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A proposal for the value of the mean anomaly M for the hyperbolic orbit calculated for the eccentric anomaly E (M = 2E)
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Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets orbiting around the Sun. After the intensive study of those conic sections during the last four hundred years it is believed that this topic is practically closed and the 21st Century cannot bring anything...
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In our approach we have combined knowledge of Old Masters (working in this field before the year 1905), New Masters (working in this field after the year 1905) and Dissidents under the guidance of Louis de Broglie and David Bohm. In our model the photon is represented as the Huygens-de Broglie’s particle on the helical path (full wave) guided by the Newton-Bohm entangled evolute (empty wave). We have formulated the concept of the Super-Elastic Photon WAVE based on the Great Works of Weber, Abbe, Voigt and Einstein. This model works with the longitudinal elasticity of that WAVE that was already very well tested experimentally. Newly, we propose to test the elastic amplitude of this WAVE for the case of the Doppler’s redshift, the Doppler’s blueshift, and the Zwicky’s redshift. We have newly used the concept of the Lorentz’ force for the description of the photon acting force and the fermion reacting force. In this model the Lorentz’ factors γ and γ3 do not describe the “transverse mass of fermions” and longitudinal mass of fermions” but the “reacting transverse force of fermions” and the “reacting longitudinal force of fermions”. (The mass of photons and fermions does not change with their speed). It is very well-known that the cylindrical helix observed from different angles forms shadows in the Plato’s Cave as circle, sine, cosine, trochoid, cochleoid, hyperbolic spiral. Therefore, the resulting shape depends on the observer position in the Plato’s Cave-this is the famous Rashomon effect between observers. Based on the Newton-Bohm helical evolute and the Huygens-de Broglie helical path of the particle we have derived interesting formula known as the quantum of the magnetic flux. When we work further with this concept based on the Mathematical Beauty developed by Dirac, Gell-Mann, Schwinger, Polchinski, Witten and many others, we will obtain possible properties of the magnetic monopole. This photon quantum of the magnetic flux can be experimentally evaluated in the known tests with superconductors and micro-WAVES and infrared-WAVES. Can it be that Nature cleverly works with the magnetic monopole hidden in plain sight? We want to pass this concept into the hands of Readers of this Journal better educated in the Mathematics and Physics.
Galileo’s Parabola describing the projectile motion passed through hands of all scholars of the classical mechanics. Therefore, it seems to be impossible to bring to this topic anything new. In our approach we will observe the Galileo’s Parabola from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola. For the description of events on this Galileo’s Parabola (this conic section parabola was discovered by Menaechmus) we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the Galileo’s empty focus that plays an important function, too. We will study properties of this MAG Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. For the visible Galileo’s Parabola in the Aristotelian World, there might be hidden curves in the Plato’s Realm behind the mechanism of that Parabola. The analysis of these curves could reveal to us hidden properties describing properties of that projectile motion. The parabolic path of the projectile motion can be described by six expressions of projectile speeds. In the Dürer-Simon’s Parabola we have determined tangential and normal accelerations with resulting acceleration g = 9.81 msec-2 directing towards to Galileo’s empty focus for the projectile moving to the vertex of that Parabola. When the projectile moves away from the vertex the resulting acceleration g = 9.81 msec-2 directs to the center of the Earth (the second focus of Galileo’s Parabola in the “infinity”). We have extracted some additional properties of Galileo’s Parabola. E.g., the Newtonian school correctly used the expression for “kinetic energy E = ½ mv2 for parabolic orbits and paths, while the Leibnizian school correctly used the expression for “vis viva” E = mv2 for hyperbolic orbits and paths. If we will insert the “vis viva” expression into the Soldner’s formula (1801) (e.g., Fengyi Huang in 2017), then we will get the right experimental value for the deflection of light on hyperbolic orbits. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?
Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia…” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician). We will study properties of this PAN Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. In the Plato’s Realm some curves carrying hidden information might be waiting for our research. One such curve - the evolute of parabola - discovered Newton behind his famous gravitational law. We have used the Castillon’s cardioid as the curve describing the tangent velocity of objects on the parabolic orbit. In the PAN Parabola we have newly used six parameters introduced by Gottfried Wilhelm Leibniz - abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal. We have obtained formulae both for the tangent and normal velocities for objects on the parabolic orbit. We have also obtained the moment of tangent momentum and the moment of normal momentum. Both moments are constant on the whole parabolic orbit and that is why we should not observe the precession of parabolic orbit. We have discovered the Ptolemy’s Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting objects. In this case the Ptolemy’s Circle plays a role of the hodograph rotating on the parabolic orbit without sliding. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?
