Институт Физики Вакуума - Теория

Shipov G.I.
Descartes Mechanics:
The Fourth Generalization of Newton's Mechanics
shipov@aha.ru,
website http://www.shipov.com
Introduction

For 317 years we have been applying Newton's mechanics to explain non-relativistic mechanical experiments on the "bench table". Although Newton's mechanics has been generalized three times: the special relativity theory, general relativity theory, and quantum mechanics, there remains a possibility for its further generalization.

1 Frenet's Oriented Point

Newton's mechanics as well as all its generalizations, mentioned above, have been based upon the concept of the material point, substituting all the material bodies in this theory. The exception is Quantum Mechanics, where the material particles demonstrate both their corpuscular and wave properties. In a three-dimensional reference frame the material point has three degrees of freedom (according to the number of coordinates). In 1847 F. Frenet introduced for the first time the concept of "oriented point", connected with three orthogonal unit vectors, orienting it. In a three-dimensional coordinate space the oriented point has got six degrees of freedom - three translational and three rotational [1].

In arbitrary coordinate system and in modern notations, Frenet's motion equations for the three-dimensional oriented point could be written as [2]
 (1)

where vectors' induces and induces A,B,C... - denote vectors of Frenet's triad,
 (2)

- the square of the element of the curve's length, where the oriented point moves along, D

-absolute differential relatively of Cristoffel's symbols
 (3)

Magnititudes
 (4)

had been introduced by F. Ricci [3] and named later as a Ricci rotation coefficients and the geometric object
 (5)

- connection of absolute parallelism [4].

The Ricci rotation coefficients describes the change of the orientation of basic vectorsand define the rotational metric [2]
 (6)

If we select the right triadso that unit vectors and will be correspondingly a tangent, normal and binormal to the curve, then the equations (1), written in Descartes' reference frame, will lead to Frenet's equations
 (7)
 (8)
 (9)

where - curvature and - torsion of curve are connected with Ricci rotational coefficients in the following way
 (10)

From the equations (7)-(9) we will get the translational motion equations of the oriented point (motion equations of the origin of triad)
 (11)
 (12)

If we multiply equations (11) by total mass т of the oriented point, then we shall get similar to the motion equations of Newton's mechanics
 (13)

where
 (14)

- force, causing translational acceleration. From the above we can see, that the mechanics of the oriented point can generalize Newton's mechanics as well, allowing us:

a) To view the dynamics of the physical objects as rotation (Descartes' idea):

b) To consider the "inner" degrees of freedom, connected with its own rotation of the oriented point, that are not addressed in Newton's mechanics.

See PDF (4329Кб)

Shipov G.I. Descartes Mechanics: The Fourth Generalization of Newton's Mechanics // «Академия Тринитаризма», М., Эл № 77-6567, публ.12404, 06.09.2005

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