Integrability of the Tolman-Oppenheimer-Volkoff equation
The paper studies a nonlinear model for a static star in general relativity from the view of the point of integrability. The considered model can be reduced to a planar system \[ \frac{d x}{d t}=y-x, \quad \frac{d y}{d t}=\frac{y}{1-2 x}\left(2-\frac{(5 \gamma-4) x}{\gamma-1}-\gamma y\right). \] After time scaling, it becomes a polynomial differential system \[ x^{\prime}=(y-x)(1-2 x), \quad y^{\prime}=y\left(2-\frac{5 \gamma-4}{\gamma-1} x-\gamma y\right).\tag{1} \] There are three main results. Firstly, they prove that system (1) with \(\gamma\neq\{0,1 \}\) has no global analytic first integrals, by analyzing the resonance relation of eigenvalues at the equilibrium. Secondly, they show that system (1) has a a Liouville first integral if and only if \(\gamma\in \{0,2/3,4/5,5/6\}\). To this end, the authors transform system (1) into a quadratic Lotka-Volterra system and apply the classification of Liouville first integrals for Lotka-Volterra system in [\textit{J. Moulin Ollagnier}, Qual. Theory Dyn. Syst. 2, No. 2, 307--358 (2002; Zbl 1084.34001)]. Finally, they further prove that it has no Puiseux first integrals with the help of a recent work in [\textit{M. V. Demina} et al., Qual. Theory Dyn. Syst. 21, No. 2, Paper No. 35, 35 p. (2022; Zbl 07489700)]. The methods used in this work may be used to study other physical ODE models in \(\mathbb{R}^2\).
scientific article; zbMATH DE number 7614817
scientific article