Jacobi Dynamics Astrophysics and Space Science Library EDITORIALBOARD Chairman W.B.BURTON,NationalRadioAstronomyObservatory,Charlottesville,Virginia, U.S.A.([email protected]);UniversityofLeiden,TheNetherlands ([email protected]) F.BERTOLA,UniversityofPadua,Italy J.P.CASSINELLI,UniversityofWisconsin,Madison,U.S.A. C.J.CESARSKY,CommissionforAtomicEnergy,Saclay,France P.EHRENFREUND,LeidenUniversity,TheNetherlands O.ENGVOLD,UniversityofOslo,Norway A.HECK,StrasbourgAstronomicalObservatory,France E.P.J.VANDENHEUVEL,UniversityofAmsterdam,TheNetherlands V.M.KASPI,McGillUniversity,Montreal,Canada J.M.E.KUIJPERS,UniversityofNijmegen,TheNetherlands H.VANDERLAAN,UniversityofUtrecht,TheNetherlands P.G.MURDIN,InstituteofAstronomy,Cambridge,UK F.PACINI,IstitutoAstronomiaArcetri,Firenze,Italy V.RADHAKRISHNAN,RamanResearchInstitute,Bangalore,India B.V.SOMOV,AstronomicalInstitute,MoscowStateUniversity,Russia R.A.SUNYAEV,SpaceResearchInstitute,Moscow,Russia Forothertitlespublishedinthisseries,goto http://www.springer.com/series/5664 V.I. Ferronsky S.A. Denisik S.V. Ferronsky l l Jacobi Dynamics A Unified Theory with Applications to Geophysics, Celestial Mechanics, Astrophysics and Cosmology Second edition V.I.Ferronsky S.V.Ferronsky(deceased) WaterProblemsInstitute oftheRussianAcademyofSciences Gubkinst.3,119333Moscow Russia [email protected] S.A.Denisik WaterProblemsInstitute oftheRussianAcademyofSciences Gubkinst.3,119333Moscow Russia Revisedandupdatededitionofthebook“JacobiDynamics:ManyBodyProbleminIntegral Characteristics”, by V.I. Ferronsky, S.A. Denisik and S.V. Ferronsky, Reidel D. Publishing Company,1986. ISSN0067-0057 ISBN978-94-007-0497-8 e-ISBN978-94-007-0498-5 DOI10.1007/978-94-007-0498-5 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2011926117 #SpringerScience+BusinessMediaB.V.2011 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Coverillustration:RealpictureofmotionofabodyAintheforcefieldofabodyB.Digitsidentify successionofturnsofthebodyAmovingaroundbodyBalongtheopenorbitC. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface to the Second Edition Thefirsteditionofthisbookwaspublishedin1987.Sincethattimenewscientific andpracticalresultsinapplicationofJacobidynamicswereobtained.Theresults, whichweincludeintothesecondedition,areasfollows. The oldest scientific problem of the origin and evolution of the Solar System bodies on the basis of the Jacobi’s oscillating dynamics has found its resolution. Wediscoveredbycalculationthatthepresentdayorbitalvelocityofeachplanetis equal to the fist cosmic velocity of the contracting Protosun having its radius reducedtothesemi-majoraxisoftheplanet’sorbit.Andalsotheorbitalvelocities ofthe planet’s moonsare equal tothe first cosmic velocitiesofthecorresponding protoplanets having their radiuses successively reduced to the semi-major axes of the moon’s orbits. It looks like the protoplanets and their protosatellites were createdfromtheseparatinguppershellsoftheircontractingparents.TheSunitself andthe small bodieslike thecomets, asteroids, meteorsandsoon,have the same historyofcreation. Thediscoveredmechanism oftheSolar System bodycreationprovestobe the physicalbasisoftheKepler’slawsandtheinversesquaredistancelawoftheouter force field distribution of a self-gravitating body. It also proves the idea that the ‘heavenspowerofgravity’ofbody’sorbitalmotionhastheelectromagneticnature andisinducedbytheparentalbody. WesucceededinunderstandingphysicalmeaningoftheJacobi’svirialequation obtainedfromtheNewton’sequationsofmotionwrittenforasystemofninteracting mass-particles.Itwasfoundthatinhisn-bodyproblem,Jacobi,whiletransforming theinitialequations,hasconvertedtheNewton’sforcesandmomentsoftheinter- actingparticlesintotheirvolumetricvalues,namely,intotheenergyandoscillating momentofinertia.Doinglikethis,Jacobihaschangedtheforceasameasureofthe particlesinteractionbyenergyoftheirinteraction.FromphysicalviewpointJacobi’s approach opens the way to search the