arXiv subjects

Jan Jerzy Sławianowski

Publications and source records attributed to Jan Jerzy Sławianowski.

7 recordsLinked to original sources

arXiv2016

Symmetries and constraints in mechanics of continua

In spite of its long history and classical character which goes back even to d'Alembert and Lagrange, the problems of constraints in mechanics of continua is still mysterious and full of misunderstandings. Let us mention the problem of difference between special solutions of unconstrained systems and constraints dynamics, holonomic and nonholonomic constrained, various versions of so-called vakonomy, sub-Riemannian structures, servoconstraints, and programme motion. There are some strange confusions between field theory and mechanics of continua; many problems call for conceptual cleaning, although both the mechanics of continua and field theory are formally based on partial differential equations. The author is deeply indebted to professor Czes\law Woźniak for his help during many years, and deep understanding and support.

math-ph↗

arXiv2016

Space-time as a structured relativistic continuum

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.

math-ph↗

arXiv2016

Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media

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.

math-ph↗

arXiv2016

Quantized Mechanics of Affinely-Rigid Bodies

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.

math-ph↗

arXiv2016

Constraints and symmetry in mechanics of affine motion

The aim of this paper is to perform a deeper geometric analysis of problems appearing in dynamics of affinely rigid bodies. First of all we present a geometric interpretation of the polar and two-polar decomposition of affine motion. Later on some additional constraints imposed on the affine motion are reviewed, both holonomic and non-holonomic. In particular, we concentrate on certain natural non-holonomic models of the rotation-less motion. We discuss both the usual d'Alembert model and the vakonomic dynamics. The resulting equations are quite different. It is not yet clear which model is practically better. In any case they both are different from the holonomic constraints defining the rotation-less motion as a time-dependent family of symmetric matrices of placements. The latter model seems to be non-geometric and non-physical. Nevertheless, there are certain relationships between our non-holonomic models and the polar decomposition.

math-ph↗

arXiv2010

Geometric Nonlinearities in Field Theory, Condensed Matter and Analytical Mechanics

There are two very important subjects in physics: Symmetry of dynamical models and nonlinearity. All really fundamental models are invariant under some particular symmetry groups. There is also no true physics, no our Universe and life at all, without nonlinearity. Particularly interesting are essential, non-perturbative nonlinearities which are not described by correction terms imposed on some well-defined linear background. Our idea in this paper is that there exists some mysterious, not yet understood link between essential, physically relevant nonlinearity and dynamical symmetry, first of all, large symmetry groups. In some sense the problem is known even in soliton theory, where the essential nonlinearity is often accompanied by the infinite system of integrals of motion, thus, by infinite-dimensional symmetry groups. Here we discuss some more familiar problems from the realm of field theory, condensed matter physics, and analytical mechanics, where the link between essential nonlinearity and high symmetry is obvious, even if not yet fully understood.

physics.gen-ph↗

arXiv2009

Symmetries and geometrically implied nonlinearities in mechanics and field theory

Discussed is relationship between nonlinearity and symmetry of dynamical models. The special stress is laid on essential, non-perturbative nonlinearity, when none linear background does exist. This is nonlinearity essentially different from ones given by nonlinear corrections imposed onto some linear background. In a sense our ideas follow and develop those underlying Born-Infeld electrodynamics and general relativity. We are particularly interested in affine symmetry of degrees of freedom and dynamical models. Discussed are mechanical geodetic models where the elastic dynamics of the body is not encoded in potential energy but rather in affinely-invariant kinetic energy, i.e., in affinely-invariant metric tensors on the configuration space. In a sense this resembles the idea of Maupertuis variational principle. We discuss also the dynamics of the field of linear frames, invariant under the action of linear group of internal symmetries. It turns out that such models have automatically the generalized Born-Infeld structure. This is some new justification of Born-Infeld ideas. The suggested models may be applied in nonlinear elasticity and in mechanics of relativistic continua with microstructure. They provide also some alternative models of gravitation theory. There exists also some interesting relationship with the theory of nonlinear integrable lattices.

math-ph↗