Operator manifold approach to geometry and particle physics

The question that guides our discussion ishow did the geometry and particles come into being? To explore this query the present theory reveals primordial deeper structures underlying fundamental concepts of contemporary physics. We begin with a drastic revision of a role of local internal symmetries in phys ical concept of curved geometry. Under the reflection of fields and their dynamics from Minkowski space to Riemannian a standard gauge principle of local internal symmetries was generalized. The gravitation gauge group is proposed, which was generated by hidden local internal symmetries. In all circumstances, it seemed to be of the most importance for understanding of physical nature of gravity. Last two parts of this paper address to the question of physical origin of geometry and basic concepts of particle physics such as the fundamental fields of quarks with the spins and various quantum numbers, internal symmetries and so forth; also four basic principles of Relativity, Quantum, Gauge and Color Confinement, which are, as it was proven, all derivative and come into being simultaneously. The substance out of which the geometry and particles are made is a set of new physical structures —the goyaks, which are involved into reciprocal linkage establishing processes. The most promising aspect of our approach so far is the fact that many of the important anticipated properties, basic concepts and principles of particle physics are appeared quite naturally in the framework of suggested theory.     

New exact solutions of Bianchi I,Bianchi III and Kantowski–Sachs spacetimes in scalar-coupled gravity theories via Noether gauge symmetries

The Noether symmetry approach is useful tool to restrict the arbitrariness in a gravity theory when the equations of motion are underdetermined due to the high number of functions to be determined in the ansatz. We consider two scalar-coupled theories of gravity, one motivated by induced gravity, the other more standard; in Bianchi I, Bianchi III and Kantowski–Sachs cosmological models. For these models, we present a full set of Noether gauge symmetries, which are more general than those obtained by the strict Noether symmetry approach in our recent work. Some exact solutions are derived using the first integrals corresponding to the obtained Noether gauge symmetries.     

Noether gauge symmetry approach in quintom cosmology

In literature usual point like symmetries of the Lagrangian have been introduced to study the symmetries and the structure of the fields. This kind of Noether symmetry is a subclass of a more general family of symmetries, called Noether gauge symmetries (NGS). Motivated by this mathematical tool, in this paper, we study the generalized Noether symmetry of quintom model of dark energy, which is a two component fluid model with quintessence and phantom scalar fields. Our model is a generalization of the Noether symmetries of a single and multiple components which have been investigated in detail before. We found the general form of the quintom potential in which the whole dynamical system has a point like symmetry. We investigated different possible solutions of the system for diverse family of gauge function. Specially, we discovered two family of potentials, one corresponds to a free quintessence (phantom) and the second is in the form of quadratic interaction between two components. These two families of potential functions are proposed from the symmetry point of view, but in the quintom models they are used as phenomenological models without clear mathematical justification. From integrability point of view, we found two forms of the scale factor: one is power law and second is de-Sitter. Some cosmological implications of the solutions have been investigated.     

Symetries du Probleme desN Corps,Regularisation des Solutions Centrales Singulieres

The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS ₁ andS ₂ changing the eccentricity of a solution. The family of symmetries,S ₁, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S ₂, is related to a possibleaffinity of the configurations space. Via a symmetry of theS ₂ family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS ₁.S ₂, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC ² H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:

1. The two families of symmetriesS ₁ andS ₂ are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
2. TheS ₁ family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
3. TheS ₂ family of symmetries may regularize multiple collisions among central solutions of the same type.

Therefore any central solution, via a symmetryS ₁ orS ₂ orS ₁.S ₂, is equivalent to a central circular solution of the same type. That is a form of regularization.     

On the eigenvalues of kinematic α-effect dynamos

The kinematic α-effect dynamo problem is investigated in the case of an exterior perfect conductor. It is shown that certain approximate symmetries discovered in the numerical analysis of ROBERTS (1972) are exact for this case. As an illustration, an exact solution is given in a cylindrical geometry, where the equations can be written in terms of one variable. The implications for the earth's dynamo are discussed.     

The linear problem for the five-dimensional projective field theory

A linear problem is given for the five-dimensional projective theory when the metric depends only on two coordinates. The symmetries of this problem are investigated.     

Groups of symmetries in the two‐body problem associated to Einstein's PN field

The two‐body problem associated to a spherical post‐Newtonian (PN) field with Einsteinian parameterization is revisited from the single standpoint of symmetries. The corresponding vector fields, in Hamiltonian and standard polar coordinates, or in collision‐blow‐up and infinity‐blow‐upMcGehee‐type coordinates, present symmetries that form diffeomorphic commutative groups endowed with a Boolean structure. The existence of such symmetries is of much help in understanding characteristics of the global flow, or in finding symmetric periodic orbits in more complex problems depending on a small parameter.     

On magnetostatic equilibrium in a stratified atmosphere

This is a study of the relationship between a magnetic field and its embedding plasma in static equilibrium in a uniform gravity. The ideal gas law is assumed. A system invariant in a given direction is treated first. We show that an exact integral of the equation for force balance across field lines can be derived in a closed form. Using this integral, exact solutions can be generated freely by integrating directly for the distributions of pressure, density and temperature necessary to keep a given magnetic field in equilibrium. Particular solutions are presented for illustration with the solar atmosphere in mind. Extending the treatment to the general system depending on all three spatial coordinates, we arrive at the general form of a theorem of Parker that a magnetic field in static equilibrium must possess certain symmetries. We derive an equation involving the Euler potentials of the magnetic field stipulating these necessary symmetries. Only those magnetic fields satisfying this equation can be in static equilibrium and for these fields, the endowed symmetries make the construction of exact solutions an essentially two dimensional problem as exemplified by the special case of invariance in a given direction.     

