Chaplygin Equations Research Articles

Study of wormhole stability in the framework of black hole surrounded by the pseudo-isothermal dark matter halo

This work is devoted to exploring the formation and stability of thin-shell wormholes developed through the two similar copies of black holes bounded by the pseudo-isothermal dark matter halo. It is found that the horizon radius of a black hole decreases in the appearance of a pseudo-isothermal dark matter halo. The primary goal of the work is to investigate the stable composition of such wormholes using the analysis of linearized radial perturbation. It is worth mentioning that the existence of a pseudo-isothermal dark matter halo reduces the violation of energy bounds for the developed thin-shell wormholes. We investigate the impact of variable equations of state, such as barotropic, variable Chaplygin, and phantom-like equations of state, on the stability of the wormholes. The inquiry highlights that the appearance of a pseudo-isothermal dark matter halo portrays remarkable importance in preserving the stable compositions of thin-shell wormholes. The wormholes show maximal stable conduct for the selection of pseudo-isothermal dark matter halo as compared to already published research charged as well as regular thin-shell wormholes. The results reveal light on the interplay between wormholes and pseudo-isothermal dark matter halo, which increases our understanding of both conjectures and their potential implications for further space travel.

Impact of cold dark matter and variable equations of state on the stability of thin-shell wormholes

The goal of this research is to better understand how two identical black hole copies, bounded by a cold dark matter halo, produce and stabilize thin-shell wormholes. The occurrence of a cold dark matter halo is discovered to increase the black hole’s horizon radius. Investigating the stable geometry of these wormholes by linearized radial perturbation analysis is the main objective of the work. It is noteworthy that the development of thin-shell wormholes with reduced energy limitations is associated with the presence of a cold dark matter halo. We study how the stability of the wormholes is affected by variable equations of state, including phantom-like, variable Chaplygin, and barotropic equations of state. The question emphasizes how crucial the existence of a cold dark matter halo is to preserving stable thin-shell wormhole configurations. The findings shed light on the interactions between a cold dark matter halo and wormholes, which improved our knowledge of both phenomena and their possible consequences for extraterrestrial travel.

Charged anisotropic composite stellar object with strange, polytropic and gaseous matter

In this study, we build a charged star model with three layers with distinct equations of state. The core, intermediate layer, and envelope layer obey linear, polytropic, and Chaplygin equations of state, respectively. The model satisfies the physical requirements for the matter variables, gravitational potentials, and stability conditions. We have generated within-acceptable range masses and radii for stars. We also recover the masses and radii for the stars PSRJ1903+0327, PSRJ1614-2230, SAXJ1808.4-3658, Vela X-1, and EXO1785-248 from earlier investigations which shows astrophysical significance of our model.

Exploring thin-shell dynamics in regular charged black hole through T-duality

We present a thin-shell in the background of a regular charged static black hole solution in the presence of T-duality effects for the Einstein–Maxwell system. The thin-shell are developed by matching the inner and outer manifolds through the well-known Visser’s cut-and-paste approach. We produce three types of thin-shells with an outer regular charged black hole from T-duality and different choices of inner manifolds named: 1. Thin-shell (inner flat spacetime), 2. Thin-shell Garvastars (inner de Sitter spacetime), and 3. Thin-shell wormholes (inner regular charged black hole from T-duality) We evaluate the surface stresses of a thin layer of matter contents at a thin-shell through Lanczos equations and also develop equations of motion of these constructed configurations. We also explore thin-shell stability through linearized radial perturbations at equilibrium shell radius with barotropic and generalized Chaplygin equations of state. We show that the stability of respective configurations enhances in the presence of exterior charged BH from T-duality. We conclude that thin-shell Garvastars are more stable than thin-shell and thin-shell wormholes.

Herglotz type conservation laws for nonconservative nonholonomic systems

The Herglotz variational principle offers an effective method for studying nonconservative system dynamics. The aim of this paper is to study the conservation laws of nonholonomic systems by using the Herglotz type generalized variational principle and establish Noether’s theorem and its inverse theorem for this system. In deriving the equations of motion, we use the Suslov definition of the reciprocity relation between differential and variational operations. First, the Herglotz type generalized variational principle is listed, and the Herglotz type Chaplygin equations for nonconservative nonholonomic systems are deduced. Second, Noether’s theorem and Noether’s inverse theorem are established, and the Herglotz type conservation laws are given. Finally, an example is provided to illustrate the practical implementation of the findings.

