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    Classical Dynamics
    Classical Dynamics
    Classical Dynamics
    Ebook568 pages5 hours

    Classical Dynamics

    By Donald T. Greenwood

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    Since Lagrange laid the foundation of analytical dynamics some two centuries ago, the discipline has continued to evolve and develop, embracing the theories of Hamilton and Jacobi, Einstein's relativity theory and advanced theories of classical mechanics.
    This text proposes to give graduate students in science and engineering a strong background in the more abstract and intellectually satisfying areas of dynamical theory. It is assumed that students are familiar with the principles of vectorial mechanics and have some facility in the use of this theory for analysis of systems of particles and for rigid-body rotation in two and three dimensions.
    After a concise review of basic concepts in Chapter 1, the author proceeds from Lagrange's and Hamilton's equations to Hamilton-Jacobi theory and canonical transformations. Topics include d'Alembert's principle and the idea of virtual work, the derivation of Langrange's equation of motion, special applications of Lagrange's equations, Hamilton's equations, the Hamilton-Jacobi theory, canonical transformations and an introduction to relativity.
    Problems included at the end of each chapter will help the student greatly in solidifying his grasp of the principal concepts of classical dynamics. An annotated bibliography at the end of each chapter, a detailed table of contents and index, and selected end-of-chapter answers complete this highly instructive text.

    LanguageEnglish
    PublisherDover Publications
    Release dateMay 4, 2012
    ISBN9780486138794
    Classical Dynamics

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      Classical Dynamics - Donald T. Greenwood

      INDEX

      PREFACE

      Nearly two hundred years have elapsed since Lagrange published his Mécanique Analytique (1788) which laid the foundations of analytical dynamics. Later discoveries, most notably those of Hamilton and Jacobi, contributed to a theory of dynamics having unusual elegance and beauty. The next great advance came early in this century when Einstein presented a new view of the physical world with the publication of the first of his papers on relativity in 1905. In the intervening years, these foundations have been studied and improved upon, particularly with respect to the mathematical methods and notation, as well as in their logical and experimental basis. Furthermore, a wider use has been made of the advanced theories of classical mechanics as our society has become more sophisticated in its technology. It is appropriate, then, that graduate students in science and engineering should have a strong background in these more abstract and intellectually satisfying areas of dynamical theory. This is the subject of the present textbook.

      The topics covered represent a somewhat expanded version of the material in an advanced dynamics course at the University of Michigan. It is assumed that the incoming students are familiar with the principles of vectorial mechanics and have some facility in the use of this theory for the analysis of systems of particles and for rigid body rotation in two and three dimensions. Furthermore, they should be familiar with Lagrange's equation and preferably have had some experience in its application to relatively difficult problems. At present, the typical incoming student in this course at Michigan has had some exposure to dynamics in a freshman physics course, as well as an introductory dynamics course of four semester hours and an intermediate course, also of four hours. The intermediate and advanced courses form a two-term sequence.

      Because the student is assumed to be familiar with the more elementary topics in dynamics, these are not covered in any detail in this book. The first chapter, Introductory Concepts, is included, however, in order to establish some of the notation and the more important definitions to be used later. For example, the ideas of virtual work and d'Alembert's principle are introduced here. If the student has already attained a proficiency in these areas, then little time needs to be spent on this chapter.

      Chapter 2 uses d'Alembert's principle as a starting point for the derivation of Lagrange's equations of motion. The explicit form and nature of these equations are discussed in detail for holonomic and nonholonomic systems. The discussion of particular applications of Lagrange's equations is continued in Chapter 3 where the idea of impulsive constraints is introduced. The study of impulsive motion also provides the opportunity for a brief discussion of quasi-coordinates. Further topics include discussions and comparisons of gyroscopic systems and dissipative systems, as well as velocity-dependent potentials. Some of these applications may be omitted without a loss of continuity in the remaining chapters.

      In Chapter 4 the calculus of variations is introduced in the study of dynamics. The most emphasis is given to Hamilton's principle, but other results such as the principle of least action are studied. Noncontemporaneous variations, as well as the usual contemporaneous variations, are considered in a general evaluation of the canonical integral. The Hamiltonian viewpoint of dynamics is given careful consideration in the discussions of the canonical equations of motion and phase space. Much of the groundwork for the theory of the next two chapters is presented here.

      Chapters 5 and 6 are concerned primarily with the theory of canonical transformations. It was decided to consider the Hamilton-Jacobi theory before a more general discussion of canonical transformations, rather than attempting the reverse order. This allows the student to obtain some rather concrete procedures for solving problems, and it is hoped that there will be a further motivation for the rather extensive theory of canonical transformations which follows.

