I WOULD LIKE TO EXPRESS MY GRATITUDE TO THE ORGANIZERS , AND IN PARTICULAR TO THE
REVIEWER(S) – HAVING IN MIND THE ‘SENSITIVITY’ OF THIS TITLE – FOR THIS OPPORTUNITY TO CONVEY
CERTAIN INSIGHTS AND FINDINGS I HAVE BEEN ARRIVING DURING LAST FIVE TO SIX YEARS, IN EXTENDING
PROFESSOR’S GORDIC ORBITAL MOTION APPROACH WHICH INVOLVES BOTH THE GRAVITATIONAL
ATTRACTING AND THE ANTI-GRAVITATIONAL – THERMAL CENTRAL FORCES. IN DOING SO, I CAME TO
FIND IMPORTANT SUPPORT IN PROFESSOR’S VUJICIC ‘STEPPING-OUT’ FROM THE CONVENTIONAL
‘DOCTRINES’ WITH THE ESSENTIALLY WEBERIAN ALTERNATIVE TO THE NEWTON’S FORMULA FOR THE
GRAVITATIONALLY INTERACTING FORCE, HIS DYNAMICS OF RHEONOMIC SYSTEMS , AS WELL AS THE
PERSEVERANCE IN PROMOTING THE FEASIBILITY OF INTEGRATION OF TOTAL DIFFERENTIAL IN VECTOR
AND TENSOR CALCULI.
THE ULTIMATE MOTIVATION AND URGENCY TO COME UP WITH RE-EXAMINATION AND QUESTIONING OF
THE FOUNDATIONAL PRINCIPLES OF THE CLASSICAL, IN THE FIRST PLACE – ORBITAL MECHANICS,
REPRESENTING BASIS FOR PHYSICS AND ALL BRANCHES OF SCIENCES, HAS BEEN COMING FROM ITS
IMPOSITION OF THE PRINCIPLES OF CONSERVATIVENESS , SYMMETRY , REVERSIBILITY AND ENTROPYSM,
PRESENTING THE OBSTACLE FOR THE EFFICIENT PRODUCTION AND UTILIZATION OF THE OTHERWISE
ABUNDANT AND ENVIRONMENTALLY SAFE, CLEAN ENERGY RESOURCES, THE LACK OF WHICH SERVES AS
TACIT JUSTIFICATION FOR THE EMERGING SOCIAL AND CIVILIZATIONAL ENGINEERING ‘UNDERTAKINGS’,
BESIDES LIKELY BEING INCAPABLE TO PROTECT LIFE ON EARTH FROM CATASTROPHIC EVENTS. FROM THE
PURELY SCIENTIFIC POINT OF VIEW, THE EXITING FORMALISMS HARDLY CAN PROVIDE THE BASIS FOR
MODELING AND ANALYSIS OF THE COMPLEX – TECHNICAL, BIOLOGICAL AND SOCIETAL – EMERGENT
DYNAMICAL SYSTEMS, FEATURING IRREVERSIBILITY AND MORPHISM, AND EVEN MAKE PEOPLE STAY SHY
FROM ADDRESSING THE ENERGETIC ‘BILANCING’ WITHIN THE RECENTLY AGAIN – MORE THAN A
HUNDRED OF YEARS OF WORKS OF LORD RAYLEIGH, VAN DER POLE, POINCARE, DUFFING AND TESLA,
INCLUDING IN THE MEANTIME ANDRONOV AND HIS COLLABORATORS – ACTUALIZED “SELF-EXCITED”
AND “SUSTAINING-SUSTAINING” NON-LINEAR DYNAMICAL SYSTEMS.
WHEN IT GOES ABOUT PROBLEMS IN THE CONTEXT OF EXISTING AND DEVELOPING THEORIES, I HAVE
FOUND THIS GOETHE’S QUOTATION QUITE APPROPRIATE, AND THIS PRESENTATION IN ESSENCE IS
ABOUT THE TWO BEING INTERRELATED, THUS BOILING DOWN TO JUST ONE – THE CONSERVATIVENESS,
ALLOWING ONLY FOR ENERGY DISSIPATION IN TERMS OF ITS BEING LOST FROM THE SYSTEMS (POSITIVE
FRICTION), FORBIDDING ITS INFLOW (“NEGATIVE FRICTION”) , TO POSSIBLY USE WORD ‘PRECIPITATION’
FOR LIKELY SOMETHING THAT PRIGOGINE WAS CONSIDERING THE COINED TERM “DISSIPITATION”.
1
I BORROWED THIS DRAWING TO ILLUSTRATE THAT THE ENTIRE EDIFICE OF SCIENCE AND
TECHNOLOGY STANDS ON JUST TWO RATHER FRIABLE ‘PILLARS’ – THE FIRST INTEGRALS
OF ENERGY AND ANGULAR MOMENTUM, WITH THE PRINCIPLE OF SYMMETRIES AS
THEIR COUNTERPART. IF NOTHING ELSE, THE INTERDEPENDENCE OF THE TWO FIRST OR
PRIME INTEGRALS SHOULD HAVE BEEN SOMETHING ENOUGH RAISE THE RED FLAG …
!?!
