This paper introduces a new evolutionary system which is uniquely suitable for the description of nonconservative systems in field theories,including quantum mechanics,but is not limited to it only.This paper also int...This paper introduces a new evolutionary system which is uniquely suitable for the description of nonconservative systems in field theories,including quantum mechanics,but is not limited to it only.This paper also introduces a new exact method of solution for such nonconservative systems.These are significant contributions because the vast majority of nonconservative systems with several independent variables donothave self-adjoint Frechet derivatives and because of that cannotbenefit from the exact methods of the classical calculus of variations.The new evolutionary system is rigorously mathematically derived and the new method for solution is mathematically proved to be applicable to systems of PDEs of second order for nonconservative process.As examples of applications,the method is applied to several nonconservative systems:the propagation of electromagnetic fields in a conductive medium,the nonlinear Schrodinger equation with electromagnetic interactions,and others.展开更多
In this paper,the parametric equations with multipliers of nonholonomic nonconservative sys- tems in the event space are established,their properties are studied,and their explicit formulation is obtained. And then th...In this paper,the parametric equations with multipliers of nonholonomic nonconservative sys- tems in the event space are established,their properties are studied,and their explicit formulation is obtained. And then the field method for integrating these equations is given.Finally,an example illustrating the appli- cation of the integration method is given.展开更多
This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian m...This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.展开更多
Focusing on the exploration of symmetry and conservation laws in event space, this paper studies Noether theorems of Herglotz-type for nonconservative Hamilton system. Herglotz’s generalized variational principle is ...Focusing on the exploration of symmetry and conservation laws in event space, this paper studies Noether theorems of Herglotz-type for nonconservative Hamilton system. Herglotz’s generalized variational principle is first extended to event space,and on this basis, Hamilton equations of Herglotz-type in event space are derived. The invariance of Hamilton-Herglotz action is then studied by introducing infinitesimal transformation, and the definition of Herglotz-type Noether symmetry in event space is given, and its criterion is derived. Noether theorem of Herglotz-type and its inverse for event space nonconservative Hamilton system are proved. The application of Herglotz-type Noether theorem we obtained is introduced by taking Emden-Fowler equation and linearly damped oscillator as examples.展开更多
In order to study discrete nonconservative system,Hamilton's principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well a...In order to study discrete nonconservative system,Hamilton's principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well as the nonconservative system with dynamic constraint are established within fractional difference operators of Riemann-Liouville type from the view of time scales. Firstly,time scale calculus and fractional calculus are reviewed.Secondly,with the help of the properties of time scale calculus,discrete Lagrange equation of the nonconservative system within fractional difference operators of Riemann-Liouville type is presented. Thirdly,using the Lagrange multipliers,discrete Lagrange equation of the nonconservative system with dynamic constraint is also established.Then two special cases are discussed. Finally,two examples are devoted to illustrate the results.展开更多
For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of tile theory of i...For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of tile theory of invariance of differential equations of motion under general infinitesimal transformations, we construct the relativistic Noether symmetry, Lie symmetry and the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations. By using the Noether symmetry, a new relativistic non-Noether conserved quantity is given which only depends on the variables t, qs and qs. An example is given to illustrate the application of the results.展开更多
In this paper Mei symmetry is introduced for a nonconservative system. The necessary and sufficient condition for a Mei symmetry to be also a Lie symmetry is derived. It is proved that the Mei symmetry leads to a non-...In this paper Mei symmetry is introduced for a nonconservative system. The necessary and sufficient condition for a Mei symmetry to be also a Lie symmetry is derived. It is proved that the Mei symmetry leads to a non-Noether conservative quantity via a Lie symmetry, and deduces a Lutzky conservative quantity via a Lie point symmetry.展开更多
In this paper,we prove that for holonomic nonconservative dynamical system the Poincare and Poincaré-Cartan integral invariants do not exist.Instead of them,we introduce the integral variants of Poincaré Car...In this paper,we prove that for holonomic nonconservative dynamical system the Poincare and Poincaré-Cartan integral invariants do not exist.Instead of them,we introduce the integral variants of Poincaré Cartan's type and of Poincaré's type for holonomie noneonservative dynamical systems,and use these variants to solve the problem of nonlinear vibration.We also prove that the integral invariants intro- duced in references[1]and[2]are merely the basic integral variants given by this paper.展开更多
The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are g...The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.展开更多
In this paper,we consider the following systems x″+ Bx′+ gradG(x,t) = 0.Weak sufficient condition for the existence of a unique 2π-periodic solution of the systems is given and the results in [1]~ [3],[7]~ [8]are c...In this paper,we consider the following systems x″+ Bx′+ gradG(x,t) = 0.Weak sufficient condition for the existence of a unique 2π-periodic solution of the systems is given and the results in [1]~ [3],[7]~ [8]are consequences of Theorem 2 in this paper if‖B‖2<1.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
