Damping oscillatory models in general theory of relativity

Abstract

In my thesis we have studied the universe models as dynamical systems which can be represented by harmonic oscillators. For example, a harmonic oscillatior equation is constructed by the transformation of the Riccati differential equations for the anisotropic and homogeneous metric. The solution of the Friedman equations with the state equation satisfies both bosonic expansion and fermionic contraction in Friedman Robertson Walker universe with different curvatures is studied as a conservative system with the harmonic oscillator equations. Apart from the oscillator representations mentioned above (constructed from the universe models), we showed that the linearization of the Einstein field equations produces harmonic oscillator equation with constant frequency and the linearization of the metric on the de-Sitter background produces damped harmonic oscillator system. In addition to these, we have constructed the doublet and the Caldirola type oscillator equations with time dependent damping and frequency terms in the light of the Sturm Liouville form. The Lagrangian and Hamiltonian functions are calculated for all particular cases of the Sturm Liouville form. Finally, we have shown that zeros of the oscillator equations constructed from the particular cases can be transformed into pole singularities of the Riccati equations.