Jacobi Dynamics: A Unified Theory with Applications to Geophysics, Celestial Mechanics, Astrophysics and Cosmology

345 Pages·2011·1.67 MB·English

by  V.I. Ferronsky

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