Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Abstract

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

Conserved Quantity for Fractional Constrained Hamiltonian System

“…( 48)) exists for the fractional constrained Hamiltonian system (31). Theorem 3 For the infinitesimal generators ξ 0 , ξ i , η i and η α i , which meet the determined equation ( 44), the limited equation (45) and the additional limited equation (46), if there exists a gauge function G ( tqpp α ) satisfying the structural equation (47), then a strong Lie symmetry conserved quantity (Eq. ( 48)) exists for the fractional constrained Hamiltonian system (31).…”

“…Taking calculation, we find that ξ 0 = 1, ξ 1 = ξ 2 = 0, η 1 = η 2 = 0, η α 1 = η α 2 = 0, G = 0 (58) satisfy the structural equation (47) and they also meet the determined equation (55) under the condition p α 1 ( t 2 ) = p α 2 ( t 2 ) = 0. Therefore, Lie symmetry conserved quantity can be obtained from Theorem 1 as…”

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

“…Fractional Noether theorem has been extended to the fractional Lagrange system, Hamilton system, generalized Hamilton system, Birkhoff system, and so on. [23][24][25][26][27][28][29][30][31][32][33] However, there is no research on fractional Noether symmetry in mechano-electrophysiological coupling equations of neuron dynamics. Considering the visco-elasticity of neuron membranes, we will adopt a fractional derivative of variable orders.…”

Further Research for Lagrangian Mechanics within Generalized Fractional Operators

“…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.…”