Uncertainty Quantification And Analysis Of Dynamical Systems With Invariants

TL;DR

This paper develops methods to analyze how statistical moments of invariants like energy and angular momentum evolve over time in stochastic dynamical systems, demonstrated through rigid body and two-body problem case studies.

Contribution

It introduces a framework for describing the evolution of moments of invariants in stochastic systems, providing closed-form solutions and bounds for specific physical models.

Findings

Closed-form evolution of kinetic energy moments in rigid body dynamics

Bounds on angular momentum moments in two-body problem

Effective uncertainty quantification in stochastic dynamical systems

Abstract

This paper considers uncertainty quantification in systems perturbed by stochastic disturbances, in particular, Gaussian white noise. The main focus of this work is on describing the time evolution of statistical moments of certain invariants (for instance total energy and magnitude of angular momentum) for such systems. A first case study for the attitude dynamics of a rigid body is presented where it is shown that these techniques offer a closed form representation of the evolution of the first and second moments of the kinetic energy of the resulting stochastic dynamical system. A second case study of a two body problem is presented in which bounds on the first and second moments of the angular momentum are presented.