2
THAT IT DOESN’T HOLD HAS BEEN NOT SEEN AS OF YET, BECAUSE IT APPARENTLY WAS
LOOKED AT WITH BLURRED SIGHT, DUE TO VARIOUS KINDS OF ‘CONFUSIONS, “WISHFUL
THINKING” (INCL. MINE), MISCONCEPTIONS REGARDING THE VARIOUS FORMS OF
COORDINATE SYSTEMS USED AND THEIR MUTUAL TRANSFORMATIONS, AND/OR EVEN –
DECEPTIONS, AS IS FOUND AT THIS WEBSITE, WHERE THE “NON-INERTIAL’” POLAR
COORDINATES HAVE BEEN ACCOUNTED FOR IN DERIVATIONS OF DIFFERENTIAL
EXPRESSIONS FOR VELOCITIES AND ACCELERATIONS, BUT THE ILLUSTRATION GIVEN FOR
THE TRADITIONALLY TERMED “CURVILINEAR (TANGENT-NORMAL) COORDINATES”
SYSTEM TIED TO THE ORBITAL. STILL, ONE MIGHT BE SUSPECTING THAT THERE MUST BE
A NON-ZERO TRANSVERSE ACCELERATION. AT LEAST IN PRINCIPLE, WHENEVER IS NONZERO TANGENTIAL ACCELERATION, ONE SHOULD EXPECT THE PRESENCE OF THE
TRANSVERSE ALONG THE NORMAL COMPONENT, ALTHOUGH IN THE MATTERS OF THE
NATURAL ORBITAL MOTION, DUE MAINLY TO PRESENCE OF THE SINGLE-COMPONENT
(CENTRIPETAL ONLY) OF THE ACTING FORCE, THETHINGS BECOME QUITE PECULIAR AND
ECCESSIVLY ‘INVOLVED’.
BY ALL MEANS, BASED ON THE INVARIANCE OF A POINT, DISTANCE BETWEEN THE TWO
POINTS, VELOCITY OF ITS CHANGING WITH TIME, VELOCITY OF IT – ACCELERATION,
JERK, ETC. HAVE TO HAVE THE SAME DIRECTIONS AND ABSOLUTE VALUES, WHEN
CORRECTLY REPRESENTED IN EACH OF THE NON-INERTIAL SYSTEMS AND CORRECTLY
TRANSFORMED , MUTUALLY AND/OR DIRECTLY TO THE RECTANGULAR ONE!?!
3
COULD IT BE THAT ALONG THESE DIFFICULTIES THE THOROUGH EXAMINATION OF THE
ORBITAL MECHANICS FOUNDATIONAL TWO FIRST INTEGRALS WOULD HAVE LED TO
OTHERWISE LEGITIMATE TO INQUIRY AS TO WHETHER THE ATTRIBUTES WHICH THE
CONVENTIONAL SCIENCE ASSIGNS TO THE REALITY CORRESPOND TO, QUITE POSSIBLY
THE LEIBNIZ’S “BEST OF ALL POSSIBLE WORDS,” SO THAT EVERYTHING TURNS OUT TO
BE JUST THE OPPOSITE, AS HINTED BY THIS QUOTATION ABOVE: THE LEFT-, INSTEAD OF
THE RIGHT-HAND ‘ITEMIZATION’, ETC, … IN PLACE OF …, ETC. …
4
IN THE FIRST PART OF MY TALK – IN THE YEAR OF 450TH KEPLER’ BIRTHDAY – I WILL
ADDRESS ON ONE SIDE THE ESSENTIAL MISINTERPRETATION AND THE UNFORTUNATE
CONSEQUENCE OF THE KEPLER’S AREA-LAW, AND – ON THE OTHER – THE
UNPARDONABLE FAILURE TO APPRECIATE KEPLER’S PHYSICAL CONSIDERATIONS OF THE
NEED FOR BOTH ATTRACTION AND REPULSION MECHANISMS, FOR WHICH I WILL
PROVIDE SUPPORT IN THE SECOND PART, AND WILL CONCLUDE WITH THE TANGIBLE
CONSEQUENCES
5
6
THIS A KIND OF TWO-COLUMNS TABLE – TO BE RAD BOTTOM-UP – THAT ENCAPSULATES
DEVELOPMENT OF MECHANICS AND PHYSICS BRANCHES FROM KEPLER INTO TWO
TRACKS: THE OFFICIAL ONE ON RIGHT-HAND SIDE, INHERITING ONE PART OF THE
ARISTOTLES’ TEACHING, AND THE LARGELY SUPPRESSED ANOTHER WHICH CAN BE
TRACED BACK TO ARISTOTLE’S “IMPLICIT FORCE” AND THE “UNMOVED MOVER”, WHICH
FOR THAT MATTER CAN READILY BE RELATED TO THE – BY THE MAIN-STREAM SCIENCE
SINCE LONG ABANDONED – AETHER SUBSTRATE.
KEPLER ESSENTIALLY DID OVERCOME BOTH PLATONIC PRECEDENCE AND SUPREMACY
OF GEOMETRY OVER THE EMPIRICAL EVIDENCE AND ARISTOTLE’S STRICT SEPARATION
AND NON-RELATEDNESS BETWEEN MATHEMATICS AND PHYSICS, WHILE HE ACTUALLY
USED THEM INTERCHANGEABLY: FOR ONE THING THE OBSERVATIONS HAD TO BE
REPRESENTABLE BY THE ‘IDEAL’ GEOMETRICAL FORMS, AND FOR THE OTHER – THE
IDEAL GEOMETRICAL FORMS AND THE PERTAINING GEOMETRY HAD TO BE
SUPPORTABLE BY VIABLE PHYSICAL MECHANISMS OF THE PHENOMENA. KEPLER THUS
BECAME FOUNDER OF THE PHYSICAL MATHEMATICS – A DISCIPLINE WHOSE MODERN
FORM OF THE “MATHEMATICAL PHYSICS” LARGELY IS ITS OPPOSITE !?!
7
THIS BOOK’S COVER PAGE IS EXTREMELY WELL ILLUSTRATIVE OF KEPLER’S ‘SITUATION’
DURING THE FOUR YEARS OF HIS WORK ON DETERMINING THE ORBITAL TRAJECTORY
OF THE PLANET MARS, THE TASK WHICH HE HAD THOUGHT TO BE ABLE TO FINISH
WITHIN A WEEK, AS - IT IS SAID THAT – HE HAD BET WITH THE MANAGER OF TYCHO
BRAHE’S ESTATE, AND …
8
…TO FINALLY RESULT IN THE EQUATION RELATING THE TIME TO THE ECCENTRIC
ANOMALY, AND FURTHER TO THE TRUE ONE; YET, THE DIFFICULTIES IN SOLVING IT
THROUGHOUT SOME FOUR CENTURIES FOR EVER INCREASING PARAMETERS OF THE
ECCENTRICITY FACTOR MIGHT VERY WELL NOT HAVE BEEN DUE TO ITS TRANSCENDENT
NATURE, BUT RATHER TO ITS NON-PHYSICALITY AND ESSENTIAL INADEQUACY – AS IT
WILL BE SHOWN IN THE SEQUEL.
