论文标题
对称组不具有光纤的ODE
ODEs whose symmetry groups are not fiber-preserving
论文作者
论文摘要
我们观察到，为了结合，大多数对称的高阶ODE（普通微分方程）和ODE系统仅具有纤维的点对称性。通过利用Lie对向量场的Lie代数的分类，我们将在一对函数上的标量ODE和ODES系统中描述所有例外。 可以通过绝对和相对标量差异不变剂表示其对称代数的标量odes可以表达纤维，而对ode系统的相似描述也要求我们还调用条件差异不变性剂，并调用矢量值相对不变的不变剂来处理动作奇异的动作。 调查了行动的延长，我们观察到了代数的不同实现之间的一些有趣的关系。我们还注意到，可能会发生在差分方程上的有限维谎言代数的延长永远不会自由。这种现象发生的一个不确定的ODE系统的一个示例显示了移动框架方法的局限性。
We observe that, up to conjugation, a majority of symmetric higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector fields, we describe all the exceptions to this in the case of scalar ODEs and systems of ODEs on a pair of functions. The scalar ODEs whose symmetry algebra is not fiber preserving can be expressed via absolute and relative scalar differential invariants, while a similar description for ODE systems requires us to also invoke conditional differential invariants and vector-valued relative invariants to deal with singular orbits of the action. Investigating prolongations of the actions, we observe some interesting relations between different realizations of Lie algebras. We also note that it may happen that the prolongation of a finite-dimensional Lie algebra acting on a differential equation never becomes free. An example of an underdetermined ODE system for which this phenomenon occurs shows limitations of the method of moving frames.