For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numer...For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.展开更多
For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carrie...For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.展开更多
Non-Noether symmetries and conservative quantities of nonholonomic nonconservative dynamical systems are investigated in this paper. Based on the relationships among motion, nonconservative forces, nonholonomic constr...Non-Noether symmetries and conservative quantities of nonholonomic nonconservative dynamical systems are investigated in this paper. Based on the relationships among motion, nonconservative forces, nonholonomic constrained forces and Lagrangian, non-Noether symmetries and Lutzky conservative quantities are presented for nonholonomic nonconservative dynamical systems. The relation between non-Noether symmetry and Noether symmetry is discussed and it is further shown that non-Noether conservative quantities can be obtained by a complete set of Noether invariants. Finally, an example is given to illustrate these results.展开更多
Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Fi...Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Firstly, the Lie symmetry of the system is given. Then, the necessary and sumeient condition under which the Lie symmetry is a Mei symmetry is presented and the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is obtained. Lastly, an example is given to illustrate the application of the result.展开更多
The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative prope...The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative properties of the conservative systems can be formulated in the mathematical form. But, the nonconservative characteristics are the nature of the systems existing in engineering. In this letter, the structure-preserving approach for the infinite dimensional nonconservative systems is proposed based on the generalized multi-symplectic method to broaden the application fields of the current structure-preserving idea. In the numerical examples,two nonconservative factors, including the strong excitation on the string and the impact on the cantilever, are considered respectively. The vibrations of the string and the cantilever are investigated by the structure-preserving approach and the good long-time numerical behaviors as well as the high numerical precision of which are illustrated by the numerical results presented.展开更多
This paper analyzes the symmetry of Lagrangians and the conserved quantity for the holonomic non-conservative system in the event space. The criterion and the definition of the symmetry are proposed first, then a quan...This paper analyzes the symmetry of Lagrangians and the conserved quantity for the holonomic non-conservative system in the event space. The criterion and the definition of the symmetry are proposed first, then a quantity caused by the symmetry and its existence condition are given. An example is shown to illustrate the application of the result at the end.展开更多
This paper investigates the momentum-dependent symmetries for nonholonomic nonconservative Hamilton canonical systems. The definition and determining equations of the momentum-dependent symmetries are presented, based...This paper investigates the momentum-dependent symmetries for nonholonomic nonconservative Hamilton canonical systems. The definition and determining equations of the momentum-dependent symmetries are presented, based on the invariance of differential equations under infinitesimal transformations with respect to the generalized coordinates and generalized momentums. The structure equation and the non-Noether conserved quantities of the systems are obtained. The inverse issues associated with the momentum-dependent symmetries are discussed. Finally, an example is discussed to further illustrate the applications.展开更多
In this paper Liapunov's second method is used to analyze the plastic dynamic stability of a column under nonconservative forces. The column is in a viscous medium, and under the action of uniformly distributed ta...In this paper Liapunov's second method is used to analyze the plastic dynamic stability of a column under nonconservative forces. The column is in a viscous medium, and under the action of uniformly distributed tangential follower forces. The strain-rate effect on the stress-strain relation of materials is included in the analysis. A condition of stability is derived, and the critical buckling load is obtained. The strain-rate effect on the stability of the column is discussed.展开更多
In this paper,we investigate whether the Lie symmetry can induce the Mei conserved quantity directly in a nonconservative Hamilton system and a theorem is presented.The condition under which the Lie symmetry of the sy...In this paper,we investigate whether the Lie symmetry can induce the Mei conserved quantity directly in a nonconservative Hamilton system and a theorem is presented.The condition under which the Lie symmetry of the system directly induces the Mei conserved quantity is given.Meanwhile,an example is discussed to illustrate the application of the results.The present results have shown that the Lie symmetry of a nonconservative Hamilton system can also induce the Mei conserved quantity directly.展开更多
文摘This paper introduces a new evolutionary system which is uniquely suitable for the description of nonconservative systems in field theories,including quantum mechanics,but is not limited to it only.This paper also introduces a new exact method of solution for such nonconservative systems.These are significant contributions because the vast majority of nonconservative systems with several independent variables donothave self-adjoint Frechet derivatives and because of that cannotbenefit from the exact methods of the classical calculus of variations.The new evolutionary system is rigorously mathematically derived and the new method for solution is mathematically proved to be applicable to systems of PDEs of second order for nonconservative process.As examples of applications,the method is applied to several nonconservative systems:the propagation of electromagnetic fields in a conductive medium,the nonlinear Schrodinger equation with electromagnetic interactions,and others.
