The most common problems in nature are about non-conservative non-linearity. Non-conservative non-linear problems can be studied with variational problems of non-standard Lagrangians. Birkhoffian mechanics, as an extension of Hamiltonian mechanics naturally, is a sign that analytical mechanics has entered a new stage of development. Therefore, the study of dynamics based on non-standard Birkhoffians provides a new idea for solving non-conservative nonlinear dynamics problems. In this paper the dynamics models based on non-standard Birkhoffians, including exponential Birkhoffian, power law Birkhoffian, and logarithm Birkhoffian, are proposed, which are called non-standard Birkhoffian systems. Firstly, the Pfaff-Birkhoff principles with non-standard Birkhoffians are established, the differential equations of motion of non-standard Birkhoffian dynamics are also derived. Secondly, in accordance with the invariance of Pfaff action under the infinitesimal transformations, giving the definitions and criteria of Noether symmetric and quasi-symmetric transformations of non-standard Birkhoffian dynamics. And next, Noether’s theorems for non-standard Birkhoffian dynamics are proved, and the connections between Noether symmetries and conserved quantities of non-standard Birkhoffian dynamics are established; Finally, three examples are given to illustrate the applications of the results.

自然界中最常见的问题是关于非保守非线性的问题。可以用非标准拉格朗日方程的变分问题来研究非保守非线性问题。伯克霍夫力学自然是哈密顿力学的延伸，这标志着分析力学已进入一个新的发展阶段。因此，基于非标准伯克霍夫定律的动力学研究为解决非保守非线性动力学问题提供了新思路。本文提出了基于非标准Birkhoffian的动力学模型，包括指数Birkhoffian，幂律Birkhoffian和对数Birkhoffian，它们被称为非标准Birkhoffian系统。首先，建立具有非标准伯克霍夫主义者的普法夫-伯克霍夫原则，还推导出非标准伯克霍夫动力学的运动微分方程。其次，根据无穷小变换下Pfaff作用的不变性，给出了非标准伯克霍夫动力学的Noether对称和准对称变换的定义和判据。其次，证明了非标准伯克霍夫动力学的Noether定理，并建立了Noether对称性与非标准伯克霍夫动力学的守恒量之间的联系。最后，给出了三个例子来说明结果的应用。给出了非标准伯克霍夫动力学的Noether对称和准对称变换的定义和判据。其次，证明了非标准伯克霍夫动力学的Noether定理，并建立了Noether对称性与非标准伯克霍夫动力学的守恒量之间的联系。最后，给出了三个例子来说明结果的应用。给出了非标准伯克霍夫动力学的Noether对称和准对称变换的定义和判据。其次，证明了非标准伯克霍夫动力学的Noether定理，并建立了Noether对称性与非标准伯克霍夫动力学的守恒量之间的联系。最后，给出了三个例子来说明结果的应用。