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Dynamical systems with internal degrees of freedom in non-Euclidean speces

Sławianowski J. J., Kovalchuk V., Gołubowska B., Martens A., Rożko E. E.

Prace Instytutu Podstawowych Problemów Techniki PAN

2006

nr 8

1-121

The primary concept of Newton mechanics is that of the material point moving in three-dimensional Euclidean space. A good deal of the theory depends only on the affine sector of geometry. The metric structure becomes essential when constructing particular functional models of forces; the concepts of energy, work, and power (time rate of work) also depend in an essential way on the metric tensor. The Galilei relativity principle implies that,l as a matter of vactl, it is not three-dimencional Euclidean space but rather four-dimensional Galilean space-time that is a proper arena of mechanics. This space-time has relatively complicated structure, does not carry any natural four-dimensional metric tensor and fails to be the Cartesian product of space and time. There exists the absolute time, but the absolute space does not. In the sequel we concentrate onf the other kind of problems, so the analysis of the subtle space-time aspects will be almost absent in our treatment. Newton theory becomes essentially realistic and viable when multiparticle systems are analyzed. It is just there where metrical concepts become almost unavoidable, because it is practically impossible to construct any realistic model of interparticle forces without the explicit use of the metric tensor. Extended bodies are described as discrete or continuous systems of material points. Their motion consists of that of the center of mass, i.e., translational motion and the relative motion of constituents with respect to the center of mass. The total configuration space may be identified with the Cartesian product of the physical space (translational motion) and the configuration space of relative motion. In many physical problems the structure of mutual interactions leads to certain hierarchy of degrees of freedom of the relative motion; in particular, some constraints may appear. The effective configuration space becomes then the Cartesian product ot the physical space and some manifold of additional degrees of freedom. There are situations when this auxiliary manifold and the corresponding dynamics are postlulated as something rather primary then derived from the multiparticle models. Usually the guiding hints are based on some symmetry principles. In this way the concept of internal degrees of freedom replaces that of relative motion. Sometimes it is a merely convenient procedure, but one can also admit something like essentially internal degrees of freedom not derivable from any multiparticle mode. After all, the very concept of the material point is an abstraction of a small piece of matter.

Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom

Sławianowski J. J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E. E., Zawistowski Z. J.

Prace Instytutu Podstawowych Problemów Techniki PAN

2004

nr 7

3-164

In some of our earlier papers including rather old ones we have discussed the concept of affinely-rigid body, i.e., continuous, discrete, or simply finite system of material points subject to such constraints that all affine relations between its elements are frozen during any admissible motion. For example, all material straight lines remain straight lines in the course of evolution, and their parallelism is also a constant, non-violated property. Unlike this, the metrical features, like distances and angles, need not be preserved. In other words, such a body is restricted in its behaviour to rigid translations, rigid rotations, and homogeneous deformations. Models of this kind may be successfully applied in a very wide spectrum of physical problems like nuclear dynamics (droplet model of the atomic nuclei), molecular vibrations, macroscopic elasticity (in situations when the length of excited waves is comparable with the size of the body), in the theory of microstructured bodies (micromorphic continua), in geophisics (the theory of the shape of Earth), and even in large-scale astropysics (vibrating stars, vibrating concentrations of the cosmic substratum, like galaxies or concentrations of the interstellar dust).