基金The Project is supported by the National Natural Science Foundation of China
文摘In this paper,the parametric equations with multipliers of nonholonomic nonconservative sys- tems in the event space are established,their properties are studied,and their explicit formulation is obtained. And then the field method for integrating these equations is given.Finally,an example illustrating the appli- cation of the integration method is given.
基金supported by the National Natural Science Foundations of China (Nos. 11972241,11572212,11272227)the Natural Science Foundation of Jiangsu Province(No. BK20191454).
文摘This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.
基金Supported by the National Natural Science Foundation of China (11972241, 11572212)the Natural Science Foundation of Jiangsu Province (BK20191454)。
文摘Focusing on the exploration of symmetry and conservation laws in event space, this paper studies Noether theorems of Herglotz-type for nonconservative Hamilton system. Herglotz’s generalized variational principle is first extended to event space,and on this basis, Hamilton equations of Herglotz-type in event space are derived. The invariance of Hamilton-Herglotz action is then studied by introducing infinitesimal transformation, and the definition of Herglotz-type Noether symmetry in event space is given, and its criterion is derived. Noether theorem of Herglotz-type and its inverse for event space nonconservative Hamilton system are proved. The application of Herglotz-type Noether theorem we obtained is introduced by taking Emden-Fowler equation and linearly damped oscillator as examples.
基金supported by the National Natural Science Foundation of China(Nos.11802193, 11572212,11272227)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (18KJB130005)+1 种基金the Science Research Foundation of Suzhou University of Science and Technology(331812137)Natural Science Foundation of Suzhou University of Science and Technology
文摘In order to study discrete nonconservative system,Hamilton's principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well as the nonconservative system with dynamic constraint are established within fractional difference operators of Riemann-Liouville type from the view of time scales. Firstly,time scale calculus and fractional calculus are reviewed.Secondly,with the help of the properties of time scale calculus,discrete Lagrange equation of the nonconservative system within fractional difference operators of Riemann-Liouville type is presented. Thirdly,using the Lagrange multipliers,discrete Lagrange equation of the nonconservative system with dynamic constraint is also established.Then two special cases are discussed. Finally,two examples are devoted to illustrate the results.
文摘For a relativistic holonomic nonconservative system, by using the Noether symmetry, a new non-Noether conserved quantity is given under general infinitesimal transformations of groups. On the basis of tile theory of invariance of differential equations of motion under general infinitesimal transformations, we construct the relativistic Noether symmetry, Lie symmetry and the condition under which the Noether symmetry is a Lie symmetry under general infinitesimal transformations. By using the Noether symmetry, a new relativistic non-Noether conserved quantity is given which only depends on the variables t, qs and qs. An example is given to illustrate the application of the results.
基金Project supported by the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences and the National Natural Science Foundation of China （Grant Nos 10672143, 10471145 and 10372053） and the Natural Science Foundation of Henan Province Government of China（Grant Nos 0511022200 and 0311011400）.
文摘In this paper Mei symmetry is introduced for a nonconservative system. The necessary and sufficient condition for a Mei symmetry to be also a Lie symmetry is derived. It is proved that the Mei symmetry leads to a non-Noether conservative quantity via a Lie symmetry, and deduces a Lutzky conservative quantity via a Lie point symmetry.
基金Project supported by National Natural Science Foundation of China
文摘In this paper,we prove that for holonomic nonconservative dynamical system the Poincare and Poincaré-Cartan integral invariants do not exist.Instead of them,we introduce the integral variants of Poincaré Cartan's type and of Poincaré's type for holonomie noneonservative dynamical systems,and use these variants to solve the problem of nonlinear vibration.We also prove that the integral invariants intro- duced in references[1]and[2]are merely the basic integral variants given by this paper.
基金The project supported by the Natural Science Foundation of Heilongjiang Province of China under Grant No. 9507
文摘The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.
