摘要 : The Herglotz-type vakonomic dynamics of nonholonomic constrained systems with delayed arguments and its Noether theory are studied in this paper. First of all, the Herglotz-type equations of time-delayed vakonomic dynamics for nonholonomic systems are established, and the Herglotz-type local extremal equations are given. Secondly, on the basis of derivation of the variational formulas of Hamilton–Herglotz action with time delay, the Herglotz-type Noether symmetry criteria for time-delayed vakonomic dynamics are investigated. Thirdly, the Herglotz-type Noether's theorems and inverse theorems for time-delayed vakonomic dynamics of nonholonomic systems are deduced. Finally, an example is presented to demonstrate the application of the results.

摘要 : Conservation laws for systems of exponential, power-law and logarithmic non-standard Birkhoffian with fractional derivatives are investigated respectively, and their relationships with Noether symmetries are revealed. Corresponding to the three forms of non-standard Birkhoffians, the Pfaff–Birkhoff principles are proposed and non-standard Birkhoff's equations with fractional derivatives are derived. The variation of Pfaff action is investigated in depth, and the formulas of total variation are deduced, and on this basis the criterion equations of Noether symmetry are set up. Then, Noether's theorems for systems of non-standard Birkhoffian with fractional derivatives are proved, which reveals the relationship between symmetry and conserved quantity. Finally, three examples are given to illustrate how to calculate the symmetry and find out the conserved quantity, and the correctness of the obtained conserved quantities is shown by numerical calculation. © 2023 Elsevier B.V.

摘要 : The canonical transformation and Poisson theory for the second-order generalized mechanical systems based on non-standard power-law Lagrangians are studied. First, the Euler–Lagrange equations and the Hamilton canonical equations for the second-order generalized mechanics with the power-law Lagrangians are established. Second, the canonical transformation theory of the systems is studied by establishing the relationship between old and new variables. Four basic forms of canonical transformation are given, and the transformation formulas in each case are derived. Third, the algebraic structure of the dynamical equations of the systems is studied, and the corresponding Poisson theory is established. Finally, the corresponding examples are presented to illustrate the application of the results we obtained.

摘要 : Based on generalized operators, local conserved quantities, global conserved quantities and adiabatic invariants for the Lagrangian systems are studied. Firstly, Euler-Lagrange equations and transversality conditions are listed. Then local conserved quantities and global conserved quantities with the generalized operators are investigated. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented. Some special cases are discussed during this study. In the end, an example is given to illustrate the methods and results.

摘要 : The Herglotz variational principle offers an effective method for studying nonconservative system dynamics. The aim of this paper is to study the conservation laws of nonholonomic systems by using the Herglotz type generalized variational principle and establish Noether's theorem and its inverse theorem for this system. In deriving the equations of motion, we use the Suslov definition of the reciprocity relation between differential and variational operations. First, the Herglotz type generalized variational principle is listed, and the Herglotz type Chaplygin equations for nonconservative nonholonomic systems are deduced. Second, Noether's theorem and Noether's inverse theorem are established, and the Herglotz type conservation laws are given. Finally, an example is provided to illustrate the practical implementation of the findings.

摘要 : The motion with variable acceleration is common in daily life and engineering problems. Variable acceleration dynamics, also known as Newtonian jerky dynamics, has gained wide attention due to its application in chaos theory and nonlinear dynamics. Gauss principle is a differential variational principle with extreme value characteristics. Therefore, it is of great significance to study the generalized Gauss principle of dynamical systems with variable acceleration in both theory and application. In this paper, the generalized Gauss principle for dynamical systems with variable accelerated motion is presented and studied. Firstly, we introduce the concept of the generalized Gaussian variation in the jerky space. We take the derivative of d'Alembert principle of a particle with respect to time, and then calculate its dot product with the generalized Gaussian variation. By using the condition of ideal constraints in the sense of Gauss, we establish the generalized Gauss principle for dynamical systems with variable acceleration. On this basis, the generalized Gauss principle of least compulsion is established and proved by constructing the generalized compulsion function. The Appell form, Lagrange form and Nielsen form of the principle are given. Secondly, the extension of the principle to variable mass mechanics is explored. Starting from Meshchersky equation and taking its derivative with respect to time, and then calculating its dot product with the generalized Gaussian variation, we establish the generalized Gauss principle for variable-mass variable-acceleration dynamical systems with ideal constraints. The generalized compulsion function in the case of variable mass is constructed and the generalized Gauss principle of least compulsion for variable-mass variable-acceleration mechanical systems is established and proved. We take the Kepler-Newton problem as an example, and use the approach of the generalized Gauss least compulsion principle we presented to calculate, and verify the effectiveness of the method.

摘要 : In this paper, the approximate Noether theorems for approximate Birkhoffian systems are presented and discussed. The approximate Birkhoff equations for the systems are established. The Noether identities for approximate Birkhoffian systems are given, which based upon the Noether symmetry and quasi-symmetry, and the relationship between the approximate Noether symmetries and approximate conservation laws for the systems are established, and the approximate Noether theorems are obtained. The results show that the results under the approximate Hamiltonian systems are a special cases of the approximate Birkhoffian systems, while the results under the approximate Lagrangian systems is equivalent to that under the approximate Hamiltonian systems. Finally, two examples are given to illustrate the application of the results.

摘要 : Variable mass systems are ubiquitous in nature and engineering. The aim of this paper is to extend the Herglotz generalized variational principle to variable mass nonholonomic systems. In this paper, Herglotz conservation laws of variable mass nonholonomic systems are studied. The Herglotz generalized variational principle of variable mass nonholonomic systems is established, and the Herglotz-d'Alembert principle for nonholonomic systems with variable mass is derived, in which the Hölder definition of commutative relation is used. The transformation of the invariance condition of Herglotz-d'Alembert principle is established by introducing the generators of space and time. Herglotz conservation theorem and its inverse for variable mass nonholonomic systems are constructed based on this principle. In the end, an example is used as a proof of this application.

摘要 : The fractional Birkhoffian systems are investigated for Mei symmetry and corresponding conserved quantities. It is divided into two cases, one is standard fractional Birkhoffian systems, the other is fractional Birkhoffian systems under quasi-fractional dynamics models of El-Nabulsi type. The fractional Pfaff-Birkhoff principles are presented, and Birkhoff's equations are deduced. Mei symmetry and its determining equations are established. Mei symmetry theorems are proved and fractional conserved quantities are obtained. © 2022 Elsevier Ltd

摘要 : The time-scale non-shifted Hamiltonian dynamics are investigated, including both general holonomic systems and nonholonomic systems. The time-scale non-shifted Hamilton principle is presented and extended to nonconservative system, and the dynamic equations in Hamiltonian framework are deduced. The Noether symmetry, including its definition and criteria, for time-scale non-shifted Hamiltonian dynamics is put forward, and Noether's theorems for both holonomic and nonholonomic systems are presented and proved. The non-shifted Noether conservation laws are given. The validity of the theorems is verified by two examples. © 2023 Wuhan University Journal of Natural Sciences. All rights reserved.