Noether theorem for non-conservative systems with time delay in phase space based on fractional model

Abstract

In this paper, we focus on discussing the fractional Noether symmetries and fractional conserved quantities for non-conservative Hamilton system with time delay. Firstly, the fractional Hamilton canonical equations of non-conservative system with time delay are established; secondly, based upon the invariance of the fractional Hamilton action with time delay under the infinitesimal transformations of group, we obtain the definitions and criterion of fractional Noether symmetric transformations, fractional Noether quasi-symmetric transformations and fractional generalized Noether quasi-symmetric transformations in phase space; finally, the relationship between the fractional Noether symmetries and fractional conserved quantities with time delay in phase space is established. At the end of the paper, some examples are given to illustrate the application of the results.

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

“…Proof. Using Equations ( 40), (47), and (61), it is easy to obtain We write the fractional constrained Hamilton equation within ICRL (Equation ( 46)) in another form:…”

“…Fractional Noether theorems have been investigated on the basis of both definitions. For instance, the works [45,46] were achieved based on the former definition, and the results [30,31,[47][48][49][50][51][52] were obtained on the basis of the latter one. However, Ferreira and Malinowska [53] proved that the fractional Noether theorem given in Ref.…”

Lie symmetry analysis on time scales and its application on mechanical systems

“…The well-known Noether theorem reveals a connection between symmetries and conserved quantities (Noether, 1918). New directions of the applications for Noether theorems such as the fractional dynamical equations (Atanacković et al., 2009; Frederico and Torres, 2007; Zhai and Zhang, 2016), the dynamical equations with time delay (Frederico and Torres, 2012; Jin and Zhang, 2015; Zhai and Zhang, 2014), and the equations derived by the variational problems of Herglotz type (Santos et al., 2015; Zhang, 2016a) can be found. And the method of Noether symmetry which presents the invariance of action under the infinitesimal transformations has been used successfully to find the conserved quantities for dynamical systems on time scales.…”

Noether’s theorems for the relative motion systems on time scales

“…Using this technique, Cai and Fu studied the theories of Noether symmetries of the nonconservative and non-holonomic systems on time scales [13,14]. Noether theory of the Hamilton systems on time scales was given by Zhang [15,16]. It is worth mentioning that the dynamic systems on time scales with delta derivative have just started to originate.…”