Abstract

It is well known [1–3] that a system of differential equations can be exactly integrated if a sufficient number of its tensor invariants (not only first integrals) are found. For example, if there is an invariant differential form of the phase volume, the number of required first integrals can be reduced. For conservative systems, this fact is natural, but for systems with attracting or repelling limit sets, not only some of the first integrals, but also the coefficients of available invariant differential forms should, generally speaking, include functions with essential singularities (see also [4–6]). In this paper, for the class of dynamical systems under consideration, we present complete sets of invariant differential forms for homogeneous systems on tangent bundles of smooth finite-dimensional manifolds.

中文翻译：

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有限维流形切丛上测地线、势和耗散系统的不变形式

摘要

众所周知[1-3]，如果找到足够数量的张量不变量（不仅仅是第一积分），则微分方程组可以精确积分。例如，如果相体积存在不变的微分形式，则可以减少所需的第一积分的数量。对于保守系统，这一事实是自然的，但对于具有吸引或排斥极限集的系统，一般来说，不仅一些第一积分，而且可用的不变微分形式的系数也应包括具有本质奇点的函数（另请参见[ 4-6]）。在本文中，对于所考虑的动力系统类别，我们提出了光滑有限维流形切丛上齐次系统的完整不变微分形式集。