Three-layered super dense star with charged anisotropic fluid

We generate a three-layered super dense charged star with Van der Waals equation of state at the intermediate layer of stellar spheres. Our model comprises of three interior layers: the core, the intermediate, and the envelope described by linear, Van der Waals, and Chaplygin equations of state, respectively. The plots for the physical variables such as energy density, radial and tangential pressures, mass, anisotropy, and metric potentials are well behaved and agree with the astrophysical results. The stability and equilibrium factors are continuous and defined at some values of the parameters. We have also generated new stellar masses and radii which are in acceptable ranges. We have also regained previous masses and radii for the stars like PSRJ1903+0327, PSRJ1614-2230, SAXJ1808.4-3658, EXO1745-248, and EXO1785-248.

Reduction of Chaplygin equations for nonholonomic systems on time scales by means of the cyclic integral and energy integral

In this paper, the reduction of the Chaplygin equations on time scales by means of the cyclic integral and energy integral are proposed and studied. The cyclic integrals and generalized energy integrals for nonholonomic systems on time scales are given. The Chaplygin equations on time scales are reduced by means of the cyclic integrals and the generalized energy integrals. The results show that if the systems have a cyclic integral and an energy integral, then the Chaplygin equations on time scales can be reduced by at least one degree and the form of the Chaplygin equations holds.

Generalized Chaplygin equations for nonholonomic systems on time scales**Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province, China (Grant No. KYLX16-0414).

The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamilton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equations are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.

Conservation laws by means of a new mixed method

In this paper, by using a mixed approach, recently introduced by the authors, some conservation laws of partial differential equations are derived. The method merges the Ibragimov’s method and the one by Anco and Bluman. In particular, by applying this new mixed method, we determine all zero-th order conservation laws of Chaplygin and Shallow Water equations, as well as new conservation laws for a second order partial differential equation involving an arbitrary function.

On systems having Poincaré and Galileo symmetry

Using the wave equation in d≥1 space dimensions it is illustrated how dynamical equations may be simultaneously Poincaré and Galileo covariant with respect to different sets of independent variables. This provides a method to obtain dynamics-dependent representations of the kinematical symmetries. When the field is a displacement function both symmetries have a physical interpretation. For d=1 the Lorentz structure is utilized to reveal hitherto unnoticed features of the non-relativistic Chaplygin gas including a relativistic structure with a limiting case that exhibits the Carroll group, and field-dependent symmetries and associated Noether charges. The Lorentz transformations of the potentials naturally associated with the Chaplygin system are given. These results prompt the search for further symmetries and it is shown that the Chaplygin equations support a nonlinear superposition principle. A known spacetime mixing symmetry is shown to decompose into label-time and superposition symmetries. It is shown that a quantum mechanical system in a stationary state behaves as a Chaplygin gas. The extension to d>1 is used to illustrate how the physical significance of the dual symmetries is contingent on the context by showing that Maxwell’s equations exhibit an exact Galileo covariant formulation where Lorentz and gauge transformations are represented by field-dependent symmetries. A natural conceptual and formal framework is provided by the Lagrangian and Eulerian pictures of continuum mechanics.

Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.

Variational nature of nonlinear Chaplygin equations and their analogues

Variational nature of nonlinear Chaplygin equations and their analogues

The Tricomi and Frankl problems for generalized Chaplygin equations in multiply connected domains

The present paper deals with Tricomi and Frankl problems for generalized Chaplygin equations in multiply connected domains. We first give the representation of solutions of the Tricomi problem for the equations, and then prove the uniqueness and existence of solutions for the problem by a new method, i.e. the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used. Finally we discuss the Frankl problem for generalized Chaplygin equations in multiply connected domains.

