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.

在本文中，提出并讨论了具有两种非标准拉格朗日准坐标的动力系统的诺特对称性及其逆定理，即指数拉格朗日拉格朗日和幂律拉格朗日。对于每种情况，通过拟坐标给出了非标准拉格朗日动力系统对应的哈密顿原理，并建立了运动的微分方程。基于无穷小变换群下拟坐标对非标准拉格朗日动力系统的哈密顿作用的不变性，给出并推导出诺特对称和拟对称变换的定义和判据。通过拟坐标建立了诺特定理及其逆定理，它通过准坐标揭示了指数和幂律拉格朗日动力系统的诺特对称性与守恒量之间的关系。给出了三个例子来说明结果的应用。