In this chapter we will deal with the elements of matrix algebra which we will use throughout the rest of this book. The material presented here is selfcontained. However, the student who has little prior knowledge of most of the topics in this chapter may find it helpful to refer to some of the books indicated at the end in our “For Further Reading” list. In classical mechanics we will be mainly dealing with vectors (and matrices) whose elements will be real. Hence unless explicitly stated, we shall deal in this book with real vectors and real matrices, as opposed to vectors and matrices whose elements may be complex numbers.

The chapter is divided into two main sections. The first is entitled Preliminaries, the second is called Generalized Inverse of a Matrix. We recommend that the reader begin by going to the second section after skimming over the two subsections that precede it. The details will become clearer as greater familiarity with the topics is gained. It is then that (s)he may go back to the section entitled Preliminaries, so that a better groundwork for the chapters to follow will be laid.

Preliminaries

This section contains some of the preliminary material related to vector spaces. The reader is welcome to skip the material here if (s)he is familiar with it. For the beginner, it may be better to simply skim over this section lightly at the first reading.

In this chapter we will introduce the reader to the fundamentals of Lagrangian mechanics. The reader has by now had a fair exposure to the fundamental equation, and has seen how to describe constrained motion in the framework of Cartesian coordinates. Here we will delve deeper into the basics of mechanics and introduce some of the key concepts: the principle of virtual work, and the concept of generalized coordinates. We will be taking on a more rigorous approach and amplifying on several of the issues which we had to gloss over in earlier chapters so that the reader could grasp the big picture better. The first part of this chapter is in a sense going back to better appreciate the issues involved. The second part of the chapter deals with Lagrange's equations.

The chapter introduces, in a rigorous way, what we mean by an “unconstrained” system described in terms of the Lagrangian coordinates chosen to describe the system's configuration. We show that, for a given physical situation, the analyst may exercise considerable choice in what (s)he considers to be an unconstrained system. For any specific choice of an unconstrained system, we then show how to obtain the explicit equations of motion of the constrained mechanical system in terms of the chosen Lagrangian coordinates.

Virtual Displacements

The concept of virtual displacements plays a central role in mechanics. We have already seen that equality constraints of interest to analytical dynamics are of the holonomic type or the nonholonomic type.

This book primarily deals with developing the equations of motion for mechanical systems. It is a problem which was first posed at least as far back as Lagrange, over 200 years ago, and has been vigorously worked on since by many physicists and mathematicians. The list of scientists who have contributed and attempted this problem is truly staggering. A recent monograph on the subject by Neimark and Fufaev lists more than 500 recent references.

The first major step in the understanding of constrained motion was taken by Lagrange when he formulated and developed the technique of using, what are called today, the Lagrange multipliers. The next step took about a century in the making when Gibbs and Appell developed the Gibbs–Appell approach in the late eighteen hundreds. As mentioned in Pars's book (written in 1965) this approach is considered by most to provide the simplest and most comprehensive way of setting up the equations of motion for systems with nonintegrable constraints. In 1964, P. A. M. Dirac attempted to solve the problem anew and, for Hamiltonian systems with singular Lagrangians, developed a procedure for obtaining the equations of motion for constrained systems by ingeniously extending the concept of a Poisson bracket. Dirac considered constraints which were not explicitly dependent on time.

Dirac's approach has not been discussed in this book. It is well documented in advanced treatises on analytical mechanics and may be found in the list of references that we have provided; besides, it requires background material which would go well beyond an introductory text in mechanics.

There are many treatises in the field of analytical mechanics that have been written in this century. This book is different in that it presents a new and fresh approach to the central problem of the motion of discrete mechanical systems. A system of point masses differs from a set of point masses in that the masses of a system satisfy certain constraints. This book primarily deals with the statement and analytical resolution of the problem of constrained motion and we provide the explicit equations of motion that govern large classes of constrained mechanical systems. The simplicity of the results has encouraged us to write a text which we hope will be well within the grasp of the average college senior in science and engineering.

We assume that the student has had an elementary level course dealing with statics and dynamics, and some exposure to elementary linear algebra, though the latter is not essential, because most of what is needed is contained in Chapter 2 of this book. Being pitched at the junior/senior undergraduate level, we have tried to take pains in introducing concepts slowly, gradually building them up in depth through a continual process of revisitation. We have also restricted our “For Further Reading List” at the end of each chapter principally to two books (those by Pars and Rosenberg), though there are also many other excellent treatises on analytical dynamics.

