Lagrange’s Equations with Conservative and Non-Conservative Forces - Phase Space Introduction

Overview

Explore Lagrange's equations with conservative and non-conservative forces in this comprehensive lecture on analytical dynamics. Delve into the decomposition of applied forces, the introduction of the Lagrangian function, and the application of Lagrange's equations to systems of rigid bodies. Work through practical examples to solidify understanding. Discover the concept of phase space and phase portraits as methods for analyzing motion characteristics and solving equations of motion. Learn about non-dimensionalization techniques and their importance in dynamics analysis. Gain valuable insights into advanced topics in engineering dynamics, including generalized coordinates, degrees of freedom, and the interplay between kinetic and potential energy in mechanical systems.

Syllabus

We start with Lagrange's equations of motion for the generalized coordinates, written in generalized force form. .
We decompose the applied forces into conservative (those coming from a potential energy U) and non-conservative forces. Those which come from a potential energy U can be absorbed into a Lagrangian function L, which is T - U, total kinetic energy minus total potential energy. The remaining forces (called non-conservative) are left in generalized force form..
We then work on a couple of examples using this method..
We then write Lagrange's equations for a system of rigid bodies, where now the kinetic energy includes translational and rotational kinetic energy, and the projection vectors are of two types, one for forces and the other for moments/torques. But otherwise, the equations of motion look the same..
Some worked examples of some rigid body systems. .
We introduce the idea of phase space and phase portraits, a method for finding and classifying the possible motions (i.e., solving the equations of motion) and analyzing the characteristics of the motion (e.g., phase plane analysis). We also do some non-dimensionanlize.