Table of content:Chapter 1 IntroductionPart I Dissipative Geometry and General Relativity TheoryChapter 2 Pseudo-Riemannian Geometry and General Relativity2.1. Curvature of Space Time and Einstein Field Equations2.1.1. Einstein Field Equations2.1.2. Energy Momentum Tensor2.2. Universe as a Dynamical System2.2.1. Friedman-Robertson-Walker (FRW) Metric2.2.2. Friedman Equations2.2.3. Adiabatic Expansion and Friedman Differential EquationChapter 3 Dynamics of Universe Models3.1. Friedman Models3.1.1. Static Models3.1.2. Empty Models3.1.3. Three Non-Empty Models with = 03.1.4. Non-Empty Models with 6= 03.2. Milne ModelChapter 4 Anisotropic and Homogeneous Universe Models4.1. General Solution4.1.1. Constant Density and Zero Pressure4.1.2. Constant Pressure and Zero Density4.1.3. Absence of Pressure and DensityChapter 5 Barotropic Models of FRW Universe5.1. Bosonic FRW Model5.2. Fermionic FRW Barotropy5.3. Decoupled Fermionic and Bosonic FRW Barotropies5.4. Coupled Fermionic and Bosonic Cosmological BarotropiesChapter 6 Time Dependent Gravitational and Cosmological Constants6.1. Model and Field Equation6.2. Solution of the Field Equation6.2.1. G (t) ~ H6.2.1.1. Inflationary Phase6.2.1.2. Radiation Dominated Phase6.2.2. G (t) ~ 1/H6.2.2.1. Inflationary Phase6.2.2.2. Radiation Dominated PhaseChapter 7 Gravitational Waves in Non-Stationary Universe and Dissipative Oscillator7.1. Linear Gravitational (Metric) Waves in Flat Space Time7.2. Linear Gravitational (Metric) Waves in Non-Stationary Universe7.2.1. Hyperbolic Geometry of Damped Oscillator and Double UniversePart II Variational Principles for Time Dependent Oscillations and DissipationsChapter 8 Lagrangian and Hamilton Description8.1. Generalized Co-ordinates and Velocities8.2. Principle of Least Action8.3. Hamilton's Equations8.3.1. Poisson BracketsChapter 9 Damped Oscillator: Classical Quantum8.4. Damped Oscillator8.5. Bateman Dual Description8.6. Caldirola Kanai Approach for Damped Oscillator8.7. Quantization of Caldirola-Kanai Damped Oscillator with Constant Frequency and Constant DampingChapter 10 Sturm Liouville Problem as Damped Parametric Oscillator10.1. Sturm Liouville Problem in Doublet Oscillator Representation and Self-Adjoint Form10.1.1. Particular Cases for Non-Self Adjoint Equation10.1.2. Variational Principle for Self Adjoint Operator10.1.3. Particular Cases for Self Adjoint Equation10.2. Oscillator Equation with Three Regular Singular Points10.2.1. Hypergeometric Functions10.2.2. Confluent Hypergeometric Function10.2.3. Bessel Equation10.2.4. Legendre Equation10.2.5. Shifted-Legendre Equation10.2.6. Associated-Legendre Equation10.2.7. Hermite Equation10.2.8. Ultra-Spherical (Gegenbauer) Equation10.2.9. Laguerre10.2.10. Associated Laguerre Equation10.2.11. Chebyshev Equation I10.2.12. Chebyshev Equation II10.2.13. Shifted Chebyshev Equation IChapter 11 Riccati Representation of Time Dependent Damped Oscillators11.1. Hypergeometric Equation11.2. Confluent Hypergeometric Equation11.3. Legendre Equation11.4. Associated-Legendre Equation11.5. Hermite Equation11.6. Laguerre Equation11.7. Associated Laguerre Equation11.8. Chebyshev Equation I11.9. Chebyshev Equation IIChapter 12 ConclusionReferencesList of TablesAppendicesAppendix A Preliminaries for Tensor CalculusA.1. Tensor CalculusA.2. Calculating Christoffel Symbols from MetricA.3. Parallel Transport and GeodesicsA.4. Variational Method for GeodesicsA.5. Properties of Riemann Curvature TensorA.6. Bianchi Identities; Ricci and Einstein TensorsA.6.1. Ricci TensorA.6.2. Einstein TensorAppendix B Riccati Differential EquationAppendix C Hermited Differential EquationC.1. OrthogonalityC.2. Even/Odd FunctionsC.3. Recurrence RelationC.4. Special ResultsAppendix D Non-Stationary Oscillator Representation of FRW UniverseD.1. Time Dependent OscillatorD.1.1. Delta Function PotentialAppendix E Bianchi Model: An Alternative Way to Model the Present-Day UniverseE.1. BI ModelsE.2. Isotropization of BI models into FRW UniverseE3. Ev ukryj opis