      The final chapter is an introductory presentation of special relativity with the Lagrangian and Hamiltonian formulations included. Enough examples and problems are presented to encourage a good working familiarity with the subject at this level.

      The author would recommend, in general, that the problems at the end of each chapter be used as much as time will allow. The problems are of varying difficulty and should help greatly in solidifying the principal concepts of classical dynamics. Occasionally, theoretical results which were not included in the text because of limited space are presented instead as problems.

      A major portion of this book was written during a sabbatical leave at the University of California at San Diego. I wish to thank Professors J. W. Miles and R. E. Roberson of the Department of Aerospace and Mechanical Engineering Sciences for helping to make my stay there enjoyable and fruitful. I am particularly appreciative of the help of my wife who did all the typing and helped with the proofreading.

      DONALD T. GREENWOOD

      Ann Arbor, Michigan

      1

      INTRODUCTORY CONCEPTS

      Dynamics is the study of the motions of interacting bodies. It describes these motions in terms of postulated laws. Classical dynamics is restricted to those systems of interacting bodies for which quantum mechanical effects are negligible; that is, it applies primarily to macroscopic phenomena. The non-relativistic theories and methods of men such as Newton, Euler, Lagrange, and Hamilton are included, as well as the more recent relativistic dynamics of Einstein.

      In this chapter we introduce a few of the basic concepts of nonrelativistic classical dynamics and shall begin to form the notational framework to be used throughout the book. Some of the material should be familiar to the reader and is not presented in detail. Other topics are explained more carefully in order to clarify the more important definitions and assumptions.

      1-1.   THE MECHANICAL SYSTEM

      Let us consider a mechanical system consisting of N particles, where a particle is an idealized material body having its mass concentrated at a point. The motion of a particle is therefore the motion of a point in space. Since a point has no geometrical dimensions we cannot specify the orientation of a particle, nor can we associate any particular rotational motion with it. In this nonrelativistic treatment, that is, for all but the final chapter, we shall assume that the mass of each particle remains constant.

      Equations of Motion. The differential equations of motion for a system of N particles can be obtained by applying Newton's laws of motion to the particles individually. For a single particle of mass m which is subject to a force F we obtain from Newton's second law the vector equation

      or

      where the linear momentum p is given by

      and where the acceleration a ) is measured relative to an inertial frame of reference.

      The existence of an inertial, or Newtonian, reference frame is a fundamental postulate of Newtonian dynamics. As an example of an inertial reference frame, consider a frame with its origin at the sun and assume that it is non-rotating with respect to the so-called fixed stars. It can be shown that any other reference frame that is not rotating, but that is translating with a uniform velocity relative to a given inertial frame, is itself an inertial frame. Hence, the existence of a single inertial reference frame implies the existence of an infinity of other inertial frames which are equally valid (but not necessarily equally convenient) for the description of the motion of a particle, using the principles of Newtonian dynamics.

      Let us assume, then, that we have found such a suitable inertial frame and that the vector ri specifies the position of the ith particle relative to that frame. The equations of motion for the system of N particles can be written with the aid of Eq. (1-1).

      where mi is the mass of the ith particle and where we have broken the total force acting on this particle into two vector components, Fi and Ri. Fi is called the applied force and Ri is the constraint force. Briefly, Ri is that force which ensures that the geometrical constraints are followed in the motion of the ith particle. The applied force Fi represents the sum of all other forces acting on the ith particle. A more detailed discussion of constraints and the associated forces is given in Secs. 1-3 and 1-4.

      In general, the forces that act on a body may be classified according to their mode of application as follows: (1) contact forces and (2) body, or field, forces. Contact forces are transmitted to the body by a direct mechanical push or pull. Body forces, on the other hand, are associated with action at a distance and are represented by gravitational, electrical, or other fields. It frequently occurs that body forces are applied throughout a body, but contact forces are applied only at its boundary surface. The forces Ri associated with the geometrical constraints are always contact forces. However, the applied forces Fi may be of either the body or contact type, or a combination of the two.

      Instead of writing a single vector equation such as (1-4) for each particle, it is sometimes more convenient to write three scalar equations. Using the Cartesian coordinates (xi, yi, zi) to represent the position of the ith particle, we obtain

      where Fix and Rix are the x components of Fi and Ri, respectively, and where Fiy, Riy, Fiz, Riz are defined similarly.

