After 412 years of the formulation and publication (in Astronomia Nova, 1609) of the Кеpler’s Equation, which relates the eccentric (and, intermediately, the true) anomaly of the planetary trajectories to the uniformly flowing time, in accordance with his Second (surface) law, in this paper are perceived with it connected certain deficiencies of the orbital mechanics and dynamics, caused by absence of the Kepler’s accompanying physical considerations, and which are repercuted in: reliance on the so-called Invariants – the First integrals of Energy and of the Angular Momentum, implicit Conservativeness, canonic formalism and omnipresence of the Symmetry principle, as well as the essential lacking of the explicit centrifugal force and its substitution by the fictitious one. It is given а survey of the Kepler’s strivings and the results attained, as well as of his key role in the historical development of the mechanics, physics, astronomy and astrophysics, and the science in general – through insights in the branching of the science development over Newton (instead over Descartes and Leibniz), the incomplete congruence among his physical considerations and the ultimately formulated laws, and also of the non-existent transverse acceleration ‘implied’ by the Kepler’s Second law. In support of the justification of the neglected development direction and the fundamentality of the Kepler’s insights in the need for both the attractive an repulsive interactions of the orbital and central body – the Sun, the Kepler-Ermakov second order non-linear differential equation has been reaffirmed along its adequacy for the phenomenological modeling of dynamic interactions on all the ‘scales’ in Nature: with brief reference to the “General Aetherodynamics” of V.A. Atsukovsky, as the basis for reuse and justification of insights/results of Descartes, Leibniz, Boscovich, D’Alambert, Engels, H. Strache, M. Petrović, M. Milanković, and P, Savić. Certain implications to the Elliptic Integration, the Symplectic Integration, Symplectic Geometry/Topology, as well as the connection between physical and mathematical continua in the context of the multi-level, scale-invariant mechanics and/or dynamics - will be briefly mentioned.

After 412 years of the formulation and publication (in Astronomia Nova, 1609) of the Кеpler’s Equation, which relates the eccentric (and, intermediately, the true) anomaly of the planetary trajectories to the uniformly flowing time, in accordance with his Second (surface) law, in this paper are perceived with it connected certain deficiencies of the orbital mechanics and dynamics, caused by absence of the Kepler’s accompanying physical considerations, and which are repercuted in: reliance on the so-called Invariants – the First integrals of Energy and of the Angular Momentum, implicit Conservativeness, canonic formalism and omnipresence of the Symmetry principle, as well as the essential lacking of the explicit centrifugal force and its substitution by the fictitious one. It is given а survey of the Kepler’s strivings and the results attained, as well as of his key role in the historical development of the mechanics, physics, astronomy and astrophysics, and the science in general – through insights in the branching of the science development over Newton (instead over Descartes and Leibniz), the incomplete congruence among his physical considerations and the ultimately formulated laws, and also of the non-existent transverse acceleration ‘implied’ by the Kepler’s Second law. In support of the justification of the neglected development direction and the fundamentality of the Kepler’s insights in the need for both the attractive an repulsive interactions of the orbital and central body – the Sun, the Kepler-Ermakov second order non-linear differential equation has been reaffirmed along its adequacy for the phenomenological modeling of dynamic interactions on all the ‘scales’ in Nature: with brief reference to the “General Aetherodynamics” of V.A. Atsukovsky, as the basis for reuse and justification of insights/results of Descartes, Leibniz, Boscovich, D’Alambert, Engels, H. Strache, M. Petrović, M. Milanković, and P. Savić. Certain implications to the Elliptic Integration, the Simplectic Integration, Simplectic Geometry/Topology, as well as the connection between physical and mathematical continua in the context of the multi-level, scale-invariant mechanics and/or dynamics - will be briefly mentioned.

