International Journal of Geometric Methods in Modern Physics

We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function $1$. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

Geometrical approach to continuum mechanics allowing an atomistic treatment is developed. It suggests a body as a surface immersed into ambient space. Four kinds of surfaces inherent to basic deformations determine classical dynamics of the atoms alongside their spatial coordinates. For uniaxial stretch, this surface is isomorphic to the cylinder. For a simple shear, it is the twisted cylinder, for a bending — a cone, and for a torsion — a helicoid. Scalar parameters of the metric of these surfaces that are stretch ratios, shear, cone, and torsion angles used instead of the strain tensor constitute extra coordinates of the particles in hypersurface. Correspondingly, vector-valued tensions are used instead of stress. In the case of pure deformation, the normal and extra coordinates of atoms obey the classical equation of motion admitting periodic boundary conditions. It is shown how the employing of artificial orders of freedom leads to a minimum of elastic energy. In boundary problems, the system of governing equations unambiguously reproduce the stress–strain–displacement relations with the help of using the tension boundary theorem instead of the Cauchy traction principle. The derived direct method of elastostatic makes the system of governing equations closed and the compatibility condition unnecessary. Commonest examples of bending problem normally following from the Airy stress function illustrate the new concept.

The equation for fluid flow in porous media is analyzed in this paper with the aid of Lie symmetry method (LSM) and invariant subspace method (ISM). Infinitesimal generators, the entire geometric fields of the vectors and the symmetry groups of the equation being considered are given. One-dimensional optimal systems of sub-algebra are reported with corresponding reduced nonlinear ordinary differential equations. By means of ISM, we determine the exact solutions and invariant subspaces (ISs) for the equation under consideration. Moreover, with the aid of the new theorem of conservation, we establish the conservation laws (CLs) for the governing equation. The construction of the conserved vectors reveals the integrability and existence of soliton solutions of the equation for fluid flow in porous media.

In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order $n$ deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.

In this paper, we find exact string cosmological solutions for FRW cosmology, by using the Hojman symmetry approach. The string cosmology under consideration includes a scalar field $\psi (t)$ with the potential $W(\psi )$, and a totally antisymmetric field strength $H_{\mu \nu \rho }$ which is specifically defined in terms of the scale factor $a(t)$. We show that for this string cosmology, Hojman conserved quantities exist using which new exact solutions for the scale factor and the scalar field are obtained for specific potentials $W(\psi )$ with some free parameters. The presence of these parameters, together with those of arising from Hojman symmetry, is an important advantage using which one can construct the solutions $a(t)$ and $\psi (t)$ with variety of cosmological behaviors.

This paper deals with the Newton–Wigner position observable for Poincaré-invariant classical systems. We prove an existence and uniqueness theorem for elementary systems that parallels the well-known Newton–Wigner theorem in the quantum context. We also discuss and justify the geometric interpretation of the Newton–Wigner position as “center of spin”, already proposed by Fleming in 1965 again in the quantum context.

If a three-dimensional $N(k)$-contact metric manifold $M$ admits a Yamabe soliton of type $(M,g,V)$, then the manifold has a constant scalar curvature and the flow vector field $V$ is Killing. Furthermore, either $M$ has a constant curvature $k$ or the flow vector field $V$ is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional $N(k)$-contact metric manifold $M$ admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature $k$. Finally, we have given an example to verify our result.

We analyze the interaction of the induced electric dipole moment of a neutral particle with an electric field in elastic medium with a charged disclination from a semiclassical point of view. We show that the interaction of the induced electric dipole moment of a neutral particle with an electric field can yield an attractive inverse-square potential, where it is influenced by the topology of the disclination. Then, by using the Wentzel, Kramers and Brillouin approximation based on the Langer transformation, we show that the centrifugal term of the radial equation must be modified due to the influence of the topology of the disclination. Besides, we obtain the bound states solutions to the Schrödinger equation.

In this paper, we have presented a cosmological model where a phantom scalar field is minimally coupled to dark matter component. Noether symmetry method was applied both to investigate the cosmological solution and to find out what is the form of the potential of scalar field and the unknown function in the considered model. By using this method, these forms are resulted as trigonometric functions. Also, the obtained cosmological solutions are compatible with observations describing the accelerated expansion of the Universe. Furthermore, it turns out that the effective equation of state parameter in the model can cross the phantom divide line.

We describe a simple approach for a possible connection between the exterior and interior of black-holes. According to our result, the Schwarzschild radius determines a $2$-dimensional sphere that separates two worlds: the exterior, where ordinary matter moves with velocities less than the light velocity, and the interior, where tachyons move with velocities greater than the light velocity. Moreover, we find a geometric map in the complex plane that connects ordinary matter with faster than light particles. Motivated by these results, we also find that the exterior and the interior structures of black-holes can be unified in a world of $(5+5)$-dimensions. Finally, we comment about the possibility that superluminal particles in the interior of a black-hole may provide an alternative solution of the rotation curves of spiral galaxies.

We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time $t$ and shown to have norm $1$ which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.

In this paper, we analyze the weak gravitational lensing in the context of Einstein-nonlinear-Maxwell–Yukawa black hole. To this desire, we derive the deflection angle of light by Einstein-nonlinear-Maxwell–Yukawa black hole using the Gibbons and Werner method. For this purpose, we obtain the Gaussian curvature and apply the Gauss–Bonnet theorem to find the deflection angle of Einstein-nonlinear-Maxwell–Yukawa black hole in weak field limits. Moreover, we derive the deflection angle of light in the influence of plasma medium. We also analyze the graphical behavior of deflection angle by Einstein-nonlinear-Maxwell–Yukawa black hole in the presence of plasma as well as non-plasma medium.

All classes of spatially homogeneous spacetime models in the generalized scalar–tensor theory of gravitation are found to allow the integration of the equations of motion of test particles and the eikonal equation by the method of separation of variables by type (3.1). Three classes of exact solutions are obtained that relate to Shapovalov wave-like spacetime models. The resulting spacetime models are of types IV, VI and VII according to the Bianchi classification and type N according to Petrov classification.

In this work, LRS Bianchi type-I cosmological model with perfect fluid source in $f(R,T)$ gravity theory, where $R$ is the Ricci scalar and $T$ is the trace of the stress energy-momentum tensor, has been studied in order to investigate early time deceleration and late time acceleration of the universe. By proposing a new special form of time-varying deceleration parameter in terms of Hubble parameter, the exact solution of the field equations has been obtained. The physical and geometric quantities of the model have been derived and their evolution has been discussed. Our model has an initial singularity and initially exhibits decelerating expansion and transits to accelerating expansion phase at last eras. The nature of the matter source of the model is consistent with the standard model in frame of the structure formation.

In this paper, we show that a Finsler warped product metric is of almost vanishing $H$-curvature if and only if it is of almost vanishing $\chi$-curvature. Furthermore, the corresponding one form is exact.

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear $(2+1)$-dimensional time-fractional Kramers equation via Riemann–Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear $(1+1)$-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov’s method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.