标题： 自治动力系统的高阶首次积分 显示英文标题

标题： Higher order first integrals of autonomous dynamical systems

摘要： 导出了一条定理，该定理能够在一般空间中确定自治的完整动力学系统的更高阶首次积分，前提是能够计算出由系统动能定义的速度度量的共形向量场和Killing张量（直到首次积分的阶数）。 这条定理被应用于牛顿自治保守的动力学系统，该系统有两个自由度，在已知和新的可积和超可积势能中确定了允许三次首次积分的情况。 显示英文摘要

摘要： A theorem is derived which determines higher order first integrals of autonomous holonomic dynamical systems in a general space, provided the collineations and the Killing tensors -- up to the order of the first integral -- of the kinetic metric, defined by the kinetic energy of the system, can be computed. The theorem is applied in the case of Newtonian autonomous conservative dynamical systems of two degrees of freedom, where known and new integrable and superintegrable potentials that admit cubic first integrals are determined.