After some more than four centuries from the formulation and publication (in Astronomia Nova) of the Kepler's Equation, which relates the eccentric (and, intermediately, the true) anomaly of the planetary trajectories to the uniformly flowing time, in accordance with his Second ("Area") law, the subsequently -- in course of development of Orbital Mechanics -- to the 2${}^{nd}$ law related and formally derived non-existent (zero-valued) transverse acceleration is refuted. Certain implications to Elliptic Integration, Symplectic Integration, Symplectic Geometry/Topology, as well as the connection between physical and mathematical continua in the context of the multi-level, scale-invariant mechanics/dynamics (with the augmented central and torquing forces) are also briefly hinted to.

Previous "Derivations" are really "Mathematical Proofs" of Orbital behaviour. This paper explains HOW Kepler's 1st and 2nd Laws are produced by the properties or physics of the space fabric and gravity. It seems impossible to separate this paper on the Conservation and Restoration of Space from the Derivation of Kepler's Law of Planetary Motion. That no doubt is why it has remained undiscovered until now. There is no discrepancy between this and Einstein's theory of Gravity. (16July 2022)

Based on the list of research topics that Milanković had conceived for the Mathematical and Astronomical institutes (one plus two), as well as for the aspirants’ and doctoral theses (eight plus fifteen) – as found out by mathematician and science-historian Dragan Trifunović back in 1978, in this article is made an attempt to establish the reasons of those which relate to the need of re-examining and further developing the very theory of gravity and the traditional formulation of the natural orbital motion, having in mind that – through the secular changes of eccentricities and other parameters of planetary trajectories – they are in direct connection with the methodology and results of his “Solar Cannon”. In the anticipation of the geologically and otherwise experimentally estblished need for the fundamental modification of the Newtonian orbital mechanics/dynamics, it is brought-up and reaffirmed – starting from as early as the Kepler’s physical considerations - the Leibniz’s two-components central acceleration, which is shown to be arrivable at by using the Milanković’s distance inverse squared dependence of planets’ temperatures within the context of Gordić’s differential formulation of orbital motion as dynamic equilibrium between the gravitational (attractive) and thermal (repulsive) components of central ‘force’. It is hinted to the important correspondence with the somewhat more than one and the half century old formulation through the Kepler-Ermakov non-linear differential equation and the relevance of its application in the context of the “archeological astronomy” and asteroids dynamics modeling, as well as on the astrophysical, astronomic and cosmogony plans, through essentially obsoleting the “dark matter” and “dark energy” problems/issues. The additional support for adequacy of the formulation of the planetary orbital motion as the Thermo-Gravitational Oscillator has been found in the foundation and implications of the Atsukovsky’s Etherodynamics.

This paper derives two of Kepler's Laws of planetary motion. These are that all orbits are conic sections and that the radius vector sweeps out equal areas in equal time increments.ard

The Newtonian mechanic and contemporary physics model the non-circular orbital systems on all scales as essentially conservative, closed-path zero-work systems and circumvent the obvious contradictions (rotor-free 'field' of 'force', in spite of its inverse proportionality to squared time-varying distance) by exploiting both energy and momentum conservation, along specific initial conditions, to be arriving at technically more or less satisfactory solutions, but leaving many of unexplained puzzles. In sharp difference to it, in recently developed thermo-gravitational oscillator approach movement of a body in planetary orbital systems is modeled in such a way that it results as consequence of two counteracting mechanisms represented by respective central forces, that is gravitational and anti-gravitational accelerations, in that the actual orbital trajectory comes out through direct application of the Least Action Principle taken as minimization of work (to be) done or, equivalently, a closed-path integral of increments (or time-rate of change) of kinetic energy. Based on the insights gained, a critique of the conventional methodology and practices reveals shortcomings that can be the cause of the numerous difficulties the modern physics has been facing: anomalies (as gravitational and Pioneer 10/11), three or more bodies problem, postulations in modern cosmology of dark matter and dark energy, the quite problematic foundation of quantum mechanics, etc. Furthermore, for their overcoming, indispensability of the Aether as an energy-substrate for all physical phenomena is gaining a very strong support, and based on recent developments in Aetherodynamics the Descartes' Vortex Physics may become largely reaffirmed in the near future.