Classical Mechanics: 2nd Edition

By H.C. Corben and Philip Stehle

Applications not usually taught in physics courses include theory of space-charge limited currents, atmospheric drag, motion of meteoritic dust, variational principles in rocket motion, transfer functions, much more. 1960 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Jan 17, 2013
• ISBN: 9780486140780

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1 KINEMATICS OF PARTICLES

1. INTRODUCTION

The sciences of physics and chemistry are concerned with the interactions between inanimate material systems. Although chemistry is related chiefly to the interactions between atoms to form molecules, physics is primarily the study of the motion induced in one system of matter by the presence of other systems and of the characterization and measurement of such states of motion by a human observer.

The situations of interest to the physicist are among the least complicated of all known phenomena. This fundamental simplicity of the subject matter allows the science to reach and even demand a degree of precision in both theory and experiment not known elsewhere. Sometimes this precision can be realized, however, only by the introduction of complexities of theory and experiment which mask the very simplicity that made it possible.

The aim of theoretical physics is to provide and develop a self-consistent mathematical structure which runs so closely parallel to the development of physical phenomena that, starting from a minimum number of hypotheses, it may be used to accurately describe and even predict the results of all carefully controlled experiments. The desire for accuracy, however, must be tempered by the need for reasonable simplicity, and the theoretical description of a physical situation is always simplified for convenience of analytical treatment. Such simplifications may be thought of as arising both from physical approximations, i.e., the neglect of certain physical effects which are judged to be of negligible importance, and from mathematical approximations made during the development of the analysis. However, these two types of approximations are not really distinct, for usually each may be discussed in the language of the other. Representing as they do an economy rather than an ignorance, such approximations may be refined by a series of increasingly accurate calculations, performed either algebraically or with an analog or digital computer.

More subtle approximations appear in the laws of motion which are assumed as a starting point in any theoretical analysis of a problem. At present the most refined form of theoretical physics is called quantum field theory, and the theory most accurately confirmed by experiment is a special case of quantum field theory called quantum electrodynamics. According to this discipline, the interactions between electrons, positrons, and electromagnetic radiation have been computed and shown to agree with the results of experiment with an over-all accuracy of one part in 10⁹. Unfortunately, analogous attempts to describe the interactions between mesons, hyperons, and nucleons are at present unsuccessful.

These recent developments are built on a solid structure which has been developed over the last three centuries and which is now called classical mechanics. Figures 1-1 and 1-2 illustrate how classical mechanics is related to other basic physical theories. A line connecting two boxes indicates that the branch of theoretical physics to which the arrow points is a particular case of the branch from which the line originates. The solid lines indicate special cases from a mathematical point of view; i.e., they lead from a theory to one of its applications. The dashed lines, on the other hand, refer to particular cases that may be obtained by neglecting one or the other of the three basic effects that have been discovered since the development of nonrelativistic classical theory-radiative effects, quantum effects, and relativistic effects. For some situations (e.g., quantum effects in atomic theory) one of these may be of dominant importance, whereas for others (e.g., celestial mechanics) the same effect may be completely negligible.

A theory that describes the motion of a particle at any level of approximation must eventually reduce to nonrelativistic classical mechanics when conditions are such that relativistic, quantum, and radiative corrections can be neglected. This fact makes the subject basic to the student’s understanding of the rest of physics, in the same way that over the centuries it has been the foundation of human , understanding of the behavior of physical phenomena.

= 1.054 × 10—27 gram-cm²/sec. Some examples are given in Table 1-1. Clearly, in all but the last case the existence of a smallest unit of angular momentum is irrelevant, and the error introduced by using the approximation of classical mechanics will be small compared with both unavoidable experimental errors and other errors and approximations made in describing theoretically the actual physical situation. However, classical mechanics should not be studied only as an introduction to the more refined theories, for despite advances made during this century it continues to be the mechanics used to describe the motion of directly observable macroscopic systems. Although an old subject, the mechanics of particles and rigid bodies is finding new applications in a number of areas, including the fields of vacuum and gaseous electronics, accelerator design, space technology, plasma physics, and magnetohydrodynamics. Indeed, more effort is being put into the development of the consequences of classical mechanics today than at any time since it was the only theory known.

fig. 1-1. The basic physical theories known in 1926 with some applications made possible within their framework.

