Institute of Fundamental Technological Research

In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions of those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic functions and elliptic integrals, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained.

action-angle analysis, mechanics of infinitesimal gyroscopes, geodesic and geodetic equations of motion, helicoid-catenoid deformation family of minimal surfaces, elliptic integrals and elliptic functions

In the present paper, we have discussed the mechanics of incompressible test bodies moving in Riemannian spaces with nontrivial curvature tensors. For Hamilton's equations of motion, the solutions have been obtained in the parametrical form and the special case of the purely gyroscopic motion on the sphere has been discussed. For the geodetic case when the potential is equal to zero, the comparison between the geodetic and geodesic solutions has been done and illustrated in details for the case of a particular choice of the constants of motion of the problem. The obtained results could be applied, among others, in geophysical problems, for example, for description of the movement of continental plates or the motion of a drop of fat or a spot of oil on the surface of the ocean (e.g., produced during some "ecological disaster"), or generally in biomechanical problems, for example, for description of the motion of objects with internal structure on different curved two-dimensional surfaces (e.g., transport of proteins along the curved biological membranes).

geodesics, geodetics, gyroscopic motion, incompressibility constraints, mechanics of infinitesimal test bodies, Riemannian spaces

This paper discusses the motion of infinitesimal gyroscopes along the two-dimensional surfaces of constant mean curvature embedded into the three-dimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding Hamilton-Jacobi equations are written and analyzed with the help of the action-angle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials.

action-angle variables, affinely-rigid bodies, d'Alembert mechanics, Delaunay surfaces, geometry of curves and surfaces, Hamilton-Jacobi equation, harmonic/anharmonic oscillator potential models, infinitesimal test bodies, residue analysis, Riemannian spaces

There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, GL(n,R), SU(2,2), GL(4,C), etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of GL(4,R)‐invariant (or rather GL(n,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 concept 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 nonaccidental 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, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a 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.

micromorphic medium, modified gravity, relativistic continuum, spherically symmetric solutions, theory of fundamental interactions.

In this paper we discuss certain dynamical models of affine bodies, including problems of partial separability and integrability. There are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the "large" affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries.

homogeneous deformation, structured media, affinely-invariant dynamics, elastic vibrations encoded in kinetic energy, Calogero-Moser and Sutherland integrable lattices

In this paper, we develop 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 configurationmanifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n,R) and the space of translations 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 2-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 elastic media and microstructured continua.

Homogeneously deformable body, Peter-Weyl analysis, Schrödinger quantization.

This paper has been inspired by ideas presented by V. V. Kozlov in his works [23, 24]. In the present work the main goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational, i.e., vakonomic ones. So far, we can see that they give quite different results. The vakonomic model from the mathematical point of view seems to be more elegant. The similar problems were examined by M. Jóźwikowski and W. Respondek in their paper [20].

Affine motion, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

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 the 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.

Affine motion, polar and two-polar decompositions, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

The main leitmotivs of this paper are the essential nonlinearities, symmetries and the mutual relationships between them. By essential nonlinearities we mean ones which are not interpretable as some extra perturbations imposed on a linear background deciding about the most important qualitative features of discussed phenomena. We also investigate some discrete and continuous systems, roughly speaking with large symmetry groups. And some remarks about the link between two concepts are reviewed. Namely, we advocate the thesis that the most important non-perturbative nonlinearities are those implied by the assumed “large” symmetry groups. It is clear that such a relationship does exist, although there is no complete theory. We compare the mechanism of inducing nonlinearity by symmetry groups of discrete and continuous systems. Many striking and instructive analogies are found, e.g., analogy between analytical mechanics of systems of affine bodies and general relativity, tetrad models of gravitation, and Born-Infeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups.

Essential nonlinearity, dynamical symmetries, affine bodies, general relativity, tetrad models and micromorphic continua

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’.

Weyl–Wigner–Moyal–Ville formalism, topological groups, classical momentum mapping, unitary projective representations, Pontryagin duality, ‘Umklapp–Prozessen’

Discussed are some problems of two (or more) mutually coupled systems with gyroscopic degrees of freedom. First of all, we mean the motion of a small gyroscope in the non-relativistic Einstein Universe RXS3(0;R); the second factor denoting the Euclidean 3-sphere of radius R in R4. But certain problems concerning two-gyroscopic systems in Euclidean space R3 are also mentioned. The special stress is laid on the relationship between various models of the configuration space like, e.g., SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R) etc. They are locally diffeomorphic, but globally different. We concentrate on classical problems, nevertheless, some quantum aspects are also mentioned.

Rigid body, gyroscopic degrees of freedom, Einstein Universe, Euclidean space, SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R)

Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than the linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. On the quantum level this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In Galilei framework it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

Galilei group, affine transformation, particle mass, projective unitary representation, Weyl-Wigner-Moyal-Ville formalism

Discussed are geometric structures underlying analytical mechanics of systems of affine bodies. Presented is detailed algebraic and geometric analysis of concepts like mutual deformation tensors and their invariants. Problems of affine invariance and of its interplay with the usual Euclidean invariance are reviewed. This analysis was motivated by mechanics of affine (homogeneously deformable) bodies, nevertheless, it is also relevant for the theory of unconstrained continua and discrete media. Postulated are some models where the dynamics of elastic vibrations is encoded not only in potential energy (sometimes even not at all) but also (sometimes first of all) in appropriately chosen models of kinetic energy (metric tensor on the configuration space), like in Maupertuis principle. Physically, the models may be applied in structured discrete media, molecular crystals, fullerens, and even in description of astrophysical objects. Continuous limit of our affine-multibody theory is expected to provide a new class of micromorphic media.

