A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether me...A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether method and the Poisson method, are given to solve a differential equation of first order, of which the way may be called the mechanical methodology in mathematics.展开更多
The theory of rotational relativistic mechanics is discussed and the theory of relativistic analytical mechanics' of the rotational systems is constructed. The relativistic generalized kinetic energy function for ...The theory of rotational relativistic mechanics is discussed and the theory of relativistic analytical mechanics' of the rotational systems is constructed. The relativistic generalized kinetic energy function for the rotational systems [GRAPHICS] and the generalized acceleration energy function [GRAPHICS] are constructed, and furthermore, the Hamilton principle and three kinds of D'Alembert principles are given. For the systems with holonomic constraints, the relativistic Lagrange equation, Nielsen equation, Appell equation and Hamilton canonical equation of the rotational systems are constructed; For the systems with nonholonomic constraints, the relativistic Routh equation, Chaplygin equation, Nielsen equation and Appell equation of the rotational systems are constructed; the relativistic Noether conservation law of the rotational systems are given too.展开更多
In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables: time t and arc coordinate s. Starting from the Lagrange equations of Cosserat ro...In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables: time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.展开更多
Nonlinear mechanics for a super-thin elastic rod with the biological background of DNA super-coiling macromolecules is an interdisciplinary research area of classical mechanics and molecular biology. It is also a subj...Nonlinear mechanics for a super-thin elastic rod with the biological background of DNA super-coiling macromolecules is an interdisciplinary research area of classical mechanics and molecular biology. It is also a subject of dynamics and elasticity because elastic bodies are analyzed via the theory of dynamics. It is in frontiers of general mechanics (dynamics and control). This dissertation is devoted to model a constrained super-thin elastic rod and analyze its stability in equilibrium. The existing research results are summarized. Analytical mechanics is systematically applied to model the elastic rod. The Schroedinger equation for complex curvatures or complex bending moments is, respectively, extended from the case of circular crosssections to that of non-circular ones. The equilibrium of a rod constrained on a surface is investigated.展开更多
Based on the nondestructive test data of operating railway tunnels in China, this paper summarizes the basic characteristics of the complex contact behavior between the rock mass and lining structure. The contact mode...Based on the nondestructive test data of operating railway tunnels in China, this paper summarizes the basic characteristics of the complex contact behavior between the rock mass and lining structure. The contact modes are classified into dense contact, local non-contact, and loose contact. Subsequently, the corresponding mechanical model for each contact mode is developed according to its mechanical characteristics using the complex variable method. In the proposed mechanical model, a special algorithm is introduced to detect whether the local non-contact zone is re-contacted. Besides, a novel conformal mapping method is also proposed to accurately calculate the mechanical response of the concrete lining. Finally, the accuracy of the proposed method is verified by comparing it with the finite element method(FEM). Several parameter investigations are conducted to analyze the effects of different contact modes on the rock-lining interaction. The results show that:(i) the height of the local noncontact area does not have a significant effect on the contact stress distribution if no re-contact occurs;(ii) backfill grouting can reduce the local stress concentration caused by poor contact modes;and(iii) reducing the friction coefficient of the interface can lead to a more uniform distribution of internal forces in the concrete lining.展开更多
In this paper, the definition and the criterion of a unified symmetry are presented. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symm...In this paper, the definition and the criterion of a unified symmetry are presented. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, are obtained. Some examples are given to illustrate the application of the results.展开更多
By applying the framework of the tangent bundle geometry to the method of Lagrange multi- pliers,a geometric description of Chetaev's nonholonomic systems subjected to unilateral nonholonomic con- straints trod un...By applying the framework of the tangent bundle geometry to the method of Lagrange multi- pliers,a geometric description of Chetaev's nonholonomic systems subjected to unilateral nonholonomic con- straints trod unilateral holonomic constraints respectively in time-independent circumstances is presented.展开更多
By introducing the generalized quasi-symmetry of the infinitesimaltransformation for transformation group G_r, this paper studies theconservation laws and symmetries of dynamical systems with unilateralconstraints in ...By introducing the generalized quasi-symmetry of the infinitesimaltransformation for transformation group G_r, this paper studies theconservation laws and symmetries of dynamical systems with unilateralconstraints in phase space. Noether's theorem and Noether's inversetheorem for me- chanical system with unilateral constraints in phasespace are obtained and two kinds of equivalent forms of generalizedKilling equations which are used to determine the generators of theinfinitesimal group transformation are given.展开更多
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...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.展开更多
To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constr...To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.展开更多
The description of modern differential geometry for time-dependent Chetaev nonholonomic mechanical systems with unilateral constraints is studied. By using the structure of exact contact manifold, the geometric framew...The description of modern differential geometry for time-dependent Chetaev nonholonomic mechanical systems with unilateral constraints is studied. By using the structure of exact contact manifold, the geometric framework of time- dependent nonholonomic mechanical systems subject to unilateral nonholonomic constraints and unilateral holonomic constraints respectively is presented.展开更多
