Infinitesimal Transformations Of Groups Research Articles

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.

Exact solutions of Burgers equation with moving boundary

In this paper, new symmetry reductions and similarity solutions for Burgers equation with moving boundary are obtained by means of Lie’s method of infinitesimal transformation groups, for a linearly moving boundary as well as a parabolically moving boundary. By using discrete symmetries, new analytical solutions for the problem under consideration are presented, for two cases of the moving boundary: one moving with constant velocity and another one rapidly oscillating.

ANALYTICAL SOLUTION OF UNSATURATED SOIL WATER FLOW FROM A POINT SOURCE

Water flow into unsaturated porous media is governed by the Richards’ partial differential equation expressing the mass conservation and Darcy’s laws. The Richards’ equation may be written in three forms,where the dependent variable is pressure head or moisture content, and the constitutive relationships between water content and pressure head allow for conversion of one form into the other. In the present paper, the “moisture-based" form of Richards’ equation is linearized by applying Kirchhoff’s transformation, whichcombines the soil water diffusivity and soil water content. Then the similarity method is used to obtain the analytical solution of wetting front position. This exact solution is obtained by means of Lie’s method of infinitesimal transformation groups. The predicted results of the analytical solution agreed well with available results of experiments and numerical solutions.

A Study for MHD Boundary Layer Flow of Variable Viscosity over a Heated Stretching Sheet via Lie-Group Method

The present work deals with the study of Magnetohydrodynamic (MHD) boundary layer flow over a heated stretching sheet with variable fluid viscosity. The fluid viscosity is assumed to vary as a linear function of temperature in the presence of uniform transverse magnetic field. The fluid is assumed to be electric ally conducting. Lie-group method is applied for determining symmetry reductions for the MHD boundary-layer equations. Lie-group method starts out with a general infinitesimal group of tran sformations under which the given partial differential equations are invariant. The determining equations are a set of linear diffe rential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variable s. After the group has been determined, a solution to the given partial differe ntial equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the n umber of independent variables of the system. The effect of the Hartmann number (M ), the viscosity parameter ( A ) and the Prandtl number ( Pr ) on the horizontal and vertical velocities, temperature pr ofiles, wall heat transfer and the wall shear stress (skin friction) , have been studied and the results are plotted.

Adiabatic Invariants of Herglotz Type for Perturbed Nonconservative Lagrangian Systems

Both conservative and nonconservative systems can be studied simultaneously using the differential variational principle of Herglotz type. For a perturbed system, in which parameters change with time, it is useful to find adiabatic invariants. Based on the Herglotz differential variational principle, we study the perturbation of infinitesimal transformations and adiabatic invariants for perturbed nonconservative Lagrangian systems. From the generalized Euler-Lagrange equation and the invariance condition for the Hamilton-Herglotz action under the group of infinitesimal transformations, we obtain an exact invariant of Herglotz type for a holonomic nonconservative system. We propose a definition of higher-order adiabatic invariants of Herglotz type and obtain such adiabatic invariants for nonconservative Lagrangian systems with small perturbations. We prove the corresponding inverse theorem of adiabatic invariants. As examples of using the obtained results, we consider an oscillator with square damping and a system with two degrees of freedom.

Conformal Invariance and Conserved Quantities of Nonmaterial Volumes

This paper investigates conformal invariance and conserved quantities of nonmaterial volumes. An infinitesimal transformation group and infinitesimal transformations vectors of generators are proposed. The definition of conformal invariance and the determining equation for the systems are presented. The necessary and sufficient conditions of conformal invariance being Noether symmetrical simultaneously by the action of infinitesimal transformations are given. The conformal factor is deduced from the conformal invariance and Noether symmetry. The corresponding Noether conserved quantities are derived with the aid of a structure equation. Three examples are given to illustrate application of the method, the corresponding conserved quantities are obtained under the conformal invariance and Noether symmetry.

Noether’s theorems of variable mass systems on time scales

Abstract This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

Lie symmetry analysis on time scales and its application on mechanical systems

The theory of time scales that can unify and extend continuous and discrete analysis has proved to be more accurate in modeling the dynamic process. The Lie symmetry approach, which is an effective way to deal with different kinds of dynamical equations in a variety of areas of applied science, is to be analyzed on time scales. We begin with the Lie group of point infinitesimal transformations on time scales and its corresponding extensions. And the invariance of dynamical equations on time scales under the infinitesimal transformations is discussed. More specifically, the Lie symmetries for dynamical equations of mechanical systems on time scales including Lagrangian systems on time scales, Hamiltonian systems on time scales, and Birkhoffian systems on time scales are investigated as applications. Thus, the corresponding conserved quantities for mechanical systems on time scales are derived by using the Lie symmetries. Examples are given to illustrate the application of the results.

