Semi-Analytic Methods for Averaging and Rectification of Complex Dynamical Systems

Dynamical Systems Theory (DST) serves as a means to understand and describe the changes that occur over time in physical systems. It involves a detailed analysis of a model based on the particular laws governing its change. These laws are in turn derived from suitable theory: Newtonian mechanics, Lagrangian and Hamiltonian mechanics, fluid dynamics, etc. All these models can be conceptually unified in the mathematical notion of a dynamical system. Broadly, there are two approaches to study dynamical systems: a numerical approach and an analytic approach. A numerical approach involves the propagation of the dynamical equations of motion, usually a set of ordinary or partial differential equations that govern the evolution of each state of the dynamical system. Analytic approaches, in contrast, results in a closed-form solution of the dynamical system that maps the states at a particular time to those at another time. With increasing complexity of physical systems, numerical computations become increasingly expensive, and analytical solutions seldom exist. DST then provides us with tools to analyze such complex behaviors by emphasizing geometric interpretations over purely numeric solutions. Leveraging internal symmetries and examining trajectory bundles in the phase volume help us decipher qualitative dynamical behavior. To emphasize the importance of such methods, two particular dynamical systems are treated in this dissertation. The attitude motion of a rigid body in Keplerian orbit is a dynamical system where the nature of motions has a strong parametric dependence. The restricted three-body problem is another dynamical system that exhibits a wide range of complex motions. Both these systems are studied, and new techniques, metrics, and insights are obtained through the use of DST and analytical averaging techniques. We show that the use of Classical Rodrigues Parameters for the attitude motion of the rigid body subject to gravity-gradient torques enables us to characterize the equilibria associated with the rotational motion about its mass center. A parametric study of the stability of equilibria shows that large oscillations are induced due to the energy exchange between the pitch and roll-yaw motions, specifically near the 2:1 resonant commensurability regions. A visualization tool is developed to study these pitch oscillations and gain insight into the rigid body motion near the internal resonance conditions. A measure of coupling between the pitching and roll-yaw motions is developed to quantify the energy exchange utilizing information from the state transition matrix. \Poincare surface of sections, bifurcation diagrams and phase-plane plots are used to examine the attitude motion of a rigid body under various conditions. Further, an analytic treatment of the rigid body dynamics in the Serret-Andoyer variables is carried out for a fast-rotating rigid body assumption. The case for a slow-rotating rigid body is also examined, and the validity of the dynamical model is tested by developing a theory for Lunar free librations. It becomes evident that both numerical and analytic methods rely heavily on the manner in which they are described: i.e., the coordinate system used. Consequently, a judicious choice of the coordinate system dramatically simplifies the problem at hand. Through the use of Hamilton-Jacobi theory and recent advances in approximation theory, this work presents a systematic procedure to mathematically obtain the best choice of coordinates that simplify the evolution of a dynamical system through rectification. Pursuit of such techniques has culminated in the development of a novel semi-analytic method to treat general dynamical systems. Several examples demonstrate the efficacy of the proposed method in obtaining a functional form of the solution of the dynamical system. Applications to the main problem in artificial satellite theory, treatment of non-conservative dynamical systems and the two-point boundary value problem are investigated. Agreement with classical solutions establish closure with analytical methods and provides strong evidence in support of the methodology developed