The Motion Constants of Linear Autonomous Dynamical Systems

Abstract

The time-independent integrals, here referred to as motion constants, for general nth-order linear autonomous systems are developed. Although it is commonly believed that this topic has been fully addressed, close inspection of the literature reveals that a comprehensive development is missing. This paper provides a complete tutorial treatment of the calculation of these motion constants. The process involves a state transformation to a canonical form of uncoupled real subsystems. Following this, motion constants that are internal to each subsystem are found, after which motion constants that connect the subsystems to each other are computed. Complete sets of n − 1 real single-valued motion constants can be formed for all linear autonomous systems with a single exception. The exception is systems composed of undamped oscillators whose frequency ratio is irrational. Such systems are known to exhibit ergodic behavior and lack a number of analytic motion constants.