宇宙学中Hojman守恒量的评述
Comment on the Hojman conservation quantities in Cosmology
摘要：
本文评述了Capozziello和Roshan（Phys.Lett. B 726（2013）471（arXiv：1308.3910）），并且最近已经被应用于Paolella \& Capozziello（Phys.Lett. A（2015），in press（arXiv：1503.00098））。本文应用作者在最小耦合标量场中引入的近似方法$ϕ\left( t\right) =ϕ\left( a\left( t\right) \right) $，研究了约化方程的Lie和Noether点对称性。我们表明，在这个Answer模型的未知函数不能被约束的要求存在的守恒定律和Hojman守恒量产生的减少方程只不过是功能形式的Noether守恒定律的自由粒子的动量。最后，我们证明了Hojman方法的哈密顿系统，其中的哈密顿函数是所涉及的系统的方程之一，是等价的应用Noether定理的广义变换。
英文摘要：
We comment upon the application of Hojman's method for the determination of conservation laws in Cosmology, which has been introduced by Capozziello \& Roshan (Phys. Lett. B 726 (2013) 471 (arXiv:1308.3910)), and has been applied recently in the cosmological scenario of a nonminimally coupled scalar field by Paolella \& Capozziello (Phys. Lett. A (2015), in press (arXiv:1503.00098)). We apply the Ansatz, $ϕ\left( t\right) =ϕ\left( a\left( t\right) \right) $, which was introduced by the cited authors for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of the Noether conservation law of momentum for the free particle. Finally we show that Hojman's method for Hamiltonian systems, in which the Hamiltonian function is one of the involved equations of the system, is equivalent with the application of Noether's Theorem for generalized transformations.
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