In this paper, the equations of motion for nonholonomic mechanical system with unilateral holonomic constraints and unilateral nonholonomic constraints are presented, and an example to illustrate the application of th...In this paper, the equations of motion for nonholonomic mechanical system with unilateral holonomic constraints and unilateral nonholonomic constraints are presented, and an example to illustrate the application of the result is given.展开更多
We investigate the application of the Mei symmetry analysis in finding conserved quantities for the thin elastic rod statics. By using the Mei symmetry analysis, we have obtained the Jacobi integral and the cyclic int...We investigate the application of the Mei symmetry analysis in finding conserved quantities for the thin elastic rod statics. By using the Mei symmetry analysis, we have obtained the Jacobi integral and the cyclic integrals for a thin elastic rod with intrinsic twisting for both the cases of circular and non-circular cross sections. Our results can be easily reduced to the results without the intrinsic twisting that have been reported. Through calculation, we find that the Noether symmetry can be more directly and easily used than the Mei symmetry in finding the first integrals for the thin elastic rod. These first integrals will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the elastic rod model for DNA.展开更多
Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariants, i.e. generalized Lutzky adiabatic invariants, of a disturbed holonomic nonconservative mechanica...Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariants, i.e. generalized Lutzky adiabatic invariants, of a disturbed holonomic nonconservative mechanical system are obtained by investigating the perturbation of Lie symmetries for a holonomic nonconservative mechanical system with the action of small disturbance. The adiabatic invariants and the exact invariants of the Lutzky type of some special cases, for example, the Lie point symmetrical transformations, the special Lie symmetrical transformations, and the Lagrange system, are given. And an example is given to illustrate the application of the method and results.展开更多
The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates.The Lie symmetry is an invariance of the differential equations of motion under the transformati...The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates.The Lie symmetry is an invariance of the differential equations of motion under the transformations.In this paper,the relation between these two symmetries is proved definitely and firstly for mechanical systems.The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold.展开更多
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 ...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.展开更多
Noether's theory of dynamical systems with unilateral constraints by introducing the generalized quasi_symmetry of the infinitesimal transformation for the transformation group G r is presented and two examples t...Noether's theory of dynamical systems with unilateral constraints by introducing the generalized quasi_symmetry of the infinitesimal transformation for the transformation group G r is presented and two examples to illustrate the application of the result are given.展开更多
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. F...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.展开更多
This paper presents a formulation and solution for the inverse problem of nonholonomic dynamics: to find the form of nonholonomic constraints when some integrals are given and to find the generalized reactive forces o...This paper presents a formulation and solution for the inverse problem of nonholonomic dynamics: to find the form of nonholonomic constraints when some integrals are given and to find the generalized reactive forces of constraint acting on the system when the expression of the kinetic energy is given. An example is given to illustrate the application of the result.展开更多
Bertrand's theorem for the determination of the applied forces to a holonomic system from one of its first integrals, is extended to nonholonomic systems. Some interesting applications of this new result are also ...Bertrand's theorem for the determination of the applied forces to a holonomic system from one of its first integrals, is extended to nonholonomic systems. Some interesting applications of this new result are also given.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 10272021) and the Doctorate Foundation of the State Education Ministry of China (Grant No 20040007022).
文摘A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton-Noether method, the Lagrange-Noether method and the Poisson method, are given to solve a differential equation of first order, of which the way may be called the mechanical methodology in mathematics.
文摘The theory of rotational relativistic mechanics is discussed and the theory of relativistic analytical mechanics' of the rotational systems is constructed. The relativistic generalized kinetic energy function for the rotational systems [GRAPHICS] and the generalized acceleration energy function [GRAPHICS] are constructed, and furthermore, the Hamilton principle and three kinds of D'Alembert principles are given. For the systems with holonomic constraints, the relativistic Lagrange equation, Nielsen equation, Appell equation and Hamilton canonical equation of the rotational systems are constructed; For the systems with nonholonomic constraints, the relativistic Routh equation, Chaplygin equation, Nielsen equation and Appell equation of the rotational systems are constructed; the relativistic Noether conservation law of the rotational systems are given too.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11262019 and 10972143)
文摘In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables: time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.
文摘Nonlinear mechanics for a super-thin elastic rod with the biological background of DNA super-coiling macromolecules is an interdisciplinary research area of classical mechanics and molecular biology. It is also a subject of dynamics and elasticity because elastic bodies are analyzed via the theory of dynamics. It is in frontiers of general mechanics (dynamics and control). This dissertation is devoted to model a constrained super-thin elastic rod and analyze its stability in equilibrium. The existing research results are summarized. Analytical mechanics is systematically applied to model the elastic rod. The Schroedinger equation for complex curvatures or complex bending moments is, respectively, extended from the case of circular crosssections to that of non-circular ones. The equilibrium of a rod constrained on a surface is investigated.
