## The General Theory of Particle Mechanics. A Special Course

### Аннотация

This book provides insights into the tight connection between fundamental math and mechanics, the basic grounding of physics. It demonstrates that quantum, classical, and relativistic mechanics, historically (and separately) formulated upon an experimental basis, can be regarded as links of a single theoretical chain readily extracted from a simple mathematical medium. It uses mathematical tools to endow formerly abstract entities, such as quantum wave-function and classical action function, with original and clear geometric images, strongly simplifying them. The book comprises the author’s lectures, manual texts, typical problems and tests, and many illustrations, and will be of interest to students of all levels majoring in mathematics, physics and advanced engineering programs.

Professor Alexander P. Yefremov, is Head of the Chair of Physics, Director of the Institute of Gravitation and Cosmology, and Vice-Rector for Academic Affairs of the Peoples’ Friendship University of Russia. His scientific interests are the theory of gravitational field, celestial mechanics, quantum mechanics, and the math of hypercomplex numbers. He is the author of more than 150 papers, four monographs, and three textbooks. His main research achievements in theoretical physics are the “quaternion version of the theory of relativity” (1996-2008), the “conic gearing image of a complex number” (2011), and the “general theory of a particle’s mechanics” (2012-2015).

Cambridge Scholars Publishing (UK) has published the book General Theory of Particle Mechanics: A Special Course by Prof. Alexander P. Yefremov.

The book is based on results of more than 30-year intensive investigation of quaternion numbers making up the third (apart from the real and complex numbers) exclusive algebra, and its interplay with physics.

Classical, relativistic, and quantum mechanics are the three cornerstones of physics, their laws historically formulated on the experimental and heuristic grounds.

However, the book demonstrates that they seem to exist immanently in the pure math medium of quaternions, and we can readily extract them.

Moreover, these physical laws arise as links of a unique chain of equations logically (and mathematically) united within a holistic theory but in other order. At first, we deduce math (unit-less) equations of quantum mechanics, then those of classical mechanics, and finally we get relativity. Written in physical units these pure math equations become precisely the Schrodinger equation, the Hamilton-Jacobi (and Newton’s dynamic) equation, and equations of the Einstein’s relativity.

A thorough geometric analysis of the striking emergence of the math-born physics reveals a specific interior structure of the 3D world, hence of a 3D particle regarded as an object “gathered” out of fractal pre-geometrical pieces. Besides, a series of formerly abstract physical objects such as quantum mechanical wave function, spinor, action function, space-time interval in this theory acquire simple and clear geometric images.

By the way, a pre-geometric model of a complex number is built in the shape of a conic gearing couple as an original supplement to classical images on the complex plane and Riemann sphere.

This edition is constructed as a special course textbook aimed to help the readers to learn these seemingly new physical and mathematical facts. It consists of two parts.

The first part “Mathematical preliminaries” in its (four) chapters gives information of exclusive algebras, especially quaternions; investigates differential properties of a quaternion basis applied to solve problems of non-inertial classical mechanics; introduces the notion of fractal space and suggests conditions aimed to save the algebra under simple deformations of a fractal cell.

The second part “General theory of particle mechanics” in its (three) chapters identifies the algebra stability condition written in the microphysical units with the quantum mechanical equation which on the lab scale reduces to the Hamilton-Jacoby (and Newton dynamic) equation for a rotating point-like classical particle. This particle model and the speed-of-light constancy generate original “helix-line born” action functional of the relativity theory as well leading with necessity to the De-Broglie correlations.

Each chapter contains an author’s lecture, a manual text, didactic accents and competences, typical problems and questions with solutions and answers, and current tests for self-control (with answers only). Altogether there are about two hundred problems and questions, and dozens of illustrations, pictures and graphs.

The book may be of interest to teachers and students majoring in math, physics and advanced engineering programs, and to all interested in non-conventional approaches to formulation of physical theories.

An important feature of this course is that the interested readers get all suggested information, learn to solve various mechanical problems, comprehend the logic of the mechanical laws derivation, and memorize these formulas only within several weeks.