Fig. 1-2. The basic physical theories discovered, apart from classical electromagnetic theory, since 1904.

TABLE 1-1

but are moving with relative velocities small compared with that of light (c = 2.99776 × 10¹⁰ cm/sec). In Chap-ter 16, however, we give an introduction to the special theory of relativity, which is capable of describing situations in which this restriction on the velocity does not necessarily hold.

In addition, none of the above theories is applicable to the description of the interaction between systems separated by distances comparable to Gm/c², where m is the largest mass and G is the gravitational constant (G = 6.670 × 10-8 dyne-cm²/gram²). For a proton and an electron this distance is extremely small (~10—52 cm) and many unknown properties of the interaction between these particles already occur at very much larger distances (~10—13 cm). For a planet inter-acting with the sun, the critical distance is 1.47 km, so that although Mercury is much farther from the sun than this, there exists in its motion a very small anomaly that cannot be described even by relativistic classical mechanics but which is adequately explained by the general theory of relativity.

2. DEFINITION AND DESCRIPTION OF PARTICLES

The simplest mechanical system is one which may be represented in the mathematical scheme of mechanics by a point. Such a system is called a particle. How good a representation this is for a particular system can be determined only when we come to consider a more complicated representation of the system and can estimate the effects due to features neglected in the particle treatment. Intuitively it is plausible that it will be a good approximation for a given body provided that this body is small compared with its distance from other bodies and provided that the motion of the body as a whole is only slightly affected by its internal motion. Thus a particle is a body whose internal motion is irrelevant to its motion as a whole, so that it may be represented by a mass point having no extension in space.

A particle is described when its position in space is given and when the values of certain parameters such as mass, electric charge, and magnetic moment are given. By our definition of a particle, these parameters must have constant values because they describe the internal constitution of the particle. If these parameters do vary with time, we are not dealing with a simple particle. The position of a particle may, of course, vary with time.

We may specify the position of a single particle in space by giving its distances from each of three mutually perpendicular planes. These three numbers are called the cartesian coordinates of the particle. Since three mutually perpendicular planes meet in three mutually perpendicular lines, we may also consider the cartesian coordinates of a particle as the displacements in the directions of these three lines needed to move the particle from the point of intersection of the three lines to its actual position. These coordinates will be denoted by x1, x2, x3, or in general by xi (i = 1, 2, 3). To avoid the use of indices in particular problems, we shall also use the letters x, y, z as coordinates.

The coordinate system introduced in this way plays the role of the observer of the particle in the mathematical scheme. The unessential features of the observer have been completely removed, only an extreme simplification of the observer’s measuring implements in the form of a set of coordinates being left. The very complex relationship between the particle and the observer which is to be described mathematically is reduced to the relationship of a point to a set of coordinates. In this way we tend almost to forget the observer entirely. In quantum mechanics and relativity theory the role of the observer himself is stressed considerably more.

The coordinates xi may be considered as the components of a vector, the radius vector r of the particle

(2.1)

where ei is the unit vector along the i coordinate axis. Sometimes these vectors are denoted by i, j, k respectively.

We now introduce a notation that makes many formulas simpler in appearance and easier to understand. Whenever an italic index appears twice, it is to be summed over the entire range of the index. Greek indices are not to be summed unless a summation is explicitly indicated. We shall usually reserve such indices for numbering individual particles in many-particle systems. In this notation (2.1) becomes simply

(2.2)

If N particles are present, the position of each one is specified by its radius vector, that of the ρth particle being denoted by rρ. The 3N = f’ quantities xρi (i = 1, 2, 3; ρ = 1, 2, . . . , N) specify the configuration of the entire system. For many purposes it is more convenient to regard the f’ quantities xρ,i as the cartesian coordinates of a single particle which is located in an fare frequently used, mp being the mass of the pth particle.) This f’-dimensional space is called the unconstrained configuration space of the system.

Very often it is not convenient to describe the position of even a single particle in terms of rectangular cartesian coordinates referred to a particular set of coordinate axes. If, for example, the particle moves in a plane under the influence of a force which is directed toward a fixed point in the plane and which is independent of the azimuthal angle θ, it is usually more convenient to use the plane polar coordinates

(2.3)

or if the force is spherically symmetrical it is natural to use the spherical coordinates

(2.4)

These coordinates are also used if the particle is constrained to move on a fixed circle or fixed sphere, respectively.

