Jacobi and Lagrangian formulation of the Classical Cosmological Equations

Classical mechanics has been a well-established field for many years, but there are still some challenges that can be addressed using modern techniques. When dealing with classical mechanics problems, the first step is usually to create a mathematical expression called the Hamiltonian based on a known function called the Lagrangian. This involves using standard procedures to establish relationships like the Poisson bracket, canonical momenta, Euler-Lagrange equations, and Hamilton-Jacobi relations. In this paper, we focus on a specific problem related to the calculus of variations, which deals with finding the Lagrangian function that, when used in the Euler-Lagrange equation, produces a given differential equation. To tackle this problem, we employ two distinct methods to determine the Lagrangian and, subsequently, the Hamiltonian for the cosmological equations derived from General Relativity. These equations describe the motion of celestial objects in the universe and are of second-order in nature.