A Geometric Interpretation of Integrable Motions

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where Newtonian motions are seen as geodesics of suitable -mechanical- manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tensor fields correspond to hidden symmetries of non-Noetherian kind. Explicit expressions for Killing tensor fields are given for the N = 2 Toda model, and for a modified Hénon-Heiles model, recovering the already known analytic expressions of the second conserved quantity besides energy for each model respectively.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

Exact solutions of the Lane–Emden-type equation

We classify the Lane–Emden-type equation xy^″+ny^′+x^νf(y)=0 with respect to the standard Lagrangian according to the Noether point symmetries it admits. First integrals of the various cases, which admit Noether point symmetries, and reduction to quadratures for these cases are obtained. Six cases result in new solutions.     

Similarity considerations and conservation laws for magneto-static atmospheres

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A. Similarity solutions of the elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained from a consideration of the invariance group of the elliptic equation. The importance of symmetries of the elliptic equation also appears in the determination of conservation laws. It turns out that the elliptic equation can be written as a variational principle, and the symmetries of the variational functional lead (via Noether's theorem) to conservation laws for the equation. As an example of the application of the similarity solutions, we construct a model magnetostatic atmosphere in which the current density J is proportional to the cube of the magnetic potential, and falls off exponentially with distance vertical to the base, with an e-folding distance equal to the gravitational scale height. The solutions show the interplay between the gravitational force, the J × B force (B, magnetic field induction) and the gas pressure gradient.     

Symmetric planar central configurations of five bodies: Euler plus two

We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.     

Symmetries, Reduction and Relative Equilibria for a Gyrostat in the Three-body Problem

The problem of three bodies when one of them is a gyrostat is considered. Using the symmetries of the system we carry out two reductions. Global considerations about the conditions for relative equilibria are made. Finally, we restrict to an approximated model of the dynamics and a complete study of the relative equilibria is made.     

Albedo distribution on Io's surface

A visual albedo distribution model for all hemispheres of Io's surface has been synthesized from available Earth-based and spacecraft image and photometric data. The resulting model indicates some interesting patterns and symmetries on Io's surface: The dark polar caps are shifted off Io's rotational axis and are eliptical rather than circular in shape, with extensions toward the sub-Jupiter and anti-Jupiter points on Io; equatorial bright areas are located approximately on a great circle about Io, the plane of which is tilted approximately 15° toward Io longitude 60°. These and other indicated features may be clues to understanding the endogenic and exogenic processes that have resulted in Io's present observed surface characteristics.     

Noether gauge symmetry approach in f(R) gravity

We discuss the f(R) gravity model in which the origin of dark energy is identified as a modification of gravity. The Noether symmetry with gauge term is investigated for the f(R) cosmological model. By utilization of the Noether Gauge Symmetry (NGS) approach, we obtain two exact forms f(R) for which such symmetries exist. Further it is shown that these forms of f(R) are stable.     

Periodic orbits around a rotating ellipsoid

This paper studies families of symmetric periodic satellite orbits around a rotating triaxial ellipsoid. Existence of the families of orbits is established, and Morse's lemma is used to analyze their bifurcations. Several consequences of the many symmetries of the ellipsoid are discussed.     

Coupled motion of rigid bodies about their center of mass

A study is made of the motion of a system consisting of two rigid bodies coupled by a massless rigid boom. Relative translational and rotational motions are examined with the assumption that no external forces are acting on the system. For specific sets of initial conditions and assumptions on the symmetries of the two bodies, nontrivial analytic solutions are observed. The stability and the internal torques are also examined for a few selected cases.This research was conducted while the author was a senior research associate of the National Research Council at the National Aeronautics and Space Administration (NASA) Lyndon B. Johnson Space Center.     

The microwave background and (m,Z) relations in a tilted cosmological model

The structure of the cosmic microwave background temperature is studied in the context of a Bianchi type-V tilted cosmological model. First integrals of the equations for the null geodesics are found by use of the symmetries of the model, enabling the celestial temperature distribution to be found. The quadrupole and dipole moments are calculated for some models, suggesting that the observed anisotropy in the cosmic microwave background can be understood in the context of a Bianchi type-V model of the Universe. The apparent magnitude-redshift relations are also calculated for these models.     

Gravitational collapse of gas clouds with spherical,cylindrical or plane symmetry in the linear wave flow approximation

The equations of gas dynamics are solved, quasi-analytically by applying McVittie's method for spherical, cylindrical and plane configurations. The hypothesis of linear wave flow is applied and it is assumed that the final state of collapsing clouds is a hydrostatic equilibrium state, determined by complete polytropes. Complete analytical solutions are found when the generalized (to the three symmetries) Emden equation admits of analytical solutions. Otherwise the solutions are left in terms of the numerical solutions of the Emden equation. Numerical solutions to the Emden equation in the plane case are found and tabulated. A strong dependence of amplification, of density, pressure and temperature of the gas, on the symmetry is found. In addition, it is conclude that the flow remains subsonic, during the collapse, except toward the boundaries of the collapsing clouds.