The reader will recall that we began our study of constrained motion in Chapter 3 by invoking Gauss's principle. There, we simply stated the principle as a basic principle of mechanics. We obtained the fundamental equation which described the motion of systems constrained by holonomic and nonholonomic constraints using this principle. It behooves us to understand Gauss's principle in greater depth now, moving full circle, as it were, by coming around to where we started from. This will be our agenda for this chapter.

We will begin with a simple proof of Gauss's principle in Cartesian coordinates, based on equation (5.18), the basic equation of analytical mechanics which we introduced in Chapter 5. We will next interpret this equation physically to demonstrate its aesthetic beauty, and then move on to prove the principle in terms of generalized coordinates. Using this principle, we will then provide an alternative proof for the fundamental equation in generalized coordinates.

Statement of Gauss's Principle in Cartesian Coordinates

Consider a system of n particles. The inertial Cartesian coordinates of the n particles can be described by the 3n-vector x = [x1x2x3 … x3n−1x3n]T in which the first three elements of the vector x correspond to the X-, Y- and Z- components of the position of the first particle, the next three elements correspond to the X-, Y- and Z-components of the position of the second particle, etc.

In the last chapter we dealt with the use of generalized coordinates and the fundamental equation in generalized coordinates. While we did not present a proof of this equation (see equation (5.153)) we did provide some plausibility arguments in its favor. In this chapter we will prove this equation starting from the basic equation of analytical mechanics. We will amplify some of our observations related to the use of generalized coordinates, and provide more examples so that the reader acquires a greater familiarity with the concepts developed in the previous chapter through increased exposure.

We will utilize the fundamental equation, now stated in terms of generalized coordinates, to show how simple and elegant the determination of the equations of motion related to complex, constrained mechanical systems turns out to be. For this, we expand our compass of application to include rigid bodies, the description of their configurations and the determination of their kinetic energy through the use of Euler angles. We include in some detail the theory of rotations, and the concepts of infinitesimal rotations and angular velocity. Our treatment of rigid bodies may be somewhat swift for the reader who is completely unfamiliar with this topic; however, our main aim here is not to provide a comprehensive account of the kinematics and dynamics of rigid bodies; for this the reader may look at our “For Further Reading” list at the end of this chapter.

In the last seven chapters we have developed some of the basic concepts of analytical mechanics and have moved along various fundamental threads of thought, among them Gauss's principle and the basic equation (principle) of analytical mechanics. In the previous chapter we came full circle, as it were, by deriving Gauss's principle from the basic principle of analytical mechanics. We then used it to obtain, in a straightforward manner, the fundamental equation of analytical mechanics in Lagrangian coordinates.

In this, our last chapter, we shall attempt to further deepen our understanding by looking at alternative formulations of the problem of constrained motion and by developing the various interconnections between them, thereby weaving the various threads that we had previously laid out into the fabric that is analytical dynamics. This, we hope, will provide the reader with a more holistic picture of the subject.

We shall begin with some of the subtler ideas related to the fundamental equation, and provide the general form of the fundamental equation. Then we will move on to the Lagrange multiplier method and show its close connection to the fundamental equation and to the basic principle of analytical mechanics. The general form of the Lagrange multiplier vector will be obtained. We shall also develop the equations of motion through the direct use of the basic principle of analytical mechanics. This approach re-emphasizes the importance of linear algebra even when considering highly nonlinear mechanical systems.

Having taken the small detour into matrix algebra in the last chapter, we suggest to the reader at this point to go back to the first chapter and skim over it again, especially so that this time the meaning of equations (1.65) to (1.70) is better understood. In this chapter, the equation of motion for constrained mechanical systems will be formulated. The reader will find that it is not necessary to digest all the intricacies that are broached in the previous chapter. Only a few key results from it will be needed. However, for a deeper understanding of the material in this chapter, we recommend that Chapter 2 be carefully read; for, in doing so, a better grounding of the basics, which we will be using in later chapters, will be obtained.

We begin by starting our discussion on the nature of some elementary constraints. We will introduce the concepts of holonomic and nonholonomic Pfaffian constraints and, in this chapter, we will limit our discussion only to these two types of constraints for they form the backbone of Lagrangian mechanics. However, we will show that the fundamental equation is valid for a much larger class of constraints. Next we will introduce Gauss's principle, which may be taken as our starting point. This principle is somewhat less popular that the principles of Lagrange, Hamilton, Gibbs and Appell; we assure the reader that what Gauss's principle lacks in popularity, it more than makes up for in its tremendous sweep of applicability.