论文标题
通用定理的几何方法
A geometric approach to the generalized Noether theorem
论文作者
论文摘要
我们提供了\ cite {zhang2020 Generalized}的缩放对称性的广义性定理的几何扩展。我们的广义Noether定理的版本具有几个积极的特征：它是在相位空间的最自然扩展中构造的，使对称性在此类歧管上是矢量场，而相关的不变性则可以成为运动的第一个积分；它具有直接的几何证明，与Noether定理的标准相空间版本的证明并行；它会自动产生逆向定理；它也适用于大量的耗散系统；最后，它允许更大的对称性类别，而不是缩放构成代数的缩放转换，因此可以适合代数处理。
We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented in \cite{zhang2020generalized}. Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether's theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.
