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.
Highlights
Birkhoffian mechanics is a new stage in the development of analytical dynamics
The dynamics theory on a time scale unifies the dynamics of continuous systems, discrete systems, and quantum systems
Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity. Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]
Summary
Birkhoffian mechanics is a new stage in the development of analytical dynamics. It was first proposed by Birkhoff [1] and later developed by Santilli [2] and Mei et al [3]. Based on the second Euler-Lagrange equations, they proposed another method to find the Noether conserved quantity Afterwards, according to these two methods, many scholars have obtained some results have been obtained in the study of variational principle, dynamical equations, and Noether symmetries for the different mechanical systems, such as references [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].
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