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Register a new userA Close Correlation between Third Kepler Law and Titius-Bode Rule
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In this work we present a close correlation between third Kepler law and Titius-Bode empirical rule. Concretely, we demonstrate that third Kepler law, or, corresponding equilibrium condition between centrifugal and Newtonian gravitational force, implies that planet orbital momentum becomes effectively a function of the planet distance as unique variable and vice versa. Then, approximation of the planet distance by its first order Taylor expansion over planet orbital momentum holds an exponential form corresponding to Titius-Bode rule. In this way it is not necessary postulate exponential form of the planet distance (as it has been done by Scardigli) but only discrete values of its argument. Physically, it simply means that, in the linear approximation, quantized planets orbital momentums do a geometrical progression.
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Under certain conditions usually fulfilled in classical mechanics, the principle of conservation of linear momentum and Newtons third law are equivalent. However, the demonstration of this fact is usually incomplete in textbooks. We shall show here that to demonstrate the equivalence, we require the explicit use of the principle of superposition contained in Newtons second law. On the other hand, under some additional conditions the combined laws of conservation of linear and angular momentum, are equivalent to Newtons third law with central forces. The conditions for such equivalence apply in many scenarios of classical mechanics; once again the principle of superposition contained in Newtons second law is the clue.
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We show that the known expressions for the force on a point-like dipole are incompatible with the relativistic transformation of force, and in this respect we apply the Lagrangian approach to the derivation of the correct equation for force on a small electric/magnetic dipole. The obtained expression for the generalized momentum of a moving dipole predicts two novel quantum effects with non-topological and non-dynamic phases, when an electric dipole is moving in an electric field, and when a magnetic dipole is moving in a magnetic field, correspondingly. The implications of the obtained results are discussed.
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