9
IN HIS FOUR YEARS LONG AND STRUGGLING (BOTH PROFESSIONALLY AND PERSONALLY)
WORK ON DETERMINATION OF THE MARS’ ORBITAL TRAJECTORY, QUITE EARLY – AT THE
STAGE OF THE ECCENTRIC CIRCLE SHAPED ONE – KEPLER DID REALIZE THE NECESSITY OF
INVOLVEMENT OF THE DISTANCE CHANGING AGENCY, AND DID ASSIGN THE RELATED
ESSENTIALLY MAGNETIC REPULSIVE AND ATTRACTIVE FEATURE TO THE PLANET ITSELF
(VIS INSITIA), BESIDES THE SUN’S “VIRTUS MOTRIX” ACTION CAUSING THE
PRESUPPOSED PURELY CIRCULAR PLANETARY MOTION.
HE ACTUALLY DID USE THE AREAS IN ORDER TO AVOID, THAT IS AS REPLACEMENT OF
SUMMING UP A LARGE - UP TO 180, WITH ALL SUBSEQUENT INTERMEDIATE
GROUPINGS – NUMBER OF PRODUCTS OF DISTANCES AND TANGENTIAL VELOCITIES –
THE TRADITIONAL INVARIANT IN THE PTOLEMAIC EPICYCLES BUT FOR KEPLER VALID
ONLY IN PERIAPSES, SO THAT AT THAT STAGE HIS EQUATION RELATING THE ANGLES TO
EQUIDISTANT TIMES DID COVER ONLY THOSE, NARROW PARTS OF THE TRAJECTORY AT
PERIHELION AND APHELION.
HOWEVER, WHEN HE HAD BEEN FORCED TO REALIZE THAT IT WAS MORE ABOUT THE
OVAL- THAN THE CIRCLE-SHAPED TRAJECTORY, AND SUBSEQUENTLY - DUE TO ITS
GEOMETRIC INTRACTABILITY - ‘SUCCUMBING’ TO THE ELLIPTICAL ONE – AT ONCE,
BEING IN MIDST OF A HUGE DESPAIR, HE HAD NOTICED THE “DIAMETRAL DISTANCE”
RELATED REGULARITY ON THE LATUS RECTUM DIRECTION, MODIFIED ACCORDINGLY
ALL OTHER MEASURED POINTS BY A CERTAIN EXTENT, AND SUDDENLY – EVERYTHING
‘SNAPPED-IN-PLACE’ AND HIS “AREA LAW” - ALONG THE DISTANCE LAW - TURNED OUT
TO APPLY FOR THE WHOLE ELLIPTIC, ORBIT, ALTHOUGH THAT EVEN WAS NOT THE CASE
10
FOR THE ECCENTRIC CIRCLE.
10
ON THIS SLIDE, BY THE RED DOTS AND ARROWS ARE SHOWN THE MEASUREMENTS’
MODIFICATIONS KEPLER DID UNDERTAKE … (ADDED FIG./TEXT FROM THE KOESTLER’S
BOOK, WITH THE SHADOWED SIECLE AND THE REALIZATION OF THE CONECTION FOR
THE 0.0062, OR SOMETHING LIKE THAT AND SMALL ANGLE – I.E. THE “OPTICAL
EQUATION” - AT THE SEMILATUS-RECTUM POSITION … WHICH LED TO THE – IN
MODERN FORMLATIONS – CRUTIAL EXPRESSION FOR DEPENDENCE OF DISTANCE ON
ECCENTRIC ANOMALY.
11
ALTHOUGH THE BASIC ARGUMENT FOR THE ANGULAR MOMENTUM CONSERVATION,
THAT IS ITS TIME-INDEPENDENCE, CANNOT IN GENERAL BE TENABLE SINCE THE
POSITION AND ACCELERATION VECTORS ARE COLLINEAR FOR ONLY THE APSIDES, THAT
TURNS OUT TO TRULY BE THE CASE AND THE ANGULAR MOMENTUM IS CONSTANT;
HOWEVER, THAT TURNS OUT TO BE THE CASE ONLY FOR THE DISTANCE AND PHASE(S)
DEPENDENCIES DETERMINED BY THE KEPLER’S EQUATION, WHICH AT THE SAME TIME –
AS A SINGLE POSSIBLE PARAMETERIZATION – ALLOWS THAT THE ELLIPSE AREA BE
INTEGRATED TO abπ !?!
