Time-Scale Version of Generalized Birkhoffian Mechanics and Its Symmetries and Conserved Quantities of Noether Type

Abstract

The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

“…By the principle and Eq. ( 11), we have the timescale nonshifted constrained Birkhoff􀆳s equations [44] σ…”

“…Song [43] also studied the adiabatic invariants for three kinds of dynamical systems under the nonshifted variational problem. Then Chen and Zhang [44] researched Noether􀆳s quasi-symmetry theorem of three kinds of Birkhoffian systems about time-scale with nonshifted variational problems. Therefore, this paper will further research Lie symmetry theorem of three Birkhoffian systems about time-scale with nonshifted variational prob-lems, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system.…”

Symmetries and perturbations of time-scale nonshifted singular systems

“…At present, there are many existing studies on the symmetries and perturbations of Lagrangian systems, [34][35][36][37][38][39] Hamiltonian systems, [40][41][42][43] and Birkhoffian systems [44][45][46][47][48][49] on the time scale. However, more research is needed on singular systems on the time scale.…”