 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A. V. Didin, "On the mechanics in N-space," Mech. Solids. 38 (2), 13-18 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
2 |
Pages |
13-18 |
| Title |
On the mechanics in N-space |
| Author(s) |
A. V. Didin (Korolev) |
| Abstract |
The classical mechanics has been being created for 3-space,
which is a particular
case of N-space. As a consequence, a number of basic concepts
of the classical mechanics such as angular momentum, moment of force,
tensor of inertia, and angular velocity have been associated with
inadequate mathematical objects. In the author's opinion, it is just
for that reason that the classical mechanics has been displaced from
a number of fields by other theories. In particular, the definitions
of angular momentum, moment of force, tensor of inertia, and angular
velocity are known to depend on the dimension of the space. For example,
the angular momentum is a pseudovector in 3-space and a number
in 2-space; the tensor of inertia is a tensor in 3-space and a number
in 2-space. Therefore, one may not carry over properties of these objects
established in a space of a certain dimension straightforwardly to spaces
of other dimensions, since these objects have different mathematical
nature in spaces of different dimension and, hence, different physical
interpretations. It turns out that one can redefine
these concepts to make them independent of the space dimension.
This enables one to apply laws involving these concepts to subspaces
of N-space having an arbitrary dimension, irrespective of N.
In addition, such an approach allowed one to establish new properties
of the angular momentum.
The aim of the present paper is to clarify the physical meaning
and properties of the concepts of the classical mechanics
on the basis of a simple body of mathematics known to scientists
and engineers.
To avoid misunderstanding, we choose to define or explain by specific
examples all terms that will be used in what follows and that
can be interpreted inadequately when applied to a space of dimension
other than 3. These definitions do not pretend to be strict but serve
only to facilitate understanding. Many of these definitions can be found
in [1]. The reference frame is assumed to be inertial unless otherwise
stated. |
| References |
| 1. | A. I. Kostrikin and Yu. I. Manin,
Linear Algebra and Geometry [in Russian],
Nauka, Moscow, 1986. |
| 2. | E. Cartan, Theory of Spinors [Russian translation],
Izd-vo Inostr. Lit-ry, Moscow, 1947. |
| 3. | V. N. Branets and I. P. Shmyglevskii, Applications
of Quaternions in Problems of Orientation of a Rigid Body
[in Russian], Nauka, Moscow, 1973. |
|
| Received |
22 January 2001 |
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