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

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.