论文标题
关于二维自主动力系统对称性及其相位平面实现之间的对应关系
On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations
论文作者
论文摘要
我们考虑了两个共同的配方中二维自主动力系统的对称性之间的关系。作为每个状态相对于时间的衍生物的一组微分方程，以及代表限于系统状态空间的动力学的相位平面中的单个微分方程。可以根据其管理微分方程的对称性进行分析，并在相应配方的一组无限发电机之间建立对应关系。我们的主要结果是表明，自主系统对称性的每个发电机都会诱导明确定义的向量场，该矢量场在相位平面中生成对称性，相反，相反，相位平面中的每个对称发电机都可以提升为原始自主系统的对称器的发生器，这是原始自主系统的对称性，这是唯一的不变转换。提起的过程需要解决线性偏微分方程的解决方案，我们称之为提升条件。我们详细讨论了该方程的解决方案，并体现了两个通常发生的例子中对称性的提升。质量保守的线性模型和非线性振荡器模型。
We consider the relationship between symmetries of two-dimensional autonomous dynamical system in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to the symmetries of their governing differential equations, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. Our main result is to show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original autonomous system, which is unique up to constant translations in time. The process of lifting requires the solution of a linear partial differential equation, which we refer to as the lifting condition. We discuss in detail the solution of this equation in general, and exemplify the lift of symmetries in two commonly occurring examples; a mass conserved linear model and a non-linear oscillator model.
