arXiv subjects
Publications and source records attributed to Agnieszka Martens.
Discussed are Bertrand systems on two-dimensional spaces of constant curvature, i.e. sphere and pseudosphere (Lobachevsky space). These models often involve applying quantization techniques, such as those based on the Laplace-Beltrami operator, to the classical models and analyzing the resulting quantum behavior.
The paper is continuation of [6] where we have discussed some classical and quantization problems of rigid bodies of infinitesimal size moving in Riemannian spaces. Strictly speaking, we have considered oscillatory dynamical models on sphere and pseudosphere. Here we concentrate on Kepler-Coulomb potential models. We have used formulated in [6] the two-dimensional situation on the quantum level. The Sommerfeld polynomial method is used to perform the quantization of such problems. The quantization of two-dimensional problems may have something to do with the dynamics of graphens, fullerens and nanotubes. This problem is also nearly related to the so-called restricted problems of rigid body dynamic [1], [8].
It is well known that there are various models of gravitation: the metrical Hilbert-Einstein theory, a wide class of intrinsically Lorentz-invariant tetrad theories (of course, generally-covariant in the space-time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, ${\rm GL}(n,\mathbb{R})$, ${\rm SU}(2,2)$, ${\rm GL}(4,\mathbb{C})$, and so on). One believes usually in gauge models and we also do it. Nevertheless, it is an interesting idea to develop the class of ${\rm GL}(4,\mathbb{R})$-invariant (or rather ${\rm GL}(n,\mathbb{R})$-invariant) tetrad ($n$-leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus idea of affine symmetry. Formally, the obtained scheme is a generally-covariant tetrad ($n$-leg) model, but it turns out that generally-covariant and intrinsically affinely-invariant models must have a kind of non-accidental Born-Infeld-like structure. Let us also mention that they, being based on tetrads ($n$-legs), have many features common with continuous defect theories. It is interesting that they possess some group-theoretical solutions and more general spherically-symmetric solutions. It is also interesting that within such framework the normal-hyperbolic signature of the space-time metric is not introduced by hand, but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born-Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.
In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and other physical objects. Particularly interesting are dynamical models invariant under the group underlying geometry of degrees of freedom. In contrary to the single body case there exist nontrivial potentials invariant under this group (left and right acting). The concept of relative (mutual) deformation tensors of pairs of affine bodies is discussed. Scalar invariants built of such tensors are constructed. There is an essential novelty in comparison to deformation scalars of single affine bodies, i.e., there exist affinely-invariant scalars of mutual deformations. Hence, the hierarchy of interaction models according to their invariance group, from Euclidean to affine ones, can be considered.
In this paper we develope the main ideas of the quantized version of affinely-rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group ${\rm GL}(n,\mathbb{R})$ and the space of translations $\mathbb{R}^{n}$, where $n$ equals the dimension of the "physical space". In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the two-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of nuclear dynamics or even in apparently exotic phenomena in vibrating neutron stars. And, of course, some more prosaic applications in macroscopic elasticity, structured continua, molecular dynamics, dynamics of inclusions, suspensions, and bubbles are also possible.
Discussed is a model of the two-dimensional affinely-rigid body with the double dynamical isotropy. We investigate the systems with potential energies for which the variables can be separated. The special stress is laid on the model of the harmonic oscillator potential and certain anharmonic alternatives. Some explicit solutions are found on the classical, quasiclassical (Bohr-Sommerfeld) and quantum level.