论文标题
将非卡丹对称性的概括性概括为任意维度
Generalization of non-Cartan Symmetries to arbitrary dimensions
论文作者
论文摘要
可线化的二阶标量普通微分方程（{\ sc ode} s）具有特殊类型的对称性。这些是方程式线性化形式的唯一对对称性的对称性，它们被称为非cartan对称性，仅适用于标量{\ sc ode} s。我们为任意维度的{\ sc ode}系统的系统的非 - 奶油对称性提供明确的表达，并表明它们形成了Abelian Lie代数。然而，这些非 - 奶油对称对称性的自然扩展仅适用于标量二阶方程的自然扩展到更高维度，即在琐碎矢量方程的点转换下等于等效类。更准确地说，可以表明，非cartan对称性表征了{\ sc ode} s的线性系统，可通过点转换可降低到它们的琐碎对应物，我们验证它们没有表征{\ sc ode} s具有此属性的非线性系统。还表明，对称矢量的非卡丹特性不含坐标。讨论了这些结果应用的一些示例。 ％
Second order scalar ordinary differential equations ({\sc ode}s) which are linearizable possess special types of symmetries. These are the only symmetries which are non fiber-preserving in the linearized form of the equation, and they are called non-Cartan symmetries and known only for scalar {\sc ode}s. We give explicit expressions of non-Cartan symmetries for systems of {\sc ode}s of arbitrary dimensions and show that they form an abelian Lie algebra. It is however shown that the natural extension of these non-Cartan symmetries to arbitrary dimensions is applicable only to the natural extension of scalar second order equations to higher dimensions, that is, to equivalence classes under point transformations of the trivial vector equation. More precisely, it is shown that non-Cartan symmetries characterize linear systems of {\sc ode}s reducible by point transformation to their trivial counterpart, and we verify that they do not characterize nonlinear systems of {\sc ode}s having this property. It is also shown amongst others that the non-Cartan property of a symmetry vector is coordinate-free. Some examples of application of these results are discussed. %
