Abstract
The time-scale dynamic equations play an important role in modeling complex dynamical processes. In this paper, the Mei symmetry and new conserved quantities of time-scale Birkhoff’s equations are studied. The definition and criterion of the Mei symmetry of the Birkhoffian system on time scales are given. The conditions and forms of new conserved quantities which are found from the Mei symmetry of the system are derived. As a special case, the Mei symmetry of time-scale Hamilton canonical equations is discussed and new conserved quantities for the Hamiltonian system on time scales are derived. Two examples are given to illustrate the application of results.
Highlights
In 1988, Hilger [1] proposed the calculus on time scales to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. is theory is really important and plays a useful role in modeling complex dynamic processes and has a tremendous potential for applications [2,3,4].Symmetry is an important aspect in studying dynamic equations
We can find a conserved quantity or the first integral of dynamic equations. e famous Noether symmetry theory which reveals the relationships between symmetries and conversation laws has been applied to a time-scale analogue of analytical mechanics, modern theoretical physics, and engineering [5,6,7,8,9,10]
Except for the Noether symmetry method, the other two popular symmetry methods, the Lie symmetry method [11] and the Mei symmetry method [12], are widely applied in studying dynamic systems. e Mei symmetry means that when the dynamic functions are replaced by the transformed functions under the infinitesimal transformations of group, the forms of the differential equations of motion keep invariant
Summary
In 1988, Hilger [1] proposed the calculus on time scales to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. is theory is really important and plays a useful role in modeling complex dynamic processes and has a tremendous potential for applications [2,3,4]. Over the past two decades, important achievements, including the symmetries and conserved quantities of Birkhoffian systems, the Birkhoffian dynamic inverse problems, the stability of motion of Birkhoff’s equations, the Birkhoffian systems with time delay, and the fractional Birkhoffian systems, have been made [27,28,29,30,31,32,33,34,35,36,37,38]. The Mei symmetry of Birkhoffian systems on time scales has not been done It is worth going a step further to find more new conserved quantities of Birkhoff’s equations by another symmetry method.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
