Caputo $$\Delta $$-type fractional time-scales Noether theorem of Birkhoffian systems

A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems

“…where Λ s = λ β F ͂ βs are the constraint forces corresponding to the nonholonomic constraints (18). Eqs.…”

“…Time scale theory has been widely applied and achieved many achievements in various fields [4][5][6][7][8][9][10] . In last two decades, some new advances have emerged on the study of time-scale dynamics and its symmetries, such as kinetic equations [11][12][13] , optimal control problems [14,15] , fractional variational problems [16][17][18] , Noether theorems [18][19][20][21][22] , Lie symmetries [23][24][25] , Mei symmetries [26,27] , canonical transformation and Hamilton-Jacobi method [28,29] , time-delay dynamics [30] , Herglotz variational problems [31] , higher-order delta derivatives [32] , etc. However, Ref.…”

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

“…Atanacković et al [9] derived the fractional Noether theorem under the classical definition of conserved quantity, which reveals the inherent connection between Noether symmetry transformations and fractional-order conserved quantities. In recent years, the study of conserved quantities and symmetries in fractional mechanics using variational methods has made some headway [10][11][12][13][14][15][16][17][18].…”