文摘In this paper,we consider the following systems x″+ Bx′+ gradG(x,t) = 0.Weak sufficient condition for the existence of a unique 2π-periodic solution of the systems is given and the results in [1]~ [3],[7]~ [8]are consequences of Theorem 2 in this paper if‖B‖2<1.
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supported by the project"Global Changefor Regional Response"of the Important Study Project of the National Natural Science Foundation of China (Grant No.902110041)the Key Innovation Project of the Chinese Academy of Sciences (KZCX3-SW-213).
文摘For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.
文摘For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.
基金Project supported by the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences and the National Natural Science Foundation of China （Grant No 10372053） and the Natural Science Foundation of Henan Province Government, China （Grant Nos 0311011400, 0511022200）.
文摘Non-Noether symmetries and conservative quantities of nonholonomic nonconservative dynamical systems are investigated in this paper. Based on the relationships among motion, nonconservative forces, nonholonomic constrained forces and Lagrangian, non-Noether symmetries and Lutzky conservative quantities are presented for nonholonomic nonconservative dynamical systems. The relation between non-Noether symmetry and Noether symmetry is discussed and it is further shown that non-Noether conservative quantities can be obtained by a complete set of Noether invariants. Finally, an example is given to illustrate these results.
文摘Based on the total time derivative along the trajectory of the system, for noneonservative dynamical system, the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is studied. Firstly, the Lie symmetry of the system is given. Then, the necessary and sumeient condition under which the Lie symmetry is a Mei symmetry is presented and the generalized Mei conserved quantity indirectly deduced from the Lie symmetry of the system is obtained. Lastly, an example is given to illustrate the application of the result.
基金supported by the National Natural Science Foundation of China (Grant 11672241)the Seed Foundation of Qian Xuesen Laboratory of Space Technologythe Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (Grant GZ1605)
文摘The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative properties of the conservative systems can be formulated in the mathematical form. But, the nonconservative characteristics are the nature of the systems existing in engineering. In this letter, the structure-preserving approach for the infinite dimensional nonconservative systems is proposed based on the generalized multi-symplectic method to broaden the application fields of the current structure-preserving idea. In the numerical examples,two nonconservative factors, including the strong excitation on the string and the impact on the cantilever, are considered respectively. The vibrations of the string and the cantilever are investigated by the structure-preserving approach and the good long-time numerical behaviors as well as the high numerical precision of which are illustrated by the numerical results presented.
基金Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 09CX04018A)the Natural Science Foundation of Shandong Province, China (Grant No. ZR2011AM012)the Postgraduate's Innovation Foundation of China University of Petroleum (East China) (Grant No. CXYB11-12)
文摘This paper analyzes the symmetry of Lagrangians and the conserved quantity for the holonomic non-conservative system in the event space. The criterion and the definition of the symmetry are proposed first, then a quantity caused by the symmetry and its existence condition are given. An example is shown to illustrate the application of the result at the end.
基金Project supported by the National Natural Science Foundation of China （Grant No 10372053） and the Natural Science Foundation of Henan Province, China （Grant Nos 0311011400 and 0511022200） and the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences.
文摘This paper investigates the momentum-dependent symmetries for nonholonomic nonconservative Hamilton canonical systems. The definition and determining equations of the momentum-dependent symmetries are presented, based on the invariance of differential equations under infinitesimal transformations with respect to the generalized coordinates and generalized momentums. The structure equation and the non-Noether conserved quantities of the systems are obtained. The inverse issues associated with the momentum-dependent symmetries are discussed. Finally, an example is discussed to further illustrate the applications.
文摘In this paper Liapunov's second method is used to analyze the plastic dynamic stability of a column under nonconservative forces. The column is in a viscous medium, and under the action of uniformly distributed tangential follower forces. The strain-rate effect on the stress-strain relation of materials is included in the analysis. A condition of stability is derived, and the critical buckling load is obtained. The strain-rate effect on the stability of the column is discussed.
基金Projct supported by the Natural Science Foundation of Shandong Province,China (Grant No. ZR2011AM012)the Fundamental Research Funds for the Central Universities,China (Grant No. 09CX04018A)
文摘In this paper,we investigate whether the Lie symmetry can induce the Mei conserved quantity directly in a nonconservative Hamilton system and a theorem is presented.The condition under which the Lie symmetry of the system directly induces the Mei conserved quantity is given.Meanwhile,an example is discussed to illustrate the application of the results.The present results have shown that the Lie symmetry of a nonconservative Hamilton system can also induce the Mei conserved quantity directly.
