Localized Lie symmetries and conserved quantities for the finite-degree-of-freedom systems

This paper focuses on the study of localized Lie symmetries under the infinitesimal transformation of an infinite continuous group for the finite-degree-of-freedom systems. Based on an invariance of differential equation under an infinitesimal transformation, we present the localized Lie symmetries including direct and inverse problems for the finite degree-of-freedom mechanical systems. We also give the definitions, determining equations, structural equation and conserved laws of localized Lie symmetries, and further, the Lie symmetries under the infinitesimal transformation of a finite continuous group derived from localized Lie symmetry. Finally, an example is discussed to illustrate these results.