Title: On the Hojman conservation quantities in Cosmology Show Chinese title

Title: 宇宙学中的Hojman守恒量

Abstract: We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced 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 Noetherian conservation laws for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation. Show Chinese abstract

Abstract: 我们讨论了Hojman对称性方法在宇宙学中守恒定律确定中的应用，该方法已被多位作者最近应用于不同的宇宙学模型。 我们表明，对于正则哈密顿系统，其中哈密顿函数是系统方程之一的Hojman方法，等价于广义变换下诺特定理的应用。 这意味着对于最小耦合标量场宇宙学或其他与标量场宇宙学共形相关的修改理论，如$f(R)$引力，Hojman方法的应用为我们提供了与诺特定理相同的结果。 此外，我们研究了特殊假设$\phi\left( t\right) =\phi\left( a\left( t\right) \right) $，它被引入用于最小耦合标量场，并研究了约化方程的李点对称性和诺特定理。 我们证明，在此假设下，模型的未知函数不能通过守恒律的存在性来约束，而约化方程产生的Hojman守恒量不过是自由粒子的Noether守恒律的功能形式。 另一方面，对于$f(T)$互平行引力，提供$f(T)$函数特殊函数形式的不是Hojman守恒律的存在，而是要求约化二阶微分方程具有雅可比最后乘子，而新的守恒律不过是约化方程的哈密顿函数。