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In this book the author derives, under the classical non-relativistic consideration of the space-time, general forms of the most common physical laws invariant under the changes of inertial or non-inertial coordinate systems, both in the classical and the quantum regime. Important examples of such invariant physical laws are the Maxwell Equations, Newtonian gravity as well as several more complicated models of gravity and many other physical laws including many of the laws of quantum mechanics, thermodynamics and statistical physics, continuum mechanics, and optics. Moreover, several basic laws of relativistic physics, both in the classical and quantum regimes can be still formulated invariantly under the non-relativistic consideration of space-time. These include the classical relativistic Second Law of Newton and the quantum Dirac and Klein--Gordon equations for relativistic particles, including their interaction with the external gravitational field. In particular, we introduce the Hamiltonian formulation of the Dirac equation, and moreover, were able to formulate the Dirac equation for multiple particles, similarly to what was done for the Schroedinger equation of the non-relativistic quantum mechanics. One of the goals of this work is to provide a self-contained and simple mathematical formulation of the most general physical laws in a manner understandable to the reader familiar only with basic calculus, classical mechanics and basic elements of non-relativistic quantum mechanics.
List of contents
Introduction.- Notations and preliminaries.- Transformations of scalar and vector elds under the change of inertial or noninertial cartesian coordinate system.- Gravity revised.- Maxwell equations revised.- Maxwell equations in non-inertial cartesian coordinate systems.- Scalar and vectorial electromagnetic potentials.- Lagrangian of the Electromagnetic eld.- Local gravitational time and Maxwell equations in a non-rotating coordinate system.- Motion of particles in external gravitational-electromagnetic eld.- Relation between the gravitational and inertial masses and conservation laws.- Lagrangian of the unied Gravitational-Electromagnetic eld.- Covariant formulation of the physical laws in the four-dimensional non-relativistic space-time.- Relativistic-like Dirac equation.- Thermodynamics of a moving continuum medium.- Maxwell equations in the presence of Dielectrics and/or Magnetics.- Some further consequences of Maxwell equations.- Appendix.