12
ONE INDEED WOULD HARDLY EXPECT THAT THESE TWO TRIGONOMETRIC FUNCTIONS
IN THE NUMERATOR AND THE DENOMINATOR IN THE ANALYTICAL EXPRESSION FOR THE
ANGULAR MOMENTUM QUANTITY WOULD EVER BE PROPORTIONAL, EXCEPT FOR THE
CIRCULAR TRAJECTORY THAT IS THE ZERO-VALUED ECCENTRICITY , THAT SURPRISINGLY
BECOMES TO BE THE CASE …
13
IT SHOULD BE NOTED HERE THAT CONSTANCY OF THE ANGULAR MOMENTUM AND THE
ZERO-VALUEDNESS OF THE TRANSVERES ACCELERATION - AS WELL OF THE CONSTANCY
OF THE TOTAL (SUM OF KINETIC AND POTENTIAL) ENERGY REMAIN ESSENTIALLY
PRESERVED WHEN EITHER NUMMERICALLY INTEGRATIING IN THE ‘POLAR’ COORDINATE
SYSTEM, OR WHEN THE COMPONENTS OF THE VELOCITY AND ACCELRATION ARE
(ROTATIONALLY-INVARIANTLY) TRANSFORMED FROM THE RESULTS OF NUMERICAL
INTEGRATION IN THE RECTANGULAR COORINATE SYSTEM; THE ABOVE RESULTS ARE
BASED ON TRANSFORMATION OF VELOCITY FROM THE ROTATIONALY-INVARIANT
‘POLAR’ COORDINATE SYSTEM INTO THE ROTATIONALY NON-INVARIANT TANGENTIALNORMAL COORDINATES SYSTEM … SINCE – AT THE TIME BEING – THE NUMERICAL
INTEGRATION IN THE LATTER ONE HAS NOT BEEN YET MADE ‘OPERATIONAL’ …
AS A MATTER OF FACT, EVEN WITH A CENTRAL FORCE IT SHOULD/WOULD BE
LEGITIMATE TO PROJECT THAT FORCE ON THE TRANSVERSE DIRECTION AND RUN
SIMULATIONS TO ALLOW FOR THE TRANSVERSE ACCELERATION TO POSSIBLY ‘DEVELOP’
… !?!
14
INDEED, THE NON-ZERO TRANSVERSE ACCELERATION IS IMPLIED BY THE VERY PRESENCE
OF THE NON-ZERO TANGENTIAL ACCELERATION, WHICH IN TURN MEANS THAT THE –
CONTRARY TO THE CONVENTIONAL NOTION PERTINENT TO THE CENTRAL-FORCE FIELD
– WORK IS BEING DONE, AND ITS MINIMUM CORRESPONDS TO THE PARTICULAR
ELLIPTIC TRAJECTORY (FOR WHICH THE TIME-DEPENDENCE IS DETERMINED BY THE
CORREPONSING KEPLER’S EQUATION) AMONG ALL ITS VERTICALLY SCALED VERSIONS –
REVEALING A KIND OF THE ALL-POSITIVE ENERGY WELL … !!!
15
HERE IS GIVEN KEPLER’S EXPLANATION IN RELATION TO CRITIQUES OF UNEQUAL
LENGTHS OF THE ELLIPSE ARCS RESULTING FROM THE FORMAL SCALING OF THE
ECCENTRIC CIRCLE FOR WHICH THOSE ARCS WERE MUTUALLY EQUAL - AN IMPLICIT
WARNING FOR PRECAUTION …
16
BY ALL MEANS, WHEN DIRECTLY EVALUATING THE CONVENTIONALLY FORMULATED
NON-LINEAR DIFFERENTIAL EQUATION (THUS, WITHOUT THE EXPLICIT CENTRIFUGAL
FORCE) THE ANGULAR MOMENTUM TURNS OUT TO BE FAR FROM CONSTANT, WHILE
WHEN CONSTRAINING IT TO A CERTAIN INITIAL VALUE THE TRAJECTORY IMMEDITELY
TENDS TOWARDS THE CIRCULAR ONE (INSTEAD OF PRODUCING THE KEPLERIAN ELLIPSE
…).
THE SO-CALLED “SYMPLECTIC INTEGRATION” REPRESENTS VARIOUS MODIFICATIONS OF
NUMERICAL INTEGRATION WITH THE GOAL TO ALWAYS TAME THE ANGULAR
MOMENTUM TO BE A CONSTANT, AND THE RELATED WHOLE BRANCH OF
MATHEMATICS NAMED “SYMPLECTIC TOPOLOGY” HAS LONG BEEN DEVELOPED – WITH
‘CONSERVED’ SURFACE-METRICS AS COUNTERPART OF CONSERVATION OF THE
ANGULAR MOMENTUM IN CLASSICAL DYNAMICS!?
17
THIS IS SIMULATION IN BASIC FORM WHERE THERE IS NO USE/INVOLVEMENT OF THE
SECOND TIME-DERIVATIVE OF THE PHASE ϕ.
18
RESULTS OF NUMERICAL INTEGRATION WITH THE TWO-TERM EXPRESSION FOR THE
TRANSVERE ACCELERATION AND THE ‘UNTAMED’ ANGULAR MOMENTUM !?!
19
RESULTS OF NUMERICAL INTEGRATION WITH THE TWO-TERM EXPRESSION FOR THE
TRANSVERE ACCELERATION AND THE ‘TAMED’ ANGULAR MOMENTUM !?!
20
RESULTS FOR THE TWO-TERM EXPRESSION FOR THE TRANSVERSE ACCELERATION WITH
0.1% OF TORQUEING AND THE ‘UNTAMED’ ANGULAR MOMENTUM !?!
21
RESULTS FOR THE TWO-TERM EXPRESSION FOR THE TRANSVERSE ACCELERATION WITH
0.1% OF TORQUEING AND THE ‘TAMED’ ANGULAR MOMENTUM !?!
22
TURNS OUT TO INDEED BE CONSTANT – THANKS TO THE ALTERNATIVE EQUATION FOR
THE RADIUS, AS THE CONSEQUENCE OF KEPLER’S ‘LAST-MINUTE’ REALIZATION AND
SUBSEQUENT MODIFICATION OF THE MEASUREMENTS TO PRODUCE THE KEPLER’S
EQUATION …
23
INTEGRATION OF THE ELLIPTIC ORBITAL TRAJECTORY (IN AT LEAST THE ROTATIONALLY
INVARIANT COORDINATES) IS NOTHING ELSE BUT THE REPETITIVE DEFINITIONAL
RELATION OF THE POTENTIAL – TANTAMOUNT FOR THE NEWTON’S SUBSEQUENT
PULLING BACK FROM THE TANGENTIAL INERTIAL MOTION AND (AMAZINGLY!?!)
CONSISTENT WITH THE KEPLER’S SYSTEMATIC MODIFICATION OF THE T. BRAHE’S
MEASUREMENTS, AND RESULTING KEPLER’S EQUATION AND THE AREA-LAW,
REPRODUCED BY THIS!