Sometimes it is useful to look at the motion of the particle from the point of view of a set of coordinates in uniform motion in the x direction with velocity v

(2.5)

or from a uniformly accelerated system

(2.6)

In general, each transformation of the coordinate system xi to a new set qm may be expressed as a set of three equations of the form

(2.7)

For the stationary coordinate systems (2.3) and (2.4), the relations between xi and qi do not involve the time t.

If equations (2.7) are such that the three coordinates qm can be expressed as functions of the xi, we have

(2.8)

The qm are as effective as the xi in describing the position of the particle. The qm are called generalized coordinates of the particle. The generalized coordinates may themselves be rectangular cartesian coordinates [as in (2.5), (2.6)] or they may be a set of any three variables, not necessarily with the dimension of length, which between them specify unambigously the position of the particle relative to some set of axes.

To obtain the necessary and sufficient conditions that equations (2.7) ay be solved for the qm at a fixed value of time to give a set of equations of the form (2.8), we consider a small displacement of the particle defined by changes δxi in the xi, with t held fixed:

(2.9)

(This is, according to our notation, the sum of three terms!) Equation (2.9) is a set of inhomogeneous linear equations for the δqm which can be solved if and only if the determinant of the coefficients does not vanish:

(2.10)

This determinant is called the Jacobian determinant of the x’s with respect to the q’s and is denoted by [∂(x1, x2, x3)]/[∂(q1, q2, q3)]. The determinant |∂qm/∂xi| is the reciprocal of that appearing in (2.10).

If the inequality (2.10) is satisfied., we may write

(2.11)

and the integration of these equations yields the q’s as functions of the x’s. It is not necessary to make this integration, as (2.7) can be solved to yield (2.8) directly provided that (2.10) holds.

The idea of the last few paragraphs can be generalized to the case of N particles. The 3N = f’ coordinates xρ,i may be considered as functions of f’ parameters qm and perhaps time:

(2.12)

If the Jacobian determinant does not vanish, i.e.,

(2.13)

the qm can be expressed as functions of the xρ,i and time:

(2.14)

The q’s are now as effective as the x’s in describing the configuration of the system.

In the case of a system of many particles it is often necessary to introduce the generalized coordinates because of the presence in a particular problem of a number—sometimes a very large number—of constraints which prevent the coordinates from changing independently of each other. These constraints may take the form of k relations:

(2.15)

Thus, for example, the coordinates of the N particles in a rigid structure are not independent, there existing 3N—6 such relations between them (see Sec. 47). Any relation of the form (2.15) between the particle coordinates and, possibly, the time is called a holonomic constraint , whether it be due to the properties of the system itself (as in the case of a rigid body) or to the conditions under which the system is interacting with its environment (e.g., a bead moving along a smooth wire). If the time t appears explicitly in the equation describing a given constraint, that constraint is said to be moving (as when, in the case of the bead on the wire, the wire is forced to move in some specified manner relative to the observer). If t does not appear explicitly in one of the equations (2.15), that constraint is said to be fixed, or workless.

The advantage of generalized coordinates now appears in the fact that it is possible to introduce these coordinates in such a way that the constraints (2.15) become trivial. There are f’ coordinates and k constraints. Let the last k of the generalized coordinates qf+1, qf+2, · · · , qf+k (f + k = f’ = 3N) be defined by the equations

(2.16)

and let the other f generalized coordinates be defined by

(2.17)

Again the Jacobian determinant of the transformation from the 3N = f’ variables xρ,i; to the f’ variables qf+s qm must not vanish. The equations of constraint are now simply

(2.18)

and the configuration of the system depends in a nontrivial way only on the rest of the q’s, i.e., the f variables q1, q2, · · · , qf.

The number f = 3N — k is called the number of degrees of freedom of the system. The configuration of the system may be represented by the position of a point in an f-dimensional space which is called the configuration space of the system.

