Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.

Section: Introductionmentioning

confidence: 99%

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

Deng,

Zhang

2024

“…Noether revealed the potential relationship between the conserved quantity of a mechanical system and its inherent dynamic symmetry for the first time, and also established the Noether symmetry theory. Many articles have been published on the Noether theorem, such as the Bible of symmetry methods [33], a comprehensive review of Noether's theorem [34], Noether's theorem for discrete equations [35], Noether's theorem for semidiscrete equations [36,37], Noether's theorem for the fractional Lagrangian system [38][39][40][41][42][43][44][45], Noether's theorem for the fractional Hamiltonian system [46][47][48][49][50], Noether's theorem for the fractional Birkhoffian system [51][52][53][54], etc. In this paper, we aim to establish Noether's theorem within the generalized fractional operators in terms of the general kernels.…”

Section: Introductionmentioning

confidence: 99%

Further Research for Lagrangian Mechanics within Generalized Fractional Operators

Song

2023

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results.

“…After the fractional constrained Hamilton equations are established, the symmetry method is considered. The symmetry method mainly contains the Noether symmetry method, the Lie symmetry method, and the Mei symmetry method [40][41][42]. This article pays attention to the first two symmetry methods.…”

Section: Introductionmentioning

confidence: 99%

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Song

2022

Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results.