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Invariant Geodetic Problems on the Projective Group Pr (n,R)
| Content Provider | Semantic Scholar |
|---|---|
| Author | Sławianowski, Jan Jerzy Kovalchuk, Vasyl |
| Copyright Year | 2004 |
| Abstract | The concept of the metrically-rigid body have played a very important role in the theoretical and applied mechanics (see e.g. [1]), mainly because our macroscopic environment is dominated by objects which are approximately rigid. But what would happen if we went forward and got rid of the metrical properties keeping the concept of rigidness? Such problems in the application to the theory of continuous media were investigated in [2–10], where the concept of an affinely-rigid body as a medium the deformative behaviour of which is restricted to performing homogeneous deformations only was developed. Other applications are also possible, e.g. in the theory of large oscillations of molecules, small mono-crystals, atomic nuclei and even in the theory of elementary particles. In fact, an affinely-rigid body in an amorphous affine space is an obvious counterpart of the usual metrically-rigid body in a Euclidean space. But we need not to get rid of the metric once and for all, we may introduce it in our consideration at any step, and that is what makes this approach attractive. For example, to be able to introduce the notion of the kinetic energy, we should have some fixed Euclidean metric. Let us explain this more precisely and consider the classical system of points (discrete or continuous), which is placed in the physical space M . We assume that the material points are distinguishable and label them by means of points of an auxiliary space N , which is called the material space (e.g. we may choose these labels as initial positions of all points at the moment t0). Let V and U be the linear spaces of translations (free vectors) in M and N , respectively, then (M, V,→) and (N, U,→) are affine spaces. The position of the a-th material point at the time instant t is denoted by x(t, a) (x ∈ M , a ∈ N). If the system is continuous, the label a and position x become the Lagrangian and Eulerian radius-vectors (material and physical variables), respectively. If the dimensions of M and N are equal, then we can impose such constraints of affine rigidness that the connection between material and physical variables is as follows: |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ippt.pan.pl/Repository/o1618.pdf |
| Alternate Webpage(s) | http://www.slac.stanford.edu/econf/C0306234/papers/kovalchuk.pdf |
| Alternate Webpage(s) | http://www.imath.kiev.ua/~snmp2003/Proceedings/kovalchuk.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