There exists another type of constraint which involves conditions only on infinitesimal changes in the coordinates in a way which cannot be integrated to yield relations such as (2.15) between the coordinates themselves. The classic example of this kind of constraint is a sphere constrained to roll without slipping on a plane. Five coordinates are needed to specify the configuration of this system, two to locate the center of the sphere and three to give the orientation of the sphere about its center (Sec. 47). There are two constraints: both components of the velocity of the point of contact of the sphere with the plane must vanish. These constraints are of a differential character. Their nonintegrability means that the sphere can be given any orientation at any position of the center by a displacement which does not violate the constraints, and so no relation between the coordinates can exist. We shall have no occasion to discuss this nonholonomic type of constraint.

3. VELOCITY

The coordinates describing a system of particles may vary with time. Consider a system consisting of a single particle. It is described by the values of its cartesian coordinates xi at each value of the time t. The rate at which the coordinates change with time gives the velocity of the particle. Denoting the cartesian components of velocity by vi, we have

(3.1)

This may be written in vector notation as

(3.2)

If the system consists of N particles, the cartesian components of the velocity of each particle may be defined as above:

(3.3)

The quantities vρ,i may be considered the cartesian components of a vector in the f’-dimensional unconstrained configuration space of the system.

Velocity may be described in terms of the generalized coordinates of Sec. 2. From (2.12) we see that

(3.4)

where f replaces f’ because we assume the constraints to have been eliminated in the manner of Sec. 2. Then

(3.5)

ρ.i even though the number of equations may be greater than the number of unknowns, since the equations are not independent but must satisfy the constraints.

The cartesian components of velocity are seen to be linear functions of the generalized velocity components no matter how the generalized coordinates are defined. This means that it is easy to express velocities in generalized coordinates. The term ∂xρ,i/∂t appears only when there are moving constraints on the system or in the rare cases where it is convenient to introduce moving coordinate axes.

4. ACCELERATION

The velocity components of a system of particles may vary with time. The rate of change of the velocity components gives the acceleration components only when the coordinates are cartesian:

(4.1)

If there is only one particle in the system, the acceleration is represented by a vector:

(4.2)

If there is more than one particle, the aρ,i may be considered the cartesian components of a vector in the unconstrained configuration space of the system.

Acceleration components may be given in terms of the generalized coordinates. From (3.5) we obtain

(4.3)

If the x’s do not depend explicitly on the time, which is the usual situation, (4.3) reduces to

(4.4)

The cartesian acceleration components are not linear functions of the second derivatives of the generalized coordinates alone, but depend quadratically on the generalized velocity components as well. This quadratic dependence on the generalized velocity components disappears only if all the second derivatives of the x’s with respect to the q’s vanish, i.e., only if the x’s are linear functions of the q’s. The terms quadratic in the velocity components, which enter whenever the coordinate curves qm = const are not straight lines, represent effects like the centripetal and Coriolis accelerations.

Higher derivatives of the coordinates with respect to time could be named and discussed. This proves unnecessary because the laws of mechanics are stated in terms of the acceleration. Even the computation of the components (4.3) of the acceleration in terms of generalized coordinates can become tedious for relatively simple problems. The advantage of introducing generalized coordinates would then seem to be counterbalanced by the algebraic complexity of the acceleration components which are to be inserted in the dynamic law F = ma. Fortunately, a method due to Lagrange, and discussed in Chapter 6, makes it possible to avoid this difficulty and to write down equations of motion in terms of generalized coordinates without ever having to compute the second time derivatives of these coordinates.

5. SPECIAL COORDINATE SYSTEMS

We may apply the results of the last section to the commonly used special coordinate systems defined by (2.3) and (2.4). In order to describe the motion of a particle in a plane, it may be convenient to introduce the plane polar coordinates r, θ defined by (2.3), r being the length of the radius vector to the particle and θ the angle between the radius vector and the x axis. If the particle is not confined to a plane, but is subjected to forces which possess cylindrical symmetry, i.e., are independent of θ, it is convenient to supplement (2.3) with the extra coordinate

q3 = z

The set of coordinates q1, q2, q3 is then called a cylindrical coordinate system (see Fig. 5-1). (The symbols r, θ are often replaced by p, φ to avoid confusion with the quantities r, θ used for spherical polar coordinates.)