基金supported by the National Natural Science Foundation of China (Grant Nos. 51738002 and 52108376)Fundamental Research Funds for the Central Universities (Grant No. 2021CZ111)
文摘Based on the nondestructive test data of operating railway tunnels in China, this paper summarizes the basic characteristics of the complex contact behavior between the rock mass and lining structure. The contact modes are classified into dense contact, local non-contact, and loose contact. Subsequently, the corresponding mechanical model for each contact mode is developed according to its mechanical characteristics using the complex variable method. In the proposed mechanical model, a special algorithm is introduced to detect whether the local non-contact zone is re-contacted. Besides, a novel conformal mapping method is also proposed to accurately calculate the mechanical response of the concrete lining. Finally, the accuracy of the proposed method is verified by comparing it with the finite element method(FEM). Several parameter investigations are conducted to analyze the effects of different contact modes on the rock-lining interaction. The results show that:(i) the height of the local noncontact area does not have a significant effect on the contact stress distribution if no re-contact occurs;(ii) backfill grouting can reduce the local stress concentration caused by poor contact modes;and(iii) reducing the friction coefficient of the interface can lead to a more uniform distribution of internal forces in the concrete lining.
基金The project supported by the National Natural Science Foundation of China(10272021)
文摘In this paper, the definition and the criterion of a unified symmetry are presented. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, are obtained. Some examples are given to illustrate the application of the results.
基金the National Natural Science Foundation of China(No.19972010)the Qing Lan Project Foundation of Jiangsu Province of Chinathe Research Foundation of Suzhou Institute of Urban Construction & Environmental Protection of China
文摘By applying the framework of the tangent bundle geometry to the method of Lagrange multi- pliers,a geometric description of Chetaev's nonholonomic systems subjected to unilateral nonholonomic con- straints trod unilateral holonomic constraints respectively in time-independent circumstances is presented.
基金the National Natural Science Foundationthe Doctoral Programme Foundation of Institution of Higher Education of China
文摘By introducing the generalized quasi-symmetry of the infinitesimaltransformation for transformation group G_r, this paper studies theconservation laws and symmetries of dynamical systems with unilateralconstraints in phase space. Noether's theorem and Noether's inversetheorem for me- chanical system with unilateral constraints in phasespace are obtained and two kinds of equivalent forms of generalizedKilling equations which are used to determine the generators of theinfinitesimal group transformation are given.
文摘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.
文摘To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.
基金Project supported by the National Natural Science Foundation of China (Grant No 10272021), the Natural Science Foundation of High Education of Jiangsu Province, China (Grant No 04KJA130135) and the "Qing Lan" Project Foundation of Jiangsu Province, China.
文摘The description of modern differential geometry for time-dependent Chetaev nonholonomic mechanical systems with unilateral constraints is studied. By using the structure of exact contact manifold, the geometric framework of time- dependent nonholonomic mechanical systems subject to unilateral nonholonomic constraints and unilateral holonomic constraints respectively is presented.
文摘In this paper, the equations of motion for nonholonomic mechanical system with unilateral holonomic constraints and unilateral nonholonomic constraints are presented, and an example to illustrate the application of the result is given.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10972143)the Research Plan of Higher Education Institutions of Xinjiang Autonomous Region, China (Grand No. XJEDU2010S31)the Foundation for Key Subject of Theory Physics of Xinjiang Autonomous Region, China
文摘We investigate the application of the Mei symmetry analysis in finding conserved quantities for the thin elastic rod statics. By using the Mei symmetry analysis, we have obtained the Jacobi integral and the cyclic integrals for a thin elastic rod with intrinsic twisting for both the cases of circular and non-circular cross sections. Our results can be easily reduced to the results without the intrinsic twisting that have been reported. Through calculation, we find that the Noether symmetry can be more directly and easily used than the Mei symmetry in finding the first integrals for the thin elastic rod. These first integrals will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the elastic rod model for DNA.
基金Project supported by the National Natural Science Foundation of China (Grant No 10372053)
文摘Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariants, i.e. generalized Lutzky adiabatic invariants, of a disturbed holonomic nonconservative mechanical system are obtained by investigating the perturbation of Lie symmetries for a holonomic nonconservative mechanical system with the action of small disturbance. The adiabatic invariants and the exact invariants of the Lutzky type of some special cases, for example, the Lie point symmetrical transformations, the special Lie symmetrical transformations, and the Lagrange system, are given. And an example is given to illustrate the application of the method and results.
基金The project supported by the National Natural Science Foundation of China (19972010)
文摘The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates.The Lie symmetry is an invariance of the differential equations of motion under the transformations.In this paper,the relation between these two symmetries is proved definitely and firstly for mechanical systems.The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold.
文摘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.
文摘Noether's theory of dynamical systems with unilateral constraints by introducing the generalized quasi_symmetry of the infinitesimal transformation for the transformation group G r is presented and two examples to illustrate the application of the result are given.
文摘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.
文摘This paper presents a formulation and solution for the inverse problem of nonholonomic dynamics: to find the form of nonholonomic constraints when some integrals are given and to find the generalized reactive forces of constraint acting on the system when the expression of the kinetic energy is given. An example is given to illustrate the application of the result.
文摘Bertrand's theorem for the determination of the applied forces to a holonomic system from one of its first integrals, is extended to nonholonomic systems. Some interesting applications of this new result are also given.