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Exact solutions of the FLRW cosmological model via invariants of the Hamilton-Jacobi method

In this study, we proceed to solve the field equations of the spatially flat Friedman-Lemaitre-Robertson-Walker (FLRW) cosmological model in the presence of the cosmological constant \(Λ\) by making use of the Invariants of Hamilton-Jacobi method (IHJM). This method enables us to extract systematically two independent first integrals such as \(l_{\rm HJ,1}(a,\dot{a})=c_{1}\) and \(l_{\rm HJ,2}(t,a,\dot{a})=c_{2}\) associated to the transformations group keeping the form of the Hamilton's canonical equations (HCEs) of the cosmological model invariant. Extracting these invariants means not only finding the general solution of the field equations of the model, but also obtaining the Lagrangian and Hamiltonian functions for the model whose dynamics acts like the dynamics of a single particle in a one-dimensional mini-super space \(\mathbb{Q}=(a)\). In addition, to obtain the general solution of the model, the IHJM have also solved the inverse problem of calculus of variation (IPCV) without resorting to Helmholtz conditions and whether the necessary conditions for the existence of the Lagrangian function are hold or not. The main part of the IHJM is to find the generating function of the canonical transformation (CT) and then extract two independent invariants for the desired model by using the Hamilton-Jacobi equation (HJE). This study shows that there is a close relationship between the group of the CTs of the Hamiltonian function of the particle and the one-parameter Lie group of transformations keeping invariant the Einstein-Friedmann dynamical equation (EFDE) \(\ddot{a}=F(a,\dot{a})\), so that both of them lead to the same result. In this way, having both the IHJM and the invariants of the symmetry groups method (ISGM), a comprehensive integration theory by unifying them can be achieved for studying the desired models.