Local Lie Group Of Transformations Research Articles

Construction of first-order invariant differential operators

The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily.

Similarity solutions for cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

In this paper, the propagation of cylindrical shock wave in rotating non-ideal gas under adiabatic flow condition using Lie group of transformation method is investigated. The density is assumed to be constant and azimuthal fluid velocity is assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the Local Lie group of transformations bring about two different cases of solutions i.e. with a power-law and exponential-law shock paths. Numerical solutions are obtained for both the cases. Distribution of gasdynamical quantities is illustrated through figures. It is obtained that the reduced flow variables pressure and azimuthal fluid velocity decrease in general, whereas density and radial fluid velocity increase in case of power-law shock path. The reduced azimuthal fluid velocity decreases, whereas reduced density, pressure and radial fluid velocity increase in case of exponential-law shock path. Also, it is obtained that shock strength decreases with increase in value of adiabatic exponent or gas non-idealness parameter, whereas it increases due to increase in ambient azimuthal fluid velocity exponent.

Similarity solutions for magnetogasdynamic shock waves in a rotating ideal gas using the Lie group-theoretic method

The propagation of a cylindrical shock wave under the influence of an azimuthal magnetic field in a rotating medium for adiabatic flow conditions is investigated using the Lie group transformation method. The density, magnetic field, and azimuthal and axial fluid velocities are assumed to vary in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about different cases of solutions, i.e., with a power-law shock path, with an exponential-law shock path, and a particular case of a power-law shock path. Numerical solutions are obtained in the case of a power-law shock path and exponential-law shock path. The distributions of gasdynamical quantities are discussed based on figures. The effects of varying the values of the adiabatic exponent $$\gamma $$ , Alfven–Mach number $$M_\mathrm{A}^{-2}$$ , ambient azimuthal fluid velocity variation index $$\lambda _{1}$$ , and ambient density variation index $$\phi $$ on the flow variables and shock strength are studied. With an increase in the adiabatic exponent or the strength of the magnetic field, the shock strength decreases. However, an increase in the ambient density variation index or ambient azimuthal fluid velocity variation index results in an increase in the shock strength. The numerical calculations are carried out using Mathematica software.

Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method

The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.

Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating ideal gas using Lie Group theoretic method: Isothermal flow

The propagation of cylindrical shock wave under the influence of axial magnetic field in rotating medium under isothermal flow condition is investigated. The density, magnetic field and azimuthal and axial components of fluid velocity are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for infinitesimals of the Local Lie group of transformations bring about three different cases of solutions, i.e. with power law shock path, exponential law shock path and a particular case of power law shock path. Numerical solutions are obtained in the cases of power law and exponential law shock paths. Distribution of gasdynamical quantities are discussed through figures. The effects of variation in values of Alfven-Mach number [Formula: see text], ambient azimuthal velocity index [Formula: see text] and ambient density index [Formula: see text] are studied on flow variables and on shock strength. The numerical integration is done using software Mathematica. It is obtained that magnetic field has a decaying effect on shock strength. Also, increase in value of ambient density or ambient azimuthal velocity variation index in the case of power law shock path and increase in value of ambient density variation index in case of exponential law shock path have the decaying effect on shock strength.

Similarity solutions using Lie group theoretic method for cylindrical shock wave in self-gravitating perfect gas with axial magnetic field: isothermal flow

Propagation of cylindrical shock wave in a self-gravitating perfect gas under the influence of axial magnetic field using Lie group of transformation method is investigated. The flow is considered to be isothermal. Density and magnetic field are assumed to be varying in the undisturbed medium. Two different cases of solutions are brought out by the arbitrary constants appearing in the expressions of infinitesimals of local Lie group of transformations. One is with a power law shock path and the other one is with an exponential law shock path. Numerical solutions are obtained for both the cases of power law and exponential law shock paths. The effects of variation in Alfven-Mach number, gravitational parameter and ambient density variation index for power law shock path and effects of variation in Alfven-Mach number, gravitational parameter and ambient magnetic field variation index on the flow variables in the case of exponential law shock path are studied. Also the effects of increase in value of gravitational parameter and in the strength of ambient magnetic field on the shock strength are investigated. The increase in value of Alfven-Mach number leads to the increase in the density ratio which infers to the decrease in shock strength.

Similarity method for the study of strong shock waves in magnetogasdynamics

Abstract In this paper, a non-dimensional unsteady adiabatic flow of a plane or cylindrical strong shock wave propagating in plasma is studied. The plasma is assumed to be an ideal gas with infinite electrical conductivity permeated by a transverse magnetic field. A self-similar solution of the problem is obtained in terms of density, velocity and pressure in the presence of magnetic field. We use the method of Lie group invariance to determine the class of self-similar solutions. The arbitrary constants, occurring in the expressions of the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. A particular case of the collapse of an imploding shock is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of flow variables. Our results are found in good agreement with the known results. All computational work is performed by using software package MATHEMATICA.