Noether theorem for non-conservative systems with time delay on time scales

This paper deals with the Noether theorem for non-conservative systems with time delay on time scales. First, the Hamilton principle for the non-conservative system with time delay on time scales is proposed, and Lagrange equations with time delay on time scales are obtained. Second, based on the invariance of the Hamilton action for the non-conservative system with time delay under a group of infinitesimal transformations, the Noether symmetric transformations, the Noether quasi-symmetric transformations and the generalized Noether quasi-symmetric transformations of the system are defined on time scales, respectively, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for non-conservative systems with time delay on time scales are established. Some examples are given to illustrate the application of the results.

Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems

Herglotz proposed a generalized variational principle through his work on contact transformations and their connections with Hamiltonian systems and Poisson brackets, which provides an effective method to study the dynamics of nonconservative systems. In this paper, the variational problem of Herglotz type for a Birkhoffian system is presented and the differential equations of motion for the system are established. The invariance of the Pfaff–Herglotz action under a group of infinitesimal transformations and its connection with the conserved quantities of the system are studied, and Noether’s theorem and its inverse for the Herglotz variational problem are derived. The variational problem of Herglotz type for a Birkhoffian system reduces to the classical Pfaff–Birkhoff variational problem under classical conditions. Thus, it contains Noether’s theorem for the classical Birkhoffian system as a special case. And since Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics, the results we obtained contain Noether’s theorem of Herglotz variational problems for Hamiltonian systems and Lagrangian systems as special cases. In the end of the paper, we give two examples to illustrate the application of the results.

Noether’s theorem for fractional Birkhoffian systems of variable order

The Noether symmetries and conserved quantities of the Birkhoffian systems in terms of fractional derivatives of variable order are studied. Firstly, the Pfaff–Birkhoff–d’Alembert principle within fractional derivatives of variable order is obtained, and corresponding variable order fractional Birkhoff’s equations are deduced. Secondly, the invariance of the fractional Pfaff action of variable order is studied under the one-parameter group of infinitesimal transformations, and the definition of the variable order fractional conserved quantity is given. Finally, the Noether’s theorem for the fractional Birkhoffian system of variable order is established. At the end of this paper, an example is given to illustrate the application of the results.

Noether symmetries and conserved quantities for fractional forced Birkhoffian systems

In this paper Noether symmetries and conserved quantities for fractional forced Birkhoffian systems are studied. Firstly, a new fractional Pfaff–Birkhoff variational principle with Riemann–Liouville derivatives and generalized force is established. And from which the fractional forced Birkhoff equations are derived. Secondly, the definitions and the criteria of the invariance of a functional under an ε-parameter group of infinitesimal transformations are given. Finally, the fractional Noether theorems for forced Birkhoffian systems are obtained. And an illustrative example is presented.

Conformal invariance and Mei conserved quantity for generalized Hamilton systems with additional terms

Conformal invariance and Mei conserved quantity for generalized Hamilton systems with additional terms are studied. Under the infinitesimal transformations of group, the conformal invariance and Mei conserved quantity for generalized Hamilton systems with additional terms are studied. A necessary and sufficient condition of which the conformal invariance for the system is also Mei symmetry is present. Then the expression of Mei conserved quantity for the system is given. Finally, an example is given to illustrate the application of the result.

Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay

The Noether symmetries and the conserved quantities for fractional Birkhoffian systems with time delay in terms of Riemann–Liouville fractional derivatives are proposed and studied. First, the fractional Pfaff–Birkhoff principle with time delay is proposed, and the fractional Birkhoff's equations with time delay are obtained. Second, based on the invariance of the fractional Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasi-symmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for fractional Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.

Noether theorem for non-conservative systems with time delay in phase space based on fractional model

In this paper, we focus on discussing the fractional Noether symmetries and fractional conserved quantities for non-conservative Hamilton system with time delay. Firstly, the fractional Hamilton canonical equations of non-conservative system with time delay are established; secondly, based upon the invariance of the fractional Hamilton action with time delay under the infinitesimal transformations of group, we obtain the definitions and criterion of fractional Noether symmetric transformations, fractional Noether quasi-symmetric transformations and fractional generalized Noether quasi-symmetric transformations in phase space; finally, the relationship between the fractional Noether symmetries and fractional conserved quantities with time delay in phase space is established. At the end of the paper, some examples are given to illustrate the application of the results.

