Russia #E publish in Chins J. Syst. Ing. & Elct G. Lvi hysics Toay Fb. 99 p. 7 & May 99 p.7...o.e. nialu aonic J O. E. nialu an R. M. Santilli Intn. J. uantu Ch D.L. Goostin Stats of Matt ntic-all Inc.975 p W. iggs hys. Rv ; C.N. Yang an R.L. Mills hys. Rv ; T. iaggio n Elntay tatis on Diffntial quations an Thi pplications Lonon 96 p. ; an fo th gotical oigin an poptis of Riccati s quations s Eoua Gousat Cous in Mathatical nalysis ol. tanslat by E.R. ick Dov ublications Inc. NY. 959 p O. aut. Coo an G.C. Ghiai hys. Rv & Nuov. Ci ;.O.E. nialu hys. Rv. D M. Gll-Man hys. Ltts 8 96; G. Zwig CERN Rpot 89T 96...O.E. nialu aonic J R. M. Santilli Int. Jounal of hysics M.Y. an an Y. Nabu hys. Rv O. aut in Goup Tho. Mths in hysics Eit by W. iglbock. oh an T. Takasugi SLN ol ;.O. aut nn. hysik Lipzig ½ O.E. nialu Ltts Nuovo Ci C. Myung in ocings of th Thi Wokshop on aonic Mchanics an nonpotntial Intactions hl at atas Gc R. M. Santilli Counication of th JINR Dubna Russia # E publish in Chins J.Syst.Ing. \& Elct

26 8. M. Gll-Mann R.J. Oaks an. Rnn hys. Rv G. Stang Intouction to ppli Mathatics Wllsly- Cabig ss 986 p. 6.. Thoo Fankl Gavitational Cuvatu n Intouction to Einstin s Thoy W.. Fan & Co San Fancisco 979 p.9. 6