      The notation of Eq. (1-5) is somewhat unwieldy, however. In order to simplify the writing of the equations, let us denote the Cartesian coordinates of the first particle by (x1, x2, x3), of the second particle by (x4, x5, x6), and so on. Then, noting that the mass of the kth particle is

      we can write the equations of motion in the form

      where Fi and Ri are the xi components of the applied forces and the constraint forces, respectively. As particular examples of this notation, we see that F5 is the y component of the applied force acting on the second particle, and R3N is the z component of the constraint force acting on the Nth particle.

      For the case in which there are no constraints, the force components Fi are expressed as functions of position, velocity, and time; the R's are zero. Hence the system of N particles is described by 3N second-order differential equations which are, in general, nonlinear. Although these equations may be solvable, in theory, for the position of each particle as a function of time, the equations are not often completely integrable in closed form. Frequently the practical solution involves the use of a computer.

      If there are m constraints acting on the system, then the forces may be functions, not only of position, velocity, and time, but also of m additional variables known as Lagrange multipliers. In this case there are a total of (3N + m) variables to be obtained as functions of time, using the 3N differential equations of motion and the m equations of constraint.

      Thus we see that the writing of the equations of motion for a system of N particles, using Cartesian coordinates, may result in a formidable set of nonlinear ordinary differential equations. In certain cases, however, the analysis can be simplified considerably by the use of a different set of coordinates, involving fewer constraints, or perhaps eliminating the constraints altogether. The proper choice of coordinates and the use of further transformations of variables for the purpose of simplifying the analysis are topics which are discussed extensively in the remainder of the book.

      Units. The equations of motion, whether they are written in the vector form of Eq. (1-4) or in the scalar form of Eq. (1-7), require that the variables be expressed in a consistent set of units. By consistent we mean that the quantities on both sides of each equation must be expressed in the same, or equivalent, units. If we consider the dimensions of the units used in the equations of motion, we find that mass, length, time, and force are present. Because the equations of motion must exhibit dimensional homogeneity, however, these four dimensions are not independent. In fact, any one dimension can be expressed in terms of the other three.

      Certain systems of units known as absolute systems use mass, length, and time as the fundamental dimensions. For example, the mks system uses the meter as the fundamental unit of length, the kilogram as the fundamental unit of mass, and the second as the fundamental unit of time. The unit of force, the newton, is a derived unit and is equivalent to 1 kg m/sec². In general, we shall use this system whenever explicit units are mentioned.

      Another common system of units is the English gravitational system in which units having the dimensions of force, length, and time are considered to be fundamental. Here the foot is the fundamental unit of length, the second is the fundamental unit of time, and the pound is the fundamental unit of force. The fundamental unit of mass is the slug and is a derived unit. It is equal to 1 lb sec²/ft.

      1-2.   GENERALIZED COORDINATES

      Degrees of Freedom. An important characteristic of a given mechanical system is its number of degrees of freedom. The number of degrees of freedom is equal to the number of coordinates minus the number of independent equations of constraint. For example, if the configuration of a system of N particles is described using 3N Cartesian coordinates, and if there are l independent equations of constraint relating these coordinates, then there are (3N − l) degrees of freedom.

      To illustrate the idea of degrees of freedom, suppose that three particles are connected by rigid rods to form a triangular body with the particles at its corners. The configuration of the system is specified by giving the locations of the three particles, that is, by 9 Cartesian coordinates. But each rigid rod is represented mathematically by an independent equation of constraint. So 3N − l = 9 − 3 = 6, and the system has six degrees of freedom.

      The triangular body is an example of a rigid body, and has the same number of degrees of freedom as a general rigid body. One can see this by noting that the triangle can be imagined to be embedded in any given rigid body. In this case, each possible configuration of the triangle determines the configuration of the rigid body, and vice versa.

      It is important to realize that the number of degrees of freedom is a characteristic of the system itself, and does not depend upon the particular set of coordinates used in its description. For example, the configuration of the previous triangular body might be specified by giving the three Cartesian coordinates of an arbitrary point in the body and also a set of three Eulerian angles describing its orientation. In this case, there are six coordinates and no constraints, again yielding six degrees of freedom.

      Frequently it is advantageous to search for such a set of independent coordinates with which to describe the configuration of a system. In this case, there are as many coordinates as degrees of freedom, and the analysis contains a minimum number of variables.

      Generalized Coordinates. We have seen that various sets of coordinates can be used to express the configuration of a given system. Furthermore, these sets do not necessarily have the same number of coordinates nor the same number of constraints. Nevertheless, the number of coordinates minus the number of independent equations of constraint is always equal to the number of degrees of freedom.