24
BY ALL MEANS, ALWAYS THE “PROOFS” ‘BOIL-DOWN’ TO REDUCTION TO RECTILINEAR
MOTION, AND/OR THE KEPLERIAN ELLIPSE ONE, A KIND OF ECCENTRIC CIRCLE
‘DEFORMATION’ BY SCALING –DOWN IT VERTICALLY … PROF. ROY IN HIS BOOK “ORBITAL
MOTION” HAS JUST HINTED THAT THE INTERMEDIATE EXPRESSION “MAY IMMEDIATELY
BE INTEGRATED TO …”, AND PROF. CURTIS IN HIS BOOK “ORBITAL MECHANICS FOR
ENGINEERING STUDENTS” DID IT BY EQUATING THE SCALAR PRODUCT OF THE POSITION
AND VELOCITY VECTORS TO THE PRODUCT OF THEIR ABSOLUTE VALUES – THUS,
COMPLETELY NEGLECTING THE FACT THAT THAT ANGLE IS NEVER ZERO ?!?
THIS VERY WELL – INDEED – MAY CLEARLY INDICATE THE INTERDEPENDENCE OF THE
TWO FIRST INTEGRALS, ALTHOUGH – NOMINALLY– THEY SHOULD BE INDEPENDENT …
!?!
25
NEED TO CHECK FOR THEE RESULTS ONCE MORE – TO BE SURE THAT THE Y
CORRESPOND TO THE LATEST STATUS OF THE CODE ‘DEVELOPMENT-DEBUGGING’ –
NOTABLY, TO WHICH ANGLE OF ROTATION THIS IS RELATED: THE ONE SHOWED IN THE
EXPRESSIONS FOR ROTATION BY THE ANGLE –PSI (OR WHICH), AND IF Vtang IS PROPER
ONE.
26
27
HERE IS WHAT NEWTON HAD ‘BOLDLY’ MISSED: TO PRODUCE – BASED ON THE KEPLERPTOLEMAIC DISTANCE TIMES VELOCITY INVARIANT – THE LEIBNIZ’S INVERSE CUBED
DISTANCE PROPORTIONAL CENTRIFUGAL FORCE, BESIDE THE INVERSE SQUARED
DISTANCE PROPORTIONAL ONE HE DID PRODUCE FROM THE KEPLER’S THIRD
LAW/RULE, THE HUYGENS’ CENTRIPETAL ACCELERATION AND HIS ‘OWN’ (?!) SECOND
LAW …
28
MYSELF, I HAD COME IN SITUATION TO EXTEND THE PROFESSOR VUJO GORDIC’S
DIFFERENTIAL FORMULATION – BY USING THE MILANKOVIC’S PLANETARY INSOLATION
LAW, AND SUBSEQUENTLY ARRIVING AT THE LEIBNIZ’S EXPLICIT CENTRIFUGAL FORCE,
ALONG THE FORMULATION OF THE TRAJECTORY DETERMINATION BY WORK
MINIMIZATION.
29
HERE IS SHOWN THE CORRESPONDING NON-LINEAR – FULLY OSCILLATORY –
DIFFERENTIAL EQUATION FORMULATION OF THE ORBITAL MOTION: ON THE OTHER
SIDE, THE TRADITIONAL FORMULATION – IN ABSENCE OF THE EXPLICIT CENTRIFUGAL
FORCE – HAS WRONGLY USING THE KINEMATIC, THAT IS PURELY GEOMETRICAL TERM
‘FROM’ THE LEFT-HAND SIDE AS THE SO-CALLED “FICTITIOUS” CENTRIFUGAL FORCE, AS
WELL AS THE RELATED “VIRTUAL POTENTIAL”; INSTEAD OF THE TOTALLY UNTENABLE
ZERO-VALUED TORQUE, FOR THE TRANSVERSE DIRECTION IN THE CASE OF THE ‘FULLFLEDGED” NL-DE BASED FORMULATION THE USE IS MADE OF THE PROJECTION OF THE
RADIAL FORCE ON TRAJECTORY’S TANGENTIAL DIRECTION …
30
IN THE MEANTIME I REALIZED THAT SOME ONE AND A HALF CENTURY AGO ERMAKOV
HAD FORMULATED VERY SIMILAR NL-DE, WHICH HAS AN EXACT FIRST INTEGRAL, AND
WHICH IN MY VIEW RIGHTEOUSLY WAS RELATED TO KEPLER - IN THE WAY IT IS BEING
REFERENCED AS KEPLER-ERMAKOV EQUATION – THE MORE THAT IT CONVEYS HIS
FUNDAMENTAL AND PHYSICAL ATTRACTION AND REPULSION MECHANISM, AND WHICH
– TO MENTION ONE MORE OF THE SERBIAN SCIENTISTS - WAS WORKED ON BY
MATHEMATICIAN DRAGOSLAV MITRINOVIC DURING ALL OF HIS CARRIER, AND LATER
COMPLAINING THAT HE SHOULD HAVE BETTER WORKED ON SOMETHING ELSE – AN
OPINION WHICH I ABSOLUTELY WOULDN’T SHARE :)
31
IN THE AETHERODYNAMICS, AS IS THE CASE IN RARIFIED GAS DYNAMICS IN GENERAL,
THE TOROIDAL VORTEXES ARE BASIC DYNAMICALLY STABLE STRUCTURES, INDUCED OR
SPONTANEOUSLY FORMED - WHICH HERE ARE ‘ROUTINELY’ PRODUCED BY SOME
“SMART GUYS” – JUST FOR FUN.
32
HERE IS THE TABLE OF THE DEDUCED ETHER PARAMETERS, WHERE I WOULD ONLY
POINT TO DENSITY, PRESSURE, THE TWO KINDS OF VELOCITIES OF THE LONGITUDINAL
AND TRANSVERSE PROPAGATION OF PERTURBATION, AND THE MASS AND THE SIZE.
