Mathematics

CLC number: O 316

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

¹ School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
² College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China

^† To whom correspondence should be addressed. E-mail: zhy@mail.usts.edu.cn

Received: 6 January 2022

Abstract

The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the time-scale nonshifted generalized Pfaff-Birkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the time-scale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.