On the Hojman conservation quantities in Cosmology

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• 作者：A. Paliathanasis^a ; ^{anpaliat@phys.uoa.gr" class="auth_mail" title="E-mail the corresponding author} ; P.G.L. Leach^b ; ^c ; ^d ; ^{leach.peter@ucy.ac.cy" class="auth_mail" title="E-mail the corresponding author} ; S. Capozziello^e ; ^f ; ^g ; ^h ; ^{capozziello@na.infn.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cosmology ; Scalar field ; Conservation laws
• 刊名：Physics Letters B
• 出版年：2016
• 出版时间：10 April 2016
• 年：2016
• 卷：755
• 期：Complete
• 页码：8-12
• 全文大小：257 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037026931600068X&_mathId=si1.gif&_user=111111111&_pii=S037026931600068X&_rdoc=1&_issn=03702693&md5=8cf1c09100bc9aeb4d207924dba829dc" title="Click to view the MathML source">f(R)class="mathContainer hidden">class="mathCode">$f(R)$ 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. class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037026931600068X&_mathId=si115.gif&_user=111111111&_pii=S037026931600068X&_rdoc=1&_issn=03702693&md5=78c3d100db1774ad6db2ed4c16fcc6c4" title="Click to view the MathML source">ϕ(t)=ϕ(a(t))class="mathContainer hidden">class="mathCode">$\varphi \left(t\right)=\varphi \left(a\left(t\right)\right)$, 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037026931600068X&_mathId=si3.gif&_user=111111111&_pii=S037026931600068X&_rdoc=1&_issn=03702693&md5=477456d1b24447a231c478b8f926dc2e" title="Click to view the MathML source">f(T)class="mathContainer hidden">class="mathCode">$f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S037026931600068X&_mathId=si3.gif&_user=111111111&_pii=S037026931600068X&_rdoc=1&_issn=03702693&md5=477456d1b24447a231c478b8f926dc2e" title="Click to view the MathML source">f(T)class="mathContainer hidden">class="mathCode">$f(T)$ 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.