A Quadric of Kinetic Energy in the Role of Phase Diagrams – Application to the Belinski–Khalatnikov–Lifshitz Scenario

If the kinetic energy \(E_k\) of an autonomous 3-dimensional Hamiltonian system is a quadratic form in the momenta, the corresponding quadric, together with the trajectory of \(E_k\) in the momentum space, may play the role of phase diagrams. The evolution in that space takes place inside or outside a quadric \(E_k-\mathcal {H}<0\) for each value of the Hamiltonian \(\mathcal {H}\) . We apply this tool to the Belinski–Khalatnikov–Lifshitz (BKL) scenario, which describes behavior of an anisotropic universe near the cosmological singularity. A transformation of variables turns the BKL system into a Hamiltonian model subject to constraint \(\mathcal {H}=0\) . The boundary of the quadric, \(E_k=0\) , which is a cone, corresponds to the limit \(t\to \infty \) . A comprehensive description of the asymptotics for large t is provided by this method. For systems like BKL, the method is similar to the diagrams of Misner, Thorne and Wheeler [5], but it seems to be more natural and simpler than those diagrams.