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
The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we ca...The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.展开更多
In this paper,according to the fractional factor derivative method,we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space.First,fractional calculus is calcula...In this paper,according to the fractional factor derivative method,we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space.First,fractional calculus is calculated by using the fractional factor,and the fractional equations of motion are derived by using the differential variational principle.Second,the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained.Furthermore,the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation,which conforms to the Noether criterion equation.Finally,we present an example of a calculation.The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems,but because of increased invariance restriction,the nonconservative forces do not change the form of conserved quantity;meanwhile,the fractional factor method has high natural consistency with the integral calculus,so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.展开更多
A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves ...A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.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.
文摘The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.
基金Jiangsu Key Laboratory of Green Process Equipment,Grant/Award Number:GPE202203Qing Lan Project of Universities in Jiangsu Province,Grant/Award Number:2022-29。
文摘In this paper,according to the fractional factor derivative method,we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space.First,fractional calculus is calculated by using the fractional factor,and the fractional equations of motion are derived by using the differential variational principle.Second,the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained.Furthermore,the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation,which conforms to the Noether criterion equation.Finally,we present an example of a calculation.The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems,but because of increased invariance restriction,the nonconservative forces do not change the form of conserved quantity;meanwhile,the fractional factor method has high natural consistency with the integral calculus,so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.
基金supported by the Czech Science Foundation (Grant No.GA CˇR 201/09/0981)the Czech-Hungarian Cooperation Programme "Kontakt" (Grant No. MEB041005)the IRSES project ’GEOMECH’ (Grant No. 246981) within the 7th European Community Framework Programme
文摘A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.