33
THE ELECTRIC FIELD THEN BECOMES JUST THE MANIFESTATION OF THE
EQUATORIAL/CIRCULAR FLOWS, AND THE MAGNETIC FIELD – OF THE MERIDIAN/TORAL
ONE, WHILE THE VERY MECHANISM OF ATTRACTION AND REPULSION GOES ALONG THE
BERNOULLI’S LAWS OF RESPECTIVELY INCREASED AND DECREASED VELOCITY
GRADIENTS.
THE CORRESPONDING ATTRACTION AND REPULSION FORCES, RESPECTIVELY, MATCH
THE SAME INVERSE PROPORTIONALITIES OF DISTANCE AS IS THE CASE OF GRAVITY AND
ITS MODIFICATION BY THE ORBITAL BODY’S HEAT.
34
THE GRAVITY BECOMES THE EFFECT OF THE PRESSURE GRADIENT CAUSED BY THE
COOLING OF ETHER BY THE BODY’S MASS, REPRESENTING ITS CONGLOMERATION OF
ITS ESSENTIALLY ORDERED FLOW, PRODUCING THE LESAGEAN PUSHING EFFECT, WHILE
THE BODY’S HEAT TENDS TO REDUCE THE PRESSURE GRADIENT IN ITS DOMAIN.
ALTERNATIVELY, THE INTERACTION OF SUN AND PLANETARY BODIES, OR IN GENERAL OF
CENTRAL AND ORBITING STRUCTURES CAN AS WELL BE PHENOMENOLOGICALLY
MODELED BY THE PERTAINING MAGNETIC AND ELECTRIC FIELDS.
35
36
IN TERMS OF MECHANICS AND MATHEMATICS IT TURNS OUT THAT ONE SHOULD
CONSIDER TO:
▪ STRICTLY DISTINGUISH BETWEEN KINEMATICS AND DYNAMICS ASPECTS,
▪ ALTOGETHER ABANDON THE PRINCIPLES OF CONSERVATIVENESS AND SYMMETRY,
▪ EXPLICIT CENTRIFUGAL FORCE HAS TO BE INVOLVED, BESIDES THE CENTRIPETAL ONE –
TO PROVIDE SCALE-INVARIANT INTER-A AND INTRA-SCALE DYNAMCS
▪ BE AWARE OF IMPORTANCE AND NECESSITY OF THE PARAMETRIC INTEGRATION,
▪ AND NOT BE LEAVING THE TIME-PARAMETER OUT OF THE EQUATION MODELING THE
PARTICULAR CLASS OF NON-LINEAR DYNAMICAL SYSTEMS.
37
FURTHERMORE, BESIDES QUITE STRAIGHTFORWARD OBSOLETING OF THE
CONTEMPORARY THEORIES’ POSTULATIONS AND DEDUCTIONS BEING OF RELEVANCE
TO ASTROPHYSICS AND COSMOLOGY, THERE ARE NUMEROUS QUITE PROFOUND
IMPLICATIONS TO QUESTIONABILITY AND UNTENABILITY OF VARIOUS CONCEPTS AND
ASPECTS OF THE BASIC PHYSICS, SO …
(BG-Slides – from 50 to 63 … WHEN RUNNING PRESENTATION, USE IS MADE OF
HYPERLINKS, THE ‘ARROW-ICONS’ – FOR BOTH TO THE RELATED SLIDES, AND RETURNIG
BACK TO THE NEXT AND, FINALLY, TO THE SLIDE AT WHICH IT WAS STARTED).
REFERENCES ON Slides 64 AND 65)
38
… IT THEN COMES TO EVIDENTLY BE UPON US TO DECIDE IN WHAT KIND OF WORLD TO
LIVE – THE ONE WITH CHARACTERIZATION AND ATTRIBUTES PERTINENT TO THE
CONVENTIONAL SCIENCE AND TECHNOLOGY PARADIGM, OR IN THE ONE WHICH MIGHT
VERY WELL BE RESEMBLING THE LEIBNIZ’S “BEST OF ALL POSSIBLE WORLDS”; AND –
WHETHER THIS QUESTION BELLOW WOULD AT ALL BE POSSIBLE TO ANSWER TO THE
POSITIVE !?
39
40
41
HERE IS THE OUTLINE OF SUCH PROCEDURE, BASED ON THE CONCEPT OF A VECTOR AT
A POINT, AS FOUND IN DOBROVYN’S ET AL. BOOK “SOVREMENAYA GEOMETRIYA”,
WHERE THE ADEQUATELY FORMULATED COMPONENT IN EACH OF THE REFERENCE
COORDINATE SYSTEMS ARE CROSS-TRANSFORMED BY STANDARD ROTATION BY THEIR
PERTINENT ANGLE DIFFERENCE, WHEREBY THE TRADITIONALLY USED CURVILINEAR
COORDINATES TURNED OUT TO NOT CORRSPOND TO THESE, ROTATIONALLY INVARIANT
ONES, AND THUS THE ONE IN WHICH THE KEPLERIAN ELLIPSE MAY BE INTEGRATED TO
REVEAL THE NON-ZERO TANGENTIAL ACCELERATION, SINCE IN ALL THE ROTATIONALLY
INVARIANT ONES THE NUMERICAL INTEGRATION IS TANTAMOUNT FOR THE
DEFINITIONAL INTEGRAL OF POTENTENTIAL AS THE ACCIERED/LOOSED KINTETIC
ENERGY IN RADIAL DISPLACEMENT FROM/ONTO INFINITY … WITH COORECT
DYNAMICAL LAWS FORMULATION THESE STILL SHOULD BE THE ADVANTAGEOUSSLY
USED ONE IN TERMS OF THE INTEGRATION OF TOTAL DIFFERENTIAL, AS VIABLE
ALTERNATIVE TO THE TRADITIONALLY EXCLUSIVELY USED CO- AND CONTRAVARIANT
COORDINATES AND THEIR TRANSFORMATIONS RELYING ON THE POSITION VECTOR AND
ITS DERIVATIVES ONLY … IT SHOULD BE NOTED HERE THAT IN HIS EFFORTS ON
PROMOTING THE TOTAL DIFFERENTIAL ITEGRATION WITHOUTH NECESSITY OF USING
THE CRISTOFFEL’S SYMBOLS , PROFESSOR VUJIČID AND (ALSO THE LATE) PROFESSOR
VERA NIKOLID WERE TARGETING THE ROTATIONALLY INVARIANT COORDINATE
SYSTEMS/TRANSFORMATIONS, BUT ACTUALLY DID NOT FOLLOW THE VERY “VECTOR-ATTHE-POINT” DEFFINITION !?!
