Lie symmetries and conserved quantities of fractional nonconservative singular systems

Abstract

In this paper, according to the fractional factor derivative method, we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space. First, fractional calculus is calculated by using the fractional factor, and the fractional equations of motion are derived by using the differential variational principle. Second, the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained. Furthermore, the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation, which conforms to the Noether criterion equation. Finally, we present an example of a calculation. The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems, but because of increased invariance restriction, the nonconservative forces do not change the form of conserved quantity; meanwhile, the fractional factor method has high natural consistency with the integral calculus, so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.