On the Topic of Motion Integrals
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An integral of motion is a function of the states of a dynamical system that is constant along the system’s trajectories. Integrals are known for their utility as a means of reducing the dimension of a system, effectively leaving only one differential – or in some cases algebraic – equation to be solved. Invariants of dynamical systems have also proven useful in other contexts, such as in estimation, numerical integration and optimal control. Regardless of the manner in which an integral is employed, finding an analytic form for the integrals of a system generally requires solution of a system of non-linear partial differential equations, with the exception of cases in which certain symmetries of the system are apparent. The objective of this work is to investigate a generalized method for determining motion integrals for non-linear dynamical systems. This method will not work for all nonlinear systems. Indeed, it is expected that the results will test the limitations of this method. In this we consider a method for determining integrals of motion for a small class of dynamical systems akin to the traditional series expansion method for solving partial differential equations. This method involves posing a candidate integral of motion as a series expansion in terms of some set of polynomials. The coefficients of the candidate polynomial are treated as the unknowns in a system of equations. The system of equations is constructed by sampling simulated trajectories of the dynamical system in question. Then the coefficients are solved for using singular value decomposition. There are a number of parameters that can potentially affect this method’s ability to generate an integral of motion that effectively approximates the phase space of the full nonlinear dynamical system. Part of this thesis proposes what some of these parameters might be and investigates how they affect the outcome. A couple of well-known systems are used to conduct these tests: the simple pendulum and the rotating rigid body. The simple pendulum is one of the simplest examples of a non-linear system, and examples of the rotating rigid body in aerospace engineering are ubiquitous.
Bertinato, Christopher (2013). On the Topic of Motion Integrals. Master's thesis, Texas A&M University. Available electronically from