Noether symmetry and its inverse for dynamical systems with two kinds of nonstandard Lagrangians via quasi-coordinates

In this paper, the Noether symmetries and their inverse theorems for dynamical systems with two kinds of nonstandard Lagrangians via quasi-coordinates, namely exponential and power-law Lagrangians, are presented and discussed. For each case, the corresponding Hamilton principle for the nonstandard Lagrangian dynamical systems via quasi-coordinates is given, and the differential equations of motion are established. Based upon the invariance of the Hamilton action for the nonstandard Lagrangian dynamical systems via quasi-coordinates under the group of infinitesimal transformations, the definitions and criteria of the Noether symmetric and quasi-symmetric transformations are given and derived. The Noether theorem and its inverse theorem via quasi-coordinates are established, which reveal the relationship between the Noether symmetry and conserved quantity for the exponential and power-law Lagrangian dynamical systems via quasi-coordinates. Three examples are given to illustrate the applications of the results.