A theorem is proved which determines the first integrals of the form I = Kab(t, q)q̇ q̇+Ka(t, q)q̇ +K(t, q) of autonomous holonomic systems using only the collineations of the kinetic metric which is defined by the kinetic energy or the Lagrangian of the system. It is shown how these first integrals can be associated via the inverse Noether theorem to a gauged weak Noether symmetry which admits the given first integral as a Noether integral. It is shown also that the associated Noether symmetry is possible to satisfy the conditions for a Hojman or a form-invariance symmetry therefore the so-called non-Noetherian first integrals are gauged weak Noether integrals. The application of the theorem requires a certain algorithm due to the complexity of the special conditions involved. We demonstrate this algorithm by a number of solved examples. We choose examples from published works in order to show that our approach produces new first integrals not found before with the standard methods.

中文翻译：

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没有诺特对称性的完整系统的第一积分

证明了一个定理，该定理仅使用动力学度量的共线来确定自治完整系统的 I = Kab(t, q)q̇ q̇+Ka(t, q)q̇ +K(t, q) 形式的第一积分由系统的动能或拉格朗日函数定义。显示了这些第一积分如何通过逆 Noether 定理与规范的弱 Noether 对称相关联，该对称性允许给定的第一积分作为 Noether 积分。还表明相关联的诺特对称性可能满足 Hojman 或形式不变对称性的条件，因此所谓的非诺特第一积分被衡量为弱诺特积分。由于所涉及的特殊条件的复杂性，该定理的应用需要一定的算法。我们通过一些解决的例子来演示这个算法。