42
43
44
45
Rotationally invariantly transformed – from the rectangular to ‘polar’ coordinate system
– velocities (upper and lower left pair) and acceleration (lower right), and the angular
momentum produced by direct numerical evaluation in the rectangular and the ‘polar’
(so called, “non-inertial) coordinate system (variation on 3rd decimals …
46
Rotationally invariantly transformed – from the ‘polar’ to the tangential-normal
coordinate system – velocities (upper pair) and acceleration (lower pair) for the rotation
angles of π – ψ and – ψ (respectively, left and right sides) … still is unresolved as to why
analytical expressions in both cases have zero-valued normal component of velocity …
?!?
47
HERE IS SHOWN THAT IT’S ONLY FOR THE CASE OF KEPLER’S EQUATION IMPLIED ANGLETIME DEPENDENCE THE AREA LAW DOES HOLD.
48
ACTUALLY, THE NAÏVE INTEGRATION OF THE ALLEGEDLY CONSTANT ‘AUTONOMOUS’,
THAT IS TIME-PARAMETER NOT EXPLICATING ANGULAR MOMENTUM EXPRESSION
CANNOT PRODUCE THE REQUIRED EXPRESSION FOR THE SURFACE-AREA OF AN ELLIPSE:
ONLY FOR THE TIME-DEPENDENCE OF TRUE ANOMALY DETERMINED BY THE KEPLER’S
EQUATION THE CORRECT ANSWER CAN BE PROVIDED - THIS BEING AN EXAMPLE QUITE
ILLUSTRATIVE OF THE NEED TO PROPERLY DEFINE THE ‘DYNAMICS’ OF ‘INDEPENDENT’
VARIABLE IN TERMS OF THE PARAMETRIC INTEGRATION !
49
HERE ARE ENLISTED ARGUMENTS FOR INAPPROPRIATENESS OF THE POTENTIAL ENERGY
CONCEPT, AS IT HAS BEEN INGRAINED INTO THE ORBITAL MECHANICS AND PHYSICS.
50
BY USING THE EXPLICIT CENTRIFUGAL FORCE IN LIEU (INSTEAD) OF THE POTENTIAL,
SOME HUNDRED AND TEN YEARS AGO HUGO STRACHE DID MULTIPLICATIVELY EXTEND
THE LEIBNIZIAN TWO-COMPONENT(AL) CENTRAL ACCELERATION – AS THE BOSCOVICH
CURVE-OF-FORCE: THE STABLE POINTS OF THE TWO SUBSEQUENT STAGES OF THE
PARTICLES’ MUTUAL ‘PENETRATION’ …
51
TO ONLY READ THROUGH THE LAST PARAGRAPH, AFTER HINTING TO THE
DISTINGUISHING BETWEEN THE HEAT AND WORK, AND THAT IN SOME SITUATIONS – AS
IN THE “NEGATIVE VISCOSITY” TO BE ADDRESSED LATER ON – THE VERY CHANGING OF
THE CENTER OF MASS POSITION, I.E. ITS MOVEMENT CAUSED BY THE SYSTEM’S
INTERNAL PROCESSES IS LOOKED AT AS GOING AGAINST THE SECOND LAW OF
THERMODYNAMICS ?!?
52
TO ESPECIALLY STRESS THAT BOTH TEMPERATURE AND PRESSURE CONCEPTS ARE BOTH
BASED ON KINETIC ENERGY …
53
IN RELATION TO THE LAST SHOWN EXPRESSION FOR ENTROPY INCREMENT, FROM THE
SAME BOOK TO STRESS THAT “OBVIOUSLY, (EVEN IN THE CONTEXT OF CARNOT’S CYCLE)
THERE ARE OCCASIONS WHEN ENTROPY INCREASES IN ONE SECTION, THERE MUST BE A
CORRESPONDING DECREASE IN ANOTHER TO ENSURE A SUCCESSFUL COMPLETION OF
THE CYCLE” …
54
ALONG THE CONCEPTS OF “NEGATIVE RESISTANCE”, “NEGATIVE MASS” AND THE LIKE,
SOME MORE THAN A HALF OF CENTURY AGO THERE AROSE THE NOTION OF THE
“NEGATIVE VISCOSITY” – A SPONTANEOUS TRANSITIONING OF THE FLOW ACROSS THE
OPPOSITE DIRECTION OF THE ONE ‘DICTATED’ BY THE ACTUAL VELOCITY GRADIENT …
55
THAT THE NEGATIVE VISCOSITY IS ACTUALLY RELATED TO STRUCTURING HAS BEEN, AS
THE ONLY ONE AVAILABLE THEORETICAL ACCOUNT OF BY THE TIME BEING, WORKED
OUT BY WILHELM BAUER BACK IN 1970 – IN HIS DIPLOMA WORK IN PHYSICS AT THE
FREIE UNIVERSITAET BERLIN (AS AN ELECTRICAL ENGINEER, AT AGE OF 59). IN THE
FRAMEWORK OF THERMODYNAMICS AND BY USING THE NAVIER-STOKES EQUATION HE
HAD ARRIVED AT THE CONDITION UNDER WHICH THE ENTROPY CAN BE BOTH
INCREASING AND DECREASING, AND REFERRED TO THE NEGATIVE VISCOSITY
COEFFICIENT AS PARTICULAR FORM OF THE VELOCITY FIELD AND THE RELATED LAMBVECTOR – VECTOR PRODUCT OF FLUID VELOCITY AND ITS CURLING/ROTATION …
56
ALL THEN ‘BOILS DOWN’ TO THE FUNDAMENTAL DIFFERENCE BETWEEN MECHANISMS,
THAT IS THE PROCESSES OF IMPLOSION AND EXPLOSION; THAT WAS THE CASE AT THE
VERY EARLY STAGE (NOT COUNTING THE NERON’S ‘STEAM ENGINE’:) OF THE MODERN
TECHNOLOGICAL EVOLUTION: THE NEWCOMEN’S LOW-PRESSURE SYSTEM WITH STEAM
CONDENSATION HAD TO GIVE IN TO THE MODIFIED AND LATER UP-SIZED WATT’S
DESIGN, WHICH WAS THE ‘RIGHT’ TECHNOLOGY FOR THE COAL MINING FOR WHICH
THE NEWCOMEN’S DESIGN WAS USED TO PUMP-OUT THE ACCUMULATING WATER – AN
EXAMPLE HOW DOING BUSINESS & MAKING PROFIT HAS DOMINATED.