common nature for all the known physical modelsofthematterinteraction.Inordertoprovethisidea,weshowinthebookthat theJacobi’svirialequation,besidestheNewton’sequations,isalsoderivedfromthe equationsofEuler,Hamilton,Einsteinandquantummechanics.Thisallowedusto putthesubtitleofthebookasa‘Unifiedtheoryoftheinteractingmattermotion’. v vi PrefacetotheSecondEdition We discuss in the book the main scientific discovery, made by the geodetic artificial satellite study that the Earth and the Moon do not stay in hydrostatic equilibrium. It puzzles us because the hydrostatics, up to now, is accepted as the physicalbasisindynamicsofstars,planetsandothercelestialbodies.Inadditionto this,wefoundthattheEarthhas noequilibriumbetweenthekinetic andpotential energy(K/U(cid:1)1/300).Thisisbecausetheexistingtheoriesindynamicsdonottake into account the kinetic energy of the body’s interacting mass particles. It makes incorrect formulation of the problem of the equilibrium state. The conclusion is madethattheconditionsofthedynamicalequilibriumstate,asanalternativetothe hydrostatics, in celestial mechanics, stellar dynamics and astrophysics should be introduced. Wefoundthattheabovefundamentaleffectsarewellexplainedbythetheoryof Jacobidynamicswhichconsidersjustself-gravitating,butnothydrostaticallystate systemsandwhererelationshipbetweentheenergyandthepolarmomentofinertia is the basic physical effect. In this connection this problem and physical meaning of the Jacobi’s virial equation in the new Chapter 2 are discussed. In order to demonstrate all new effects in dynamics of stars, planets and satellites, including creation and evolution of the Solar system, Chaps. 6, 7, and 8 were revised andupdatedbyapplying the above discoveries.Also,the newChap.“The Nature ofElectromagneticFieldofaCelestialBodyandMechanismofItsEnergyGener- ation”isintroduced.TheformerChaps.1,2,3,4,and5undernewnumberswereas awholepreserved.ThesecondeditionwaspreparedbyV.I.Ferronsky. V.I.Ferronsky Preface to the First Edition This book sets forth and builds upon the fundamentals of the dynamics of natural systemsinformulatingtheproblempresentedbyJacobiinhisfamouslectureseries “Vorlesungu¨berDynamik”(Jacobi1884). In the dynamics of systems described by models of discrete and continuous media, the many-body problem is usually solved in some approximation, or the behavior of the medium is studied at each point of the space it occupies. Such an approachrequiresthesystemofequationsofmotiontobewrittenintermsofspace co-ordinatesandvelocities,inwhichcasetherequirementsofaninternalobserver foradetaileddescriptionoftheprocessesaresatisfied. In the dynamics discussed here we study the time behavior of the fundamental characteristics of the physical system, i.e. the Jacobi function (polar moment of inertia)andenergy(potential,kineticandtotal),whicharefunctionsofmassdensity distribution,andthestructureofasystem.Thisapproachsatisfiestherequirements ofanexternalobserver.Itisdesignedtosolvetheproblemofglobaldynamicsand the evolution of natural systems in which the motion of the system’s individual elementswritteninspaceco-ordinatesandvelocitiesisofnointerest.Itisimportant tonotethat anintegral approach ismadetointernal andexternalinteractions ofa systemwhichresultsinradiationandabsorptionofenergy.Thiseffectconstitutes thebasiccontentofglobaldynamicsandtheevolutionofnaturalsystems. From the standpoint of methodology, the integral approach has an important advantage.Inthisapproachtheintegralcharacteroftheprincipleofleastaction– thebasicphilosophicalprincipleofmechanicsandphysics–isfullyrealized.Itis achievedbyusingacanonicalpairconsistingoftheJacobifunctionandfrequency in writing the basic equation of global dynamics. The practical use of this pair in Jacobi’svirial equationmadeitpossibletofarthergeneralizetheformsofmotion andtoshowthatthenon-linearoscillationsofasystemissuchageneralization. We note that the ten well-known integrals of motion in the many-body problem