二维自治动力系统的对称性与其相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
AI总结：
本文建立二维自治动力系统微分方程表述与相平面单方程表述之间对称生成元的对应关系，提出并求解线性偏微分形式的提升条件，且以线性质量守恒模型和非线性振子模型验证。
摘要：
我们考察二维自治动力系统在两种常见表述中的对称性之间的关系：一种表述是关于每个状态对时间导数的一组微分方程，另一种是相平面中的单个微分方程，表示限制在系统状态空间上的动力学。两种表示都可以就其支配微分方程的对称性进行分析，我们建立了各自表述的无穷小生成元集合之间的对应关系。我们的主要结果表明，自治系统的每个对称生成元都会诱导出一个定义良好的向量场，该向量场在相平面中生成一个对称；反之，相平面中的每个对称生成都可以被提升为原自治系统的一个对称生成元，该生成元在相差时间的常数平移意义下是唯一的。提升过程需要求解一个线性偏微分方程，我们将其称为提升条件。我们详细讨论了该方程的一般求解方法，并通过两个常见例子说明对称性的提升：一个质量守恒的线性模型和一个非线性振子模型。
英文摘要：
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.
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