On the dynamics of nonholonomic systems

In the development of nonholonomic mechanics one can observe recurring confusion over the very equations of motion as well as the deeper questions associated with the geometry and analysis of these equations. First of all, as far as the equations of motion themselves are concerned, the confusion mainly centered on whether or not the equations could be derived from a variational principle in the usual sense. Attempting to dissipate this confusion, in the present paper we deduce a new form of equations of motion which are suitable for both nonholonomic systems with either linear or nonlinear constraints and holonomic systems ( A -model). These equations are deduced from the principle of stationary action (or Hamiltonian principle) with nonzero transpositional relations. We show that the well-known equations of motion for nonholonomic and holonomic systems can be deduced from the A -model. For the systems which we call the generalized Vorones-Chaplygin systems we deduce the equations of motion which coincide with the Vorones and Chaplygin equations for the case in which the constraints are linear with respect to the velocity. An additional result is that the transpositional relations are different from zero only for those generalized coordinates whose variations (in accordance with the equations of nonholonomic constraints) are dependent. For the remaining coordinates, the transpositional relations may be zero.

Dark energy interacting with neutrinos and dark matter: a phenomenological theory

A model for a flat homogeneous and isotropic Universe composed of dark energy, dark matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos are supposed to interact with the dark energy. The dark energy is considered to obey either the van der Waals or the Chaplygin equations of state. The ratio between the pressure and the energy density of the neutrinos varies with the red-shift simulating massive and non-relativistic neutrinos at small red-shifts and non-massive relativistic neutrinos at high red-shifts. The model can reproduce the expected red-shift behaviors of the deceleration parameter and of the density parameters of each constituent.

General Tricomi–Rassias problem and oblique derivative problem for generalized Chaplygin equations

Many authors have discussed the Tricomi problem for some second order equations of mixed type, which has important applications in gas dynamics. In particular, Bers proposed the Tricomi problem for Chaplygin equations in multiply connected domains [L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958]. And Rassias proposed the exterior Tricomi problem for mixed equations in a doubly connected domain and proved the uniqueness of solutions for the problem [J.M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, Singapore, 1990]. In the present paper, we discuss the general Tricomi–Rassias problem for generalized Chaplygin equations. This is one general oblique derivative problem that includes the exterior Tricomi problem as a special case. We first give the representation of solutions of the general Tricomi–Rassias problem, and then prove the uniqueness and existence of solutions for the problem by a new method. In this paper, we shall also discuss another general oblique derivative problem for generalized Chaplygin equations.

On the motion of a heavy rigid body in an ideal fluid with circulation

We consider Chaplygin's equations [Izd. Akad. Nauk SSSR 3, 3 (1933)] describing the planar motion of a rigid body in an unbounded volume of an ideal fluid while circulation around the body is not zero. Hamiltonian structures and new integrable cases are revealed; certain remarkable partial solutions are found and their stability is examined. The nonintegrability of the system describing the motion of a body in the field of gravity is proved and the chaotic behavior of the system is illustrated.

A non-stationary analogue of Chaplygin's equations in one-dimensional gas dynamics

As an extension of the results obtained in [1], two equivalent uniformly divergent systems of equations are constructed in thespeedograph plane, each of which is the analogue of Chaplygin's equation in the hodograph plane. Each of the systems reduces to a linear second-order equation, in one case for the particle function (the Lagrange coordinate) ψ, and in the other for the time t. These systems possess an infinite set of exact solutions. It is shown that a uniformly divergent system of first-order equations correspond to each of these, and, related to them, the simplest non-linear homogeneous second-order equation in the modified events plane (ψ, t) and the conservation law in the events plant ( x, t). Clear relations are obtained between the velocities of the fronts of constant values of the newly constructed dependent variables and the velocity of sound. Examples are given which demonstrate the relation between the exact solutions with the uniformly divergent equations and the conservation laws of one-dimensional non-stationary gas dynamics and, simultaneously, enable one to compare the newly obtained results (the exact solutions, the equations and conservation laws, and the relations for the velocities of the front) with existing results, including those for plane steady flows. The so-called additional conservation laws, to which Godunov drew attention, are considered.

The Hess case in rigid-body dynamics

Generalizations of the Hess integral are presented for different forms of the equations of motions of rigid body. The general conditions for the existence of this integral, which is due to the presence of additional explicit symmetries of the equations of motion, are pointed out. Problems of reducing the order, of the explicit integration and the qualitative analysis of the motion of a rigid body subject to these conditions are considered. Analogues of Hess cases for a gyroscope in gimbals and the Chaplygin equations describing the fall of a rigid body in a fluid are indicated for the first time.

Chaplygin equations and an infinite set of uniformly divergent gas-dynamics equations

Chaplygin equations and an infinite set of uniformly divergent gas-dynamics equations