Symmetries and perturbations of a singular nonconservative system on time scales (Q6556496)

This paper deals with symmetries and perturbations of singular systems on the time scale \(\mathbb{T}\), where \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers \(\mathbb{R}\).\N\NIn Section 2, some notions as jump operators, the delta derivative, and the corresponding antiderivative are recalled.\N\NIn Section 3, the Euler operator on the time scale, which gives the Lagrange equation on the time scale, is introduced (also see [\textit{Y. Zhang}, J. Suzhou Univ. Sci. Technol., Nat. Sci. 37, No. 1, 6--11 (2020; Zbl 1463.70013)]).\N\NIn Section 4, some symmetries and conserved quantities are presented. Firstly, the Noether symmetry for a singular nonconservative Lagrangian system on a time scale is introduced and some Noether conserved quantities are deduced. Then, Noether conserved quantities deduced from the Lie symmetry are given. Finally, the Mei symmetry of the singular nonconservative Lagrangian system is defined. Moreover, the criterion equation for Mei symmetry is obtained. Consequently, some conserved quantities are deduced from the Mei symmetry. In each case, for \(\mathbb{T}=\mathbb{R}\), the classical results are retrieved.\N\NIn Section 5, perturbations and adiabatic invariants of the above-mentioned symmetries for a singular nonconservative Lagrangian system on a time scale are given.\N\NIn Section 6, some examples (with \(\mathbb{T}=\{2^n:n\in\mathbb{Z}\}\cup\{0\}\)) are presented.