in its classical formulation should be regarded as historically the earliest equations of the integral type. These integrals, however, reflects not the specific natureofasystemunderconsiderationbutthegeneralpropertiesofspaceandtime, i.e.homogeneityofspaceandtimeandisotropicityofspace. vii viii PrefacetotheFirstEdition Thefirstnon-trivialequationofdynamicsintermsoftheintegralcharacteristics ofasystemis Jacobi’svirialequation, which describes changesinthe momentof inertia (Jacobi function) as a function of time. The next step in this direction was takenbyChandrasekhar(1969).Heusedanddevelopedforsolutionofproblemsin mechanicsthemethodofmoments,socalledinanalogytothemethodwellknownin mathematical physics. However, the problem of non-trivial solution of the non- linearizedequationsintermsofintegralcharacteristicswasnotsolvedineitherof thesecases. Ourworkbeganin1974.Asaresult,anumberofarticlesonthetheoryofvirial oscillations of celestial bodies were published in the journal Celestial Mechanics andotherperiodicals(Ferronskyetal.1978,1979a–c,1981,1982,1984,1985).The theory was based on solution of Jacobi’s virial equation for conservative and dissipativesystems. TosolveJacobi’sinitialequation,wefirstusedtheheuristicallyfoundrelation- ship between the potential energy and moment of inertia of a system, which was expressed in terms ofthe productof the corresponding form factors. It was found thattheproductdependslittleonthelawofdistributionofmassdensityforawide range of formal, non-physical systems. It was then demonstrated that in the asymptotic limit of simultaneous collision of the particles constituting a system, theobservedconstancyoftheproductofformfactorsremainedvalidwithoutany restrictionswithintheframeworkoftheNewtonandCoulombinteractionlaws.The invariant found was also demonstrated to be valid for the widely used relativistic andnon-relativisticphysicalmodelsofnaturalsystems.Itenabledustoderivefrom Jacobi’s equation a simple form of the equation of virial oscillations with one unknownfunctionandtofinditsrigoroussolution.Theequationobtaineddescribes the dynamics of a wide class of physical systems ranging from empty space-time and collapsing stars to the atom. Thus, it was established that the theory of virial oscillationsofcelestialbodieswasvalidfarbeyondthelimitsofcelestialmechan- icsbasedonNewton’slawofequations. The work was done on concepts of Professor V.I. Ferronsky and under his supervision. Chaps. 1 and 7 were written by V.I. Ferronsky and S.A. Denisik; Chaps.2,3,4,5,6and8werewrittenbyS.V.Ferronsky. Thisbookpresentsasystematicdescriptionofourresearchwork.Itisintended for researchers, teachers and students engaged in theoretical and experimental research in the various branches of astronomy (astrophysics, celestial and stellar mechanics and radiophysics), geophysics (physics of the Earth, atmosphere and oceans),planetologyandcosmogony,andforstudentsandpostgraduatesofclassi- cal,statistical,quantumandrelativisticmechanicsandhydrodynamics. It is our pleasant duty to express sincere gratitude to Professor G.N. Duboshin of the M.V. Lomonosov Moscow State University for his constant support and encouragement. We are indebted to Professor E.P. Aksenov, Director of the Sternberg Institute of Astronomy, Moscow, who organized helpful discussions of our work at a number of seminars. We also wish to express our gratitude to Dr. L. Osipkov from Leningrad University and Drs J. Schmidt, A. Lorenz, M. Mehta and T. Akity from the Division of Research and Laboratories, PrefacetotheFirstEdition ix InternationalAtomicEnergyAgency,whoreadseveralchaptersofthemanuscript and made editorial contributions, and to Miriam Lewis, who edited the final manuscript.WealsowishtothankRenateB.Blamhoferforassistanceinpreparing thebook.WeareparticularlyindebtedtotheInternationalAtomicEnergyAgency and its Division of Publications for support and assistance in preparation of the camera-readymanuscript. Theauthors