      Now consider two sets of coordinates which describe the same system. At any given time, the values of each set of coordinates are simply a group of numbers. The process of obtaining one set of numbers from the other is known as a coordinate transformation.

      As we think of the wide variety of possible coordinate transformations, any set of parameters which gives an unambiguous representation of the configuration of the system will serve as a system of coordinates in a more general sense. These parameters are known as generalized coordinates. All the common types of coordinates can serve as generalized coordinates, but many other parameters can also be used. For example, motion of a certain generalized coordinate might involve translation of one portion of the system and rotation of another portion.

      Generalized coordinates usually have a readily visualized geometrical significance, and are often chosen on this basis. Furthermore, it is helpful in most analyses to choose a set of generalized coordinates which are independent. If the generalized coordinates specify the configuration of the system and can be varied independently without violating the constraints, then the number of generalized coordinates is equal to the number of degrees of freedom.

      Straightforward procedures, such as the use of Lagrange's equations, exist for obtaining the differential equations of motion in terms of generalized coordinates. As we shall discover in the discussion of Chap. 2, the use of independent generalized coordinates allows the analysis of the motion of most systems to be made without solving for the forces of constraint.

      Returning now to a consideration of the transformation equations relating the Cartesian coordinates x1, x2, …, x3N to the generalized coordinates q1, q2, …, qn, we will assume that these equations are of the form

      It is possible that each system of coordinates has equations of constraint associated with it. If the x's have l equations of constraint and the q's have m equations of constraint, then, equating the number of degrees of freedom in each case, we find that

      It is desirable that one and only one set of q's corresponds to each possible configuration of the system. In other words, there should be a one-to-one correspondence between points in the allowable domain of the x's and points in the allowable domain of the q's for each value of time. The necessary and sufficient condition that one can solve for the q's as functions of the x's and t is that the Jacobian determinant of the transformation be nonzero.

      As an example, suppose that the 3N x's have l equations of constraint of the form

      Let the n generalized coordinates be chosen so that they are independent, that is, the number of degrees of freedom is

      Now define an additional set of l q's and identify them with the l constant functions fj of Eq. (1–10).

      Then the transformation equations of (1-8) can be considered to be of the form

      If the Jacobian determinant is nonzero, that is, if

      then Eq. (1-8) or Eq. (1-12) can be solved for the q's as functions of the x's and time.

      The remaining constant q's for j = n + 1, …, 3N were given by Eqs. (1-10) and (1-11).

      Example 1-1. As a simple example of a transformation from Cartesian to generalized coordinates, consider a particle which is constrained to move on a fixed circular path of radius a, as shown in Fig. 1-1. The equation of constraint is

      Let a single generalized coordinate q1 represent the one degree of freedom. This polar angle can vary freely without violating the constraint. In accordance with Eq. (1-11), let us define a second generalized coordinate q2 which is constant.

      Fig. 1-1. A particle on a fixed circular path.

      The transformation equations are

      The Jacobian for this transformation is

      Hence, the q's may be expressed as functions of the x's except when the Jacobian is zero at q2 = 0. In this case the radius of the circle is zero and the angle q1 is undefined. These transformation equations are

      where we arbitrarily take 0 ≤ q1 < 2π and 0 < q2 < ∞ in order that the q's will be single-valued functions of the x's. These transformation equations apply at all points on the finite x1x2 plane except at the origin.

      Configuration Space. We have seen that the configuration of a system of N particles is specified by giving the values of its 3N Cartesian coordinates. If the system has l independent equations of constraint of the form of Eq. (1-10), then it is possible to find n independent generalized coordinates q1, q2, …, qn, where n = 3N − l. Hence a set of n numbers, namely, the values of the n q's, completely specifies the configuration of the system. It is convenient to think of these n numbers as the coordinates of a single point in an n-dimensional space known as configuration space. In other words, the configuration of any mechanical system having a finite number of degrees of freedom is represented as a single point in an n-dimensional q-space. We may also consider a vector q to be drawn from the origin to the given configuration point. This q vector has the corresponding n q's as its components in a Euclidean (rectangular) space of n dimensions.

      As a given mechanical system changes its configuration with time, the configuration point traces out a curve in q-space. For the usual case of independent q's, the curve will be continuous but otherwise unconstrained. But if there are constraints which are expressed as functions of the q's, the configuration point moves on a hypersurface having fewer than n dimensions.

      The concept of a configuration space or q-space is used frequently in our further discussions of analytical dynamics.