Systems of affine bodies, mutual deformation tensors, affine and Euclidean invariance, structured media, elastic vibrations, geometric structures

We use the mathematical structure of group algebras and $H^+$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient SO(3, R). The proposed scheme is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. Secondly, the other promising area of applications is Schroedinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schroedinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2).

Systems of angular momenta, models with symmetries, quantum dynamics, spin systems, Schroedinger quantum mechanics, quasiclassical limit

In Part I of this series we have presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment there was very “ascetic” in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Relying on differential geometry one is able to introduce explicitly representation of important physical quantities and to formulate the general ideas of quasiclassical representation and classical analogy.

Systems of angular momenta, models with symmetries, quantum dynamics, Lie group structures, quasiclassical representation, classical analogy

This is the third part of our series “Quasiclassical and Quantum Systems of Angular Momentum”. In two previous parts we have discussed the methods of group algebras in formulation of quantum mechanics and certain quasiclassical problems. Below we specify to the special case of the group SU(2) and its quotient SO(3,R), and discuss just our main subject in this series, i.e., angular momentum problems. To be more precise, this is the purely SU(2)-treatment, so formally this might also apply to isospin. However. it is rather hard to imagine realistic quasiclassical isospin problems.

Systems of angular momenta, models with symmetries, quantum dynamics, quasiclassical isospin problems

Discussed is the structure of classical and quantum excitations of internal degrees of freedom of multiparticle objects like molecules, fullerens, atomic nuclei, etc. Basing on some invariance properties under the action of isometric and affine transformations we reviewed some new models of the mutual interaction between rotational and deformative degrees of freedom. Our methodology and some results may be useful in the theory of Raman scattering and nuclear radiation.

Internal degrees of freedom, isometric and affine invariance, classical and quantum excitations, multiparticle objects, Raman scattering, nuclear radiation

Discussed is mechanics of objects with internal degrees of freedom in generally non- Euclidean spaces. Geometric peculiarities of the model are investigated in detail. Discussed are also possible mechanical applications, e.g. in dynamics of structured continua, defect theory and in other fields of mechanics of deformable bodies. Elaborated is a new method of analysis based on nonholonomic frames. We compare our results and methods with those of other authors working in nonlinear dynamics. Simple examples are presented.

affine invariance, affinely-rigid bodies, collective modes, internal degrees of freedom, nonlinear elasticity, Riemannian manifolds

Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schroedinger quantization and reduce effectively the quantized problem from $n^2$ to $n$ degrees of freedom. Some possible applications in nuclear physics and other quantum many-body problems are suggested. Discussed is also the possibility of half-integer angular momentum in composed systems of spinless particles.

Collective modes, affine invariance, Schroedinger quantization, quantum many-body problem

Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also dynamics that is affinely-invariant. The relationship with the dynamics of integrable one-dimensional lattices is discussed. It is shown that affinely-invariant geodetic models may encode the dynamics of something like elastic vibrations.

Collective modes, affine invariance, integrable lattices, nonlinear elasticity

Studied is the problem of degeneracy of mechanical systems the configuration space of which is the three-dimensional sphere, the elliptic space, i.e., the quotient of that sphere modulo the antipodal identification, and finally, the three-dimensional pseudo-sphere, namely, the Lobatchevski space. In other words, discussed are systems on groups SU(2), SO(3,R), and SL(2,R) or its quotient SO(1,2). The main subject are completely degenerate Bertrand-like systems. We present the action-angle classical description, the corresponding quasi-classical analysis and the rigorous quantum formulas. It is interesting that both the classical action-angle formulas and the rigorous quantum mechanical energy levels are superpositions of the flat-space expression, with those describing free geodetic motion on groups.

action-angle description, Bertrand systems, completely degenerate problems, elliptic space, Lobatchevski space, quasi-classical analysis, sphere

Our work has been inspired among others by the work of Arnold, Kozlov and Neihstadt. Our goal is to carry out a thorough analysis of the geometric problems we are faced with in the dynamics of affinely rigid bodies. We examine two models: classical dynamics description by d'Alembert and vakonomic one. We conclude that their results are quite different. It is not yet clear which model i practically better.

Affine motion, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

Schrödinger equation as a self-adjoint differential equation of mathematical physics is discussed. For simplicity, a finite-level system is considered. A modified Schrödinger equation with the second time derivatives is described and some direct nonlinearity is admitted. The key of our idea is the assumption that the scalar product is not fixed once for all, but is a dynamical quantity mutually interacting with the state vector. We assume that the Lagrangian term describing its dynamics has the large symmetry group, the total complex linear group. This implies the strong essential nonlinearity. There is a hope that this geometrically implied nonlinearity may explain the decoherence and measurement paradoxes in quantum mechanics.

Schrödinger equation, Krawietz-type matrices, affine rigid body