Similarity method for imploding strong shocks in a non-ideal relaxing gas

Self-similar solutions arise naturally as special solutions of system of partial differential equations (PDEs) from dimensional analysis and, more generally, from the invariance of system of PDEs under scaling of variables. Usually, such solutions do not globally satisfy imposed boundary conditions. However, through delicate analysis, one can often show that a self-similar solution holds asymptotically in certain identified domains. In the present paper, it is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems. In particular, there is a discussion of the role of symmetries in the context of self-similar dynamics. We use the method of Lie group invariance to determine the class of self-similar solutions to a problem involving plane and radially symmetric flows of a relaxing non-ideal gas involving strong shocks. The ambient gas ahead of the shock is considered to be homogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the values of the self-similarity exponent and the profile of the flow variables behind the shock. All computations are performed using the computation package Mathematica.

Self-similar solutions in a plasma with axial magnetic field (θ-pinch)

In this paper, Lie group of transformation method is used to investigate the self-similar solutions for the system of partial differential equations describing a plasma with axial magnetic field (θ–pinch). The arbitrary constants occurring in the expressions for the infinitesimals of the local Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. A particular solution to the problem in one case has been found out.

Phenomenologically symmetric local lie groups of transformations of the space R s

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.

Classification of four-dimensional transitive local Lie groups of transformations of R 4 and their two-point invariants

In the theory of physical structures the classification of metric functions (both on a single set and on two ones) plays an important role. A metric function represents a two-point invariant of a certain local Lie transformation group. Moreover, one can uniquely restore this group with the help of the invariance condition. According to this theorem, in order to find all metric functions, it suffices to construct the complete classification of local Lie transformation groups. In this paper we classify Lie algebras of simply transitive local Lie groups of local transformations of a four-dimensional space, and then we define metric functions. The obtained results admit application in physics, in particular, in thermodynamics.

Group Classification of Equations of the Form y = f(x,y)

The problem of classification of ordinary differential equations of the form y″ = f(x,y) by admissible local Lie groups of transformations is solved. “Standard” equations are listed on the basis of the equivalence concept. The classes of equations admitting a one‐parameter group and obtained from the “standard” equations by invariant extension are described.

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles $$\prod :\mathbb{R}^{n + 1} \to \mathbb{R}^n $$ endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups $$\mathbb{G}$$ (1, n) and construct $$\mathbb{G}$$ (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups $$\mathbb{G}$$ (1, n), we find all affine nonholonomic $$\Gamma _{1^ - } ,\Gamma _{2^ - } $$ , and Γ1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations $$\mathbb{G}$$ (1, n) and also describe the corresponding $$\mathbb{G}$$ (1, n)invariant planar connections.

On new realizations of the poincare groups P (1,2) and P(2, 2)

We classify realizations of the Poincare groups P (1, 2) and P (2, 2) in the class of local Lie groups of transformations and obtain new realizations of the Lie algebras of infinitesimal operators of these groups.

Similarity solutions for converging shocks in a relaxing gas

In the present paper, we use the method of Lie group invariance to determine the class of self-similar solutions to a problem concerning plane and radially symmetric flows of a relaxing gas involving shocks of arbitrary strength. The ambient gas ahead of the shock is considered to be inhomogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the effects of relaxation and the ambient density on the self-similar exponent and the flow patterns.

On Lewis' exact invariant for the linear harmonic oscillator with time-dependent frequency

In this paper we purport to show that Lewis' exact invariant for a linear harmonic oscillator with a time-dependent frequency arises as a consequence of the invariance of the problem under local Lie groups of transformations. We will also show that Lewis' invariant can be deduced via a systematic perturbation theory for the case with slowly-varying frequency as a power series in the small parameter ϵ characterizing the slow variations in the frequency.

Similarity solutions of the Euler equation in the calculus of variations

A necessary and sufficient condition for the equivalence of the conformal invariance of the Euler equation and the divergence invariance of the associated variational integral is obtained in terms of the generators of the local Lie group of transformations under which invariance is studied.

An invariance theory for second-order variational problems

This paper investigates the invariance properties of second-order variational problems when the configuration space is subjected to an r-parameter local Lie group of transformations. In particular, the recent results of Hanno Rund on first-order problems are extended to the higher order case: A new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of the Zermelo conditions and Noether’s theorem are presented. The results are applied to a variational problem whose second-order Lagrangian depends upon a scalar field in Minkowski space, and some conformal identities are obtained.

The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces

The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces