Lie symmetry and conserved quantity of a system of first-order differential equations

This paper focuses on studying the Lie symmetry and a conserved quantity of a system of first-order differential equations. The determining equations of the Lie symmetry for a system of first-order differential equations, from which a kind of conserved quantity is deduced, are presented. And their general conclusion is applied to a Hamilton system, a Birkhoff system and a generalized Hamilton system. Two examples are given to illustrate the application of the results.

Lie Symmetry and Conserved Quantity of Three-Order Lagrangian Equations for Non-conserved Mechanical System

基于无穷小并且一参数转变，三顺序的 Lagrangian 方程的谎言对称的问题被学习了。在谎言转变下面，使 Lagrangian 方程三顺序未改变的足够、必要的条件并且不变在这篇论文被获得。

Lie Symmetry and Hojman Conserved Quantity of Maggi Equations

Lie symmetry of Maggi equations is studied. Its determining equation and restriction equation of nonholonomic constraint are given. A Hojman conserved quantity can be deduced directly by using the Lie symmetry. An example is given to illustrate the application of the result.

Noether-Lie symmetry and conserved quantities of mechanical system in phase space

In this paper, a new kind of symmetry and its conserved quantities of a mechanical system in phase space are studied. The definition of this new symmetry, i.e. a Noether-Lie symmetry, is presented, and the criterion of this symmetry is also given. The Noether conserved quantity and the generalized Hojman conserved quantity of the Noether Lie symmetry of the system are obtained. The Noether-Lie symmetry contains the Noether symmetry and the Lie symmetry, and has more generalized significance.

Lie Symmetrical Hojman Conserved Quantity of Relativistic Mechanical System

In this paper, we study the Lie symmetrical Hojman conserved quantity of a relativistic mechanical system under general infinitesimal transformations of groups in which the time parameter is variable. The determining equation of Lie symmetry of the system is established. The theorem of the Lie symmetrical Hojman conserved quantity of the system is presented. The above results are generalization to Hojman's conclusions, in which the time parameter is not variable and the system is non-relativistic. An example is given to illustrate the application of the results in the last.

New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical Systems in Phase Space

The new types of conserved quantities,which are directly induced by Lie symmetry of nonholonomicmechanical systems in phase space,are studied.Firstly,the criterion of the weak Lie symmetry and the strong Liesymmetry are given.Secondly,the conditions of existence of the new type of conserved quantities induced by the weakLie symmetry and the strong Lie symmetry directly are obtained,and their form is presented.Finally,an Appell-Hamelexample is discussed to further illustrate the applications of the results.

Noether-Lie Symmetry of Generalized Classical Mechanical Systems

在这篇论文， Noether 谎言对称和概括古典机械系统的保存数量被学习。定义和为在组的一般无穷小的转变下面的系统的 Noether 谎言对称的标准被给。Noether 保存了数量， Hojman 保存了从 Noether 谎言对称推出的数量被获得。一个例子被给说明结果的申请。

The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems

This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.

The Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass

This paper studies the Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass. The discrete Euler-Lagrange equation and energy evolution equation are derived by using a total variational principle. The invariance of discrete equations under infinitesimal transformation groups is defined to be Lie symmetry. The condition of obtaining the Noether conserved quantities from the Lie symmetries is also presented. An example is discussed for applications of the results.

Lie Symmetry and Conserved Quantities for Mechanical-Electrical Systems

在这篇论文，我们为一个机械电的系统学习谎言对称和保存数量。为这个系统的谎言对称的标准被给。概括 Hojman 保存了数量并且概括了保存数量从谎言对称推出了的 Lutzky 因为系统被获得。一个例子被举说明结果。

On Form Invariance,Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.

Noether symmetry and Lie symmetry of discrete holonomic systems with dependent coordinates

The Noether symmetry, the Lie symmetry and the conserved quantity of discrete holonomic systems with dependent coordinates are investigated in this paper. The Noether symmetry provides a discrete Noether identity and a conserved quantity of the system. The invariance of discrete motion equations under infinitesimal transformation groups is defined as the Lie symmetry, and the condition of obtaining the Noether conserved quantity from the Lie symmetry is also presented. An example is discussed to show the applications of the results.

Non-Noether Conserved Quantity of Poincaré-Chetaev Equations of a Generalized Classical Mechanics

在谎言对称的 non-Nocther 保存了的现在的纸，在谎言组的一般无穷小的转变下面的概括古典力学的 Poincare-Chetaev 方程的数量被讨论。首先，我们建立方程的 Liesymmetry 的决定方程。第二，谎言对称的 non-Noether 保存了方程的数量被推出。最后，一个例子被给说明结果的申请。

The Invariance of the Differential Equationsand Conserved Quantities

In this paper,a new form conserved quantity of differential equations is presented.The conserved quantity is constructed only based on the general Lie group of transformation vector of the differential equations.The first-order and second-order differential equations are studied,respectively.Two theorems concerning conserved quantities are proved.The relations between these theorems and preious conservation laws are discussed.A condition is given to exclude trivial conserved quantities.Finally,we give two examples to illustrate the application of the results.

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.

LIE SYMMETRIES AND CONSERVED QUANTITIES OF ROTATIONAL RELATIVISTIC SYSTEMS

The Lie symmetries and conserved quantities of the rotational relativistic holonomic and nonholonomic systems were studied. By defining the infinitesimal transformations' generators and by using the invariance of the differential equations under the infinitesimal transformations, the determining equations of Lie symmetries for the rotational relativistic mechanical systems are established. The structure equations and the forms of conserved quantities are obtained. An example to illustrate the application of the results is given.

THE LIE SYMMETRIES AND CONSERVED QUANTITIES OF VARIABLE-MASS NONHOLONOMIC SYSTEM OF NON-CHETAEV'S TYPE IN PHASE SPACE

The Lie symmetries and the conserved quantities of a variable mass nonholonomic system of non-Chetaev's type are studied by introducing the infinitesimal transformations of groups in phase space. By using the invariance of the differential equations of motion under the infinitesmal transformations of groups, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equations and the conserved quantities are obtained. An example is given to illustrate the application of the result.

Lie symmetries and conserved quantities for a two-dimentional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation ot concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.

Lie symmetry analysis,explicit solutions,and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses,we consider the time-fractional Fisher equation in two-dimensional space.Through the use of the Riemann-Liouville derivative approach,the well-known Lie point symmetries of the utilized equation are derived.Herein,we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries.The diminutive equation’s derivative is in the Erdélyi-Kober sense,whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time.The conservation laws for the dominant equation are built using a novel conservation theorem.Several graphical countenances were utilized to award a visual performance of the obtained solutions.Finally,some concluding remarks and future recommendations are utilized.