      1-3.   CONSTRAINTS

      We have seen that a system of N particles may have less than 3N degrees of freedom because of the presence of constraints. These constraints put geometrical restrictions upon the possible motions of the system and result in corresponding forces of constraint. Now let us consider the classification and mathematical description of constraints in greater detail.

      Holonomic Constraints. Suppose the configuration of a system is specified by the n generalized coordinates q1, q2, …, qn and assume that there are k independent equations of constraint of the form

      A constraint which can be expressed in this fashion is known as a holonomic constraint.

      A system whose constraint equations, if any, are all of the holonomic form given in Eq. (1-15) is called a holonomic system. As an example of a holonomic system, consider the motion in the xy plane of the two particles shown in Fig. 1-2. These particles are connected by a rigid rod of length l; hence the corresponding equation of constraint is

      In this case there are four coordinates and one equation of constraint, yielding three degrees of freedom. One could use this equation to eliminate one of the variables from the equations of motion. This procedure often entails algebraic difficulties, however, and is rarely used. Instead, let us search for a set of independent generalized coordinates, since it is known that these coordinates exist for all holonomic systems. For example, we can choose the Cartesian coordinates (x, y) of the center of the rod and the angle θ between the rod and the x axis as the generalized coordinates.

      We have assumed that the length l of the rod is constant, and therefore the holonomic constraint equation does not contain time. Constraints of this sort in which the time t does not appear explicitly are known as scleronomic

      Fig. 1-2. Two particles connected by a rod of length l.

      constraints. On the other hand, if the length l had been given as an explicit function of time, the constraint would have been classed as rheonomic. In the usual case, a rheonomic constraint is a moving constraint.

      The terms scleronomic and rheonomic can also be applied to a mechanical system. A system is scleronomic if (1) none of the constraint equations contain t explicitly and (2) the transformation equations (1-8) give the x's as functions of the q's only. If any of the constraint equations or the transformation equations contain time explicitly, the system is rheonomic.

      To explain this point further, it sometimes occurs that the generalized coordinates can be chosen in such a manner that there are no equations of constraint, or perhaps only scleronomic constraints, and yet the transformation equations contain time explicitly. An example is a particle constrained to move on a rigid wire which is rotating uniformly about a fixed axis, the single generalized coordinate being the position of the particle relative to the wire. Here there are no equations of constraint, but the system is rheonomic.

      So far we have discussed two methods that can be used in the analysis of systems with holonomic constraints, namely, the elimination of variables using the constraint equations and the use of independent generalized coordinates. A third approach, which can be applied to either holonomic or nonholonomic systems, is the Lagrange multiplier method. This method represents the constraints by introducing the corresponding constraint forces which are expressed in terms of k variable parameters λj known as Lagrange multipliers. This method will be explained further in Sec. 2-1.

      Nonholonomic Constraints. Now let us consider a system of m constraints which are written as nonintegrable differential expressions of the form

      where the a's are, in general, functions of the q's and t. Constraints of this type are known as nonholonomic constraints.

      As a result of the nonintegrable nature of these differential equations, one cannot obtain functions of the form given in Eq. (1-15) and use these to eliminate some of the variables. Nor is it possible to find a set of independent generalized coordinates. Hence, nonholonomic systems always require more coordinates for their description than there are degrees of freedom.

      As an example of a nonholonomic system, consider again the two particles and rigid rod of Fig. 1-2. We assume that the particles can slide on the horizontal xy plane without friction. The system is changed, however, by the addition of a nonholonomic constraint in the form of knife-edge supports at the two particles, as in Fig. 2-6. These supports move with the system and are oriented perpendicular to the direction of the rod in such a manner that they allow no velocity component along the rod at either particle. Hence, the velocity of the center of the rod must be perpendicular to the rod, resulting in the constraint equation

      or

      This expression is not an exact differential, that is, no function Φ(x, y, θ) exists such that Eq. (1-17) is of the form

      Furthermore, Eq. (1-17) cannot be multiplied by any integrating factor to yield an exact differential. Hence, it is not integrable.

      More generally, it can be shown† that the necessary and sufficient condition for the integrability of the differential equation

      is that

      where the a's are functions of x, y, and θ. Applying this criterion to Eq. (1-17), we confirm that the expression is not integrable.

      The system consisting of two particles and a rigid rod illustrates an important kinematic difference between holonomic and nonholonomic constraints. This difference occurs with respect to accessibility. If we first consider two unconstrained particles moving on the xy plane, we note that there is a four-dimensional configuration space, corresponding to the four independent coor dinates used to describe the configuration of the system. The addition of a holonomic constraint in the form of a rigid rod connecting the particles results in a reduction of the number of degrees

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