Unified symmetry of nonholonomic mechanical systems with variable mass

Based on the total time derivative along the trajectory of the system the definition and the criterion for a unified symmetry of nonholonomic mechanical system with variable mass are presented in this paper. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, are also obtained, An example is given to illustrate the application of the results.

Lie symmetry and the generalized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion

Lie symmetry and the generalized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion are studied. The determining equation of Lie symmetry of Nielsen equations for a variable mass holonomic system of relative motion under the infinitesimal transformations of groups is given. The expression of generalized Hojman conserved quantity deduced directly from Lie symmetry for a variable mass holonomic system of relative motion is obtained. An example is given to illustrate the application of the results.

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system of relative motion

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system of relative motion are studied.The definition and criterion of the Mei symmetry of Appell equations for a variable mass holonomic system of relative motion under the infinitesimal transformations of groups are given.The structural equation of Mei symmetry of Appell equations and the expression of Mei conserved quantity deduced directly from Mei symmetry for a variable mass holonomic system of relative motion are gained.Finally,an example is given to illustrate the application of the results.

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system are investi- gated. Appell equations and differential equations of motion for a variable mass holonomic system are established. A new expression of the total first derivative of the function with respect of time t along the systematic motional track curve, and the definition and the criterion of Mei symmetry for Appell equations under the infinitesimal transformations of groups are given. The expressions of the structural equation and Mei conserved quantity for Mei symmetry in Appell are obtained. An example is given to illustrate the application of the results.

Conformal invariance and conserved quantities of a general holonomic system with variable mass

Conformal invariance and conserved quantities of a general holonomic system with variable mass are studied. The definition and the determining equation of conformal invariance for a general holonomic system with variable mass are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition under which the conformal invariance would be the Lie symmetry of the system under an infinitesimal oneparameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.

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.

Symmetry of Lagrangians of a holonomic variable mass system

The symmetry of Lagrangians of a holonomic variable mass system is studied. Firstly, the differential equations of motion of the system are established. Secondly, the definition and the criterion of the symmetry of the system are presented. Thirdly, the conditions under which there exists a conserved quantity deduced by the symmetry are obtained. The form of the conserved quantity is the same as that of the constant mass Lagrange system. Finally, an example is shown to illustrate the application of the result.

GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper, the Gibbs-Appell's equations of motion are extended to the most general variable mass nonholonomie mechanical systems. Then the Gibbs-Appell's equations of motion in terms of generalized coordinates or quasi-coordinates and an integral variational principle of variable mass nonlinear nonholonomie mechanical systems are obtained. Finally, an example is given.

THE PRINCIPLES OF LEAST ACTION OF VARIABLE MASS NONHOLONOMIC NONCONSERVATIVE SYSTEM IN NONINERTIAL REFERENCE FRAMES

This paper presents the generalized principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frame, proves the equivalence between Holder form and Suslov form, and then obtains differential equations of motion of variable mass nonholonomic nonconservative system in noninertial reference frame.

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic system of Chetaev's type with variable mass

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic, non-conservative system of Chetaev＇s type with variable mass are studied. The differential equations of motion of the Nielsen equation for the system, the definition and criterion of Mei symmetry, and the condition and the form of Mei conserved quantity deduced directly by Mei symmetry for the system are obtained. An example is given to illustrate the application of the results.

Mei symmetry and generalized Hojman conserved quantity for variable mass systems with unilateral holonomic constraints

This paper studies Mei symmetry which leads to a generalized Hojman conserved quantity for variable mass systems with unilateral holonomic constraints. The differential equations of motion for the systems are established, the definition and criterion of the Mei symmetry for the systems are given. The necessary and sufficient condition under which the Mei symmetry is a Lie symmetry for the systems is obtained and a generalized Hojman conserved quantity deduced from the Mei symmetry is got. An example is given to illustrate the application of the results.

FORM INVARIANCE AND LIE SYMMETRY OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEM

The form invariance and Lie symmetry of a variable mass nonholonomic mechanical system is studied. The definition and the criterion and the conserved quantity of form invariance and Lie symmetry for the variable mass nonholonomic mechanical system are given. The relation between the form invariance and Lie symmetry is obtained. An example is given to illustrate the application of the result.

INTEGRATION METHOD FOR THE DYNAMICS EQUATION OF RELATIVE MOTION OF VARIABLE MASS NONLINEAR NONHOLONOMIC SYSTEM

In this paper, the integration methods of dynamics equations of relative motion of variable mass nonlinear nonholonomic system are given such as the gradient method, the single-component method and the field method. Firstly, the dynamics equations are written in the canonical form and the field form. Secondly, the gradient method, the single-component method and the field method are used to integrate the dynamics equations of the corresponding constant mass holonomic system in inertial reference frame respectively. With the restriction of nonholonomic constraints to the initial conditions being considered, the solutions of the dynamics equations of variable mass nonlinear nonholonomic system in noninertial reference frame are obtained.

Conformal invariance for nonholonomic system of Chetaev's type with variable mass

Conformal invariance and conserved quantities for a nonholonomic system of Chetaev＇s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system＇s con- formal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper, Routh 's 'equations for the mechanical systems of the variable mass with nonlinear nonholonomic constraints of arbitrary orders in a noninertial reference system have been deduced not from any variational principles, but from the dynamical equations of Newtonian mechanics. And then again the other forms of equations for nonholonomic systems of variable mass are obtained from Routh's equations.

On the Quantum Mechanical Treatment of the Bateman-Morse-Feshbach Damped Oscillator with Variable Mass

The harmonic oscillator with time-dependent (indefinite and variable) mass subject to the force proportional to velocity is studied by extending Bateman’s dual Lagrangian and Hamiltonian formalism. To study the quantum analog of such a dissipative system, the Batemann-Morse-Feshback classical Hamiltonian of the damped harmonic oscillator with varying (time-dependent) mass is canonically quantized. In order to discuss the stability of the quantum dissipative system due to the influence of varying mass and the dissipative force, we derived a formula for the vacuum state of the dissipative system with the help of quantum field theoretical framework. It is shown that the formula based on this simple model could be used to study the influence of dissipation such as the instability due to the dissipative force and/or the variable mass. It is understood that the change in the oscillator mass corresponds to a control parameter in quantum dissipative systems.

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

The first integrals and their conditions of existence for variable massnonholonomic system in noninertial relerence frames are obtained,and the canonicalequations and the variation equations of the system are extended. It is proved that using the first integral we can construct the integral invariant of the system.Finally,a series of deductions and an example are 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.

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

With classical variable mass and relativistic variable mass cases being considered.the relativistic D' Alembert principles of Lagrange form Nielsen form and Appell. form for variable mass controllable mechanical system are given the relativistic Chaplygin equation. Nielsen equation and Appell equation .for variable mass controllable mechanical system in quasi-coordinates and generalized- coordinates are obtained, and the equations of motion of relativistic controllable mechanical system for holonomic system and constant mass system are diseussed

THE EQUATION OF MOTION FOR THE SYSTEM OF THE VARIABLE MASS IN THE NON－LINEAR NON－HOLONOMIC SPACE

The dot product of bases vectors on the super-surface of constraints of the nonlinear non-holonomic space and Mesherskii equations may act as the equations of fundamental dynamics of mechanical system for the variable mass.These are very simple and convenient for computation.From these known equations,the equations of Chaplygin,Nielson,Appell,Mac-Millan et al.are deriv d;it is unnecessary to introduce the definition if Appell-Chetaev or Niu Qinping for the virtual displacement.These are compatible with the D'Alembert-Lagrange's principle.