Equations of Motion of Dynamical Systems From Kinematic Characteristics of the Phase Space

Abstract

In the analysis of most engineering dynamical systems, relativistic considerations are unnecessary, allowing an absolute time and simultaneity of events to be assumed. In this article, it is first established that this simultaneity links all generalized coordinates and velocities via simple kinematic relations in the phase space. Subsequently, it is shown that equations of motion of dynamical systems can be derived by imposing these kinematic relations on a generating equation, which is a generalized form of the Jacobi’s integral. A specific process is presented for combining the kinematic relations with the generating equation to yield correct equations of motion. The process is validated by using it to prove the Lagrange’s equation. Examples are provided to demonstrate the approach. The aforementioned kinematic relations are fundamental characteristics of the phase space of general dynamical systems. They provide a novel perspective on equations of motion in analytical dynamics, which leads to a new method of deriving them.