Accurate conserved quantity and approximate conserved quantity deduced from Noether symmetry for a weakly Chetaev nonholonomic system

Accurate conserved quantity and approximate conserved quantity deduced from Noether symmetry of Lagrange equations for a weakly Chetaev nonholonomic system were studied. First, the differential equations of motion of the system were established. Second, under the infinitesimal transformations of group, the definitions of Noether symmetry for the weakly Chetaev nonholonomic system and the first approximate system were given. Third, the first approximate Noether equation was mainly obtained, and the expressions of the accurate and approximate conserved quantities were deduced from the Noether symmetry for the weakly Chetaev nonholonomic system. Finally, an example was given to illustrate the application of the results.

Conformal invariance and conserved quantity of Mei symmetry for Appell equation in a holonomic system in relative motion

For a holonomic system in relative motion, the conformal invariance and the conserved quantity of Mei symmetry with Appell equations are investigated. First, by using the infinitesimal one-parameter transformation group and the infinitesimal generator vector, the definitions of Mei symmetry and the conformal invariance with Appell equations in a holonomic system in relative motion are given, and the determining equations of the conformal invariance of Mei symmetry for the system are derived. Relationship between the conformal invariance and Mei symmetry for the system is mainly studied. Then, by means of the structural equation which the gauge function satisfies, the expression of Mei conserved quantity deduced from Mei symmetry for the system is obtained. Finally, an example is given to illustrate the application of the result.

Lie Group Method for Studying the Heat Generation Effect on Freeconvection Laminar Boundary-layer Flow Over a Vertical Flat Plate

The nonlinear equations of heat and mass transfer in two-dimensional free-convection, laminar, boundary layer flow of a viscous incompressible fluid over a vertical plate with thermophoresis and heat generation effect have been considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Liegroup method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the heat generation parameter He, the Prandtl number Pr, the Schimted number Sc, the thermophoretic parameter τ, the solutal Grashof number Gc and the thermal Grashof number Gr on velocity, concentration and temperature have been studied and the results are plotted.

Inverse problem of Mei symmetry for a general holonomic system

Inverse problems in dynamics are the basic problems in astronautics, rocket dynamics, and motion planning theory, etc. Mei symmetry is a kind of new symmetry where the dynamical function in differential equations of motion still satisfies the equation's primary form under infinitesimal transformations of the group. Mei symmetry and its inverse problem of dynamics for a general holonomic system in generalized coordinates are studied. Firstly, the direct problem of dynamics of the system is proposed and solved. Introducing a one-parameter infinitesimal transformation group with respect to time and coordinates, the infinitesimal generator vector and its first prolonged vector are obtained. Based on the discussion of the differential equations of motion for a general holonomic system determined by n generalized coordinates, their Lagrangian and non-potential generalized forces are made to have an infinitesimal transformation, the definition of Mei symmetry about differential equation of motion for the system is then provided. Ignoring the high-order terms in the infinitesimal transformation, the determining equation of Mei symmetry is given. With the aid of a structure equation which the gauge function satisfies, the system's corresponding conserved quantities are derived. Secondly, the inverse problem for the Mei symmetry of the system is studied. The formulation of the inverse problem of Mei symmetry is that we use the known conserved quantity to seek the corresponding Mei symmetry. The method is: considering a given integral as a Noether conserved quantity obtained by Mei symmetry, the generators of the infinitesimal transformations can be obtained by the inverse Noether theorem. Then the question whether the obtained generators are Mei symmetrical or not is verified by the determining equation, and the effect of generators' changes on the symmetries is discussed. It has been shown from the studies that the changes of the generators have no effect on the Noether and Lie symmetries, but have effects on the Mei symmetry. However, under certain conditions, while adjusting the gauge function, changes of generators can also have no effect on the Mei symmetry. In the end of the paper, an example for the system is provided to illustrate the application of the result.

Form invariance and Mei conserved quantity for generalized Hamilton systems after adding additional terms

Form invariance and Mei conserved quantity for generalized Hamilton systems after adding additional terms are studied. By introducing infinitesimal transformation group and its infinitesimal transformation vector of generators, the definition and determining equations of the Mei symmetry for generalized Hamilton systems after adding additional terms are provided. By means of the structure equation satisfied by the gauge function, the Mei conserved quantity corresponding to the form invariance for the system is derived. Finally an illustrative example is given to verify the results.