文档介绍：Math 439 Course Notes
Lagrangian Mechanics, Dynamics, and
Control
Andrew D. Lewis
January–April 2003
This version: 03/04/2003
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This version: 03/04/2003
Preface
These notes deal primarily with the subject of Lagrangian mechanics. Matters related to me-
chanics are the dynamics and control of mechanical systems. While dynamics of Lagrangian
systems is a generally well-founded field, control for Lagrangian systems has less of a history.
In consequence, the control theory we discuss here is quite elementary, and does not really
touch upon some of the really challenging aspects of the subject. However, it is hoped that
it will serve to give a flavour of the subject so that people can see if the area is one which
they’d like to pursue.
Our presentation begins in Chapter 1 with a very general axiomatic treatment of basic
Newtonian mechanics. In this chapter we will arrive at some conclusions you may already
know about from your previous experience, but we will also very likely touch upon some
things which you had not previously dealt with, and certainly the presentation is more
general and abstract than in a first-time dynamics course. While none of the material in
this chapter is technically hard, the abstraction may be off-putting to some. The hope,
however, is that at the end of the day, the generality will bring into focus and demystify
some basic facts about the dynamics of particles and rigid bodies. As far as we know, this is
the first thoroughly Galilean treatment of rigid body dynamics, although Galilean particle
mechanics is well-understood.
Lagrangian mechanics is introduced in Chapter 2. When instigating a treatment of
Lagrangian mechanics at a not quite introductory level, one has a difficult choice to make;
does one use differentiable manifolds or not? The choice made here runs down the middle
of the usual, “No, it is far too much machinery,” and, “Yes, the unity of the differential
geometric approach is exquisite.” The basic con