57
DERIVATION OF THE SECOND LAW OF THERMODYNAMICS IN THE CONTEXT OF
STATISTICAL MECHANICS WAS DONE BY RATHER CIRCULAR ARGUMENTS, WHILE THE
RESEMBLANCE WITH THE SHANNON’S INFORMATIONAL ENTROPY IS QUALITATIVELY
VERY DIFFERENT; BY ALL MEANS, THE CORRELATION AMONG RANDOM VARIABLES
SHOULD BE CORRESPONDING TO THE SMALLER ENTROPY, AND NOT THE OTHER WAY
AROUND – THAT THE ORGANIZED STRUCTURES BE LOOKED AT AS ‘CONTAINING’ MORE
INFORMATION AND THUS BEING MORE ‘COMPLEX’ … !?!
58
TO STRESS, AND JUST READ THROUGH (AT LEAST A PART OF) THE LAST (UNDERLINED)
PARAGRAPH ?!?
59
ONLY TO READ SOMETHING FROM THE LAST PARAGRAPH …
60
TO THE VERY RIGHT OF THE SLIDE, THE EXTENDED CANTORIAN SET OF ALEPH-S (SETS OF
DIFFERING CARDINALITIES, THAT IS THE ‘LEVELS’ IF INFINITIES) IS SHOWN – AN
EXTENSION OF HIS NOTION OF MATTER AS ALEPH 0 AND ETHER AS ALEPH 1; ALL THE
MATHEMATICS HAS STILL STICKING TO THE “CONTINUUM HYPOTHESIS” (NONEXISTENCE OF ANY SET OF NUMBERS BETWEEN THE REAL AND INTEGERS WHICH
WOULD HAVE THE CARDINALITY OF INFINITIES FALLING BETWEEN THE ONES OF THOSE
TWO), WHILE THIS DISPROVES IT … IT IS TO EXPECT THAT USING THE THERMO
GRAVITATIONAL OSCILLATOR APPROACH ON THE PHENOMENOLOGICAL LEVEL IN
COMBINATION WITH THE ‘FULL-FLEDGED’ NAVIER-STOKES EQUATION ON SUBSTANTIAL
LEVEL WOULD BE MUCH BETTER ALTERNATIVE TO THE ONE USED IN THE WORK CITED,
AND THAT IT WOULD READILY REVEAL GOLDEN-RATIO RELATIONSHIPS – IN BOTH THE
INTRA- AND THE INTER-SCALES CONTEXTS !?! (BG-Slides 70 and 71 …)
61
62
THE RECENT RECORDING OF THE “BLACK-HOLE”, WHICH ACTUALLY IS THE CENTER-AXIAL
PART OF THE UBIQUITOUS TOROIDAL VORTEX STRUCTURE WHICH IN THE CASE OF
GALAXIES AND THE LIKE STRUCTURES IS ‘CONGLOMERATION’ OF STARS, AND WHICH
STILL EMITS THE MATERIAL FOR THE LOWER SCALES OF THE OVERALL PHYSICAL
CONTINUUM, REPLENISHING THE ETHER AND OTHER LOWER SUBSTRATES … THUS, THE
CONTEMPORARY PROBLEM OF THE “INFORMATION LOSS” WITHIN THE BLACK HOLES
SHOULD BE NON-EXISTENT: THE STRUCTURES WHICH WERE RESULTS OF (I.E.
ACCOMPANIED BY) THE DECREASING ENTROPY BECOME DESTROYED, AND ON
ACCOUNT OF THAT THE ENTROPY BECOMES INCREASED IN THE SURROUNDING ZONES
…
… AN INTERESTING METAPHOR AND/OR ALLEGORY FOR THE UNIVERSE STRUCTURE AND
THE ITS ‘ORIGIN’ COULD BE THE SPECULATED ‘MECHANISM’ OF EELS’ SPAWNING, IN
THAT THEY GATHER AT CERTAIN ZONES OF THE PACIFIC OCEAN AND AFTER ENTERING
THE STEROIDAL VOTES FORMATION THEIR COMPETE BODIES GET PRESSED AND
DISMEMBERED, AND IN THAT WAY, OUT OF THE SQUEEZED-OUT FEMALES’ EGGS AND
MALES’ SPERMS ORIGINATED EMBRYOS START DRIFTING AWAY, GROWING IN A WAY
SIMILAR TO FORMATIONS OF STARS AND OTHER CELESTIAL BODIES – FOR THE
REPRODUCTION CYCLE TO BE REPEATED NEXT YEAR, AND SO ON, AND ON …
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