When solved for x and y, equations (2.3) give

(5.1)

The velocity components are found by differentiating (5.1) :

(5.2)

:

(5.3)

The acceleration components are found by differentiating (5.2). The result is

(5.4)

This is seen to involve terms quadratic in the velocity components, as was shown would be the case in (4.4). The various terms can be identified. Those in the first parentheses give the acceleration in the

Fig. 5-1. Cylindrical coordinates.

r direction, and those in the second give the acceleration in the θ direc-tion. The r acceleration consists of two parts, that due to the second derivative of r and that due to the centripetal acceleration. The latter arises because the curves of r = const are not lines but circles, and the motion of a particle along a circle with constant speed is an accelerated motion since the direction of the velocity changes continually. The θ acceleration also consists of two parts. The first is due to the second derivative of θ, which changes as r is constant, and because even a constant velocity vector has a component in the θ direction which changes as θ changes.

For a particle moving under the influence of a force which possesses spherical symmetry, i.e., is directed towards a fixed point and depends only on the distance of the particle from the point, it is convenient to introduce the spherical coordinates defined by (2.4). When solved for the x’s, these yield (see Fig. 5-2)

(5.5)

The geometrical significance of these coordinates is the following: r is the length of the radius vector of the particle; θ is the angle between

Fig. 5-2. Spherical polar coordinates.

the radius vector and the z axis, or the complement of the latitude of the particle; φ is the angle between the plane containing the radius vector and the z axis and the plane containing the x axis and the z axis, or the longitude of the particle.

The velocity components are found by differentiating (5.5):

(5.6)

, φ in terms of the coordinates x, y, z :

The acceleration components could be found by differentiating (5.6), but this would lead to a multitude of terms; as has been pointed out, it will not be necessary to perform this differentiation.

On rare occasions the analysis is simplified by the introduction of elliptical coordinates or parabolic coordinates to specify the position of a particle.

Elliptical coordinates in the plane are defined by a one-parameter family of confocal ellipses and a one-parameter family of confocal hyperbolas, the latter having the same focuses as the ellipses. Through any point in the plane there pass one and only one ellipse and one and only one hyperbola. The values of the parameters for these two curves are the coordinates of the point. Parabolic coordinates in the plane are defined by two one-parameter families of parabolas with the origin as common focus. One family opens to the right and one to the left. The values of the parameters which give the two parabolas intersecting at a point give the parabolic coordinates of that point.

The equations describing these coordinates are given in Exercises 4, 5, and 6 at the end of this chapter and the coordinate systems are discussed in more detail in Sec. 64, where we consider the problems to which they are especially applicable—the motion of a particle under the influence of two fixed centers of force, and the motion of a particle near one such attracting center in the presence of an external uniform field.

6. VECTOR ALGEBRA

In this section we are concerned with coordinate transformations of a simpler type than those discussed in Sec. 5. We consider a rectangular cartesian set of coordinate axes relative to which the position of a point is (x1, x2, x). Once the rotation is specified, the relations between the xi become known:

(6.1)

For this type of coordinate transformation, which is of the general form (2.7), the xi , and vice versa; i.e., equations (6.1) are of the form

(6.2)

where the Sij are a set of constants. Thus, for example, the reader may verify that if the rotation from the x’s to the x’s were through an

Fig. 6-1. Rotation of the coordinate axes through an angle θ in the (1-2) plane.

angle θ about the 3 axis, the transformation equations would be (see Fig. 6-1)

(6.3)

Although a detailed discussion of the transformation (6.2) is given in Appendix II, we should draw attention here to the fact that (6.2) is of basic significance in defining a vector. If three quantities have the values a1, a2, a3 when referred to the xaxes, and if

(6.4)

i.e., if the three quantities transform under a rotation of the axes in the same way that the coordinates xi transform, then the set of quantities is called a vector.

Any relation of the form a=b

between two vectors, if true in one coordinate system, is automatically true in any other coordinate system obtained from the first by a rota-tion about the origin. We may therefore write relations between vectors without specifying the orientation of the axes to which the vectors are referred. We give below a review of vector algebra with which the student should be familiar.

Given two vectors a and b, their scalar product is defined to be

a . b = |a| |b| cos θ

where |a| is read length of a, and where θ is the angle between the positive senses of the two vectors. In cartesian components,

(6.5)

Scalar multiplication is commutative and distributive; i.e.,

(6.6)

As the name implies, the result of the scalar multiplication of two vectors is a scalar. The scalar product is sometimes known as the inner product.

The vector product of two vectors, a and b, is defined by

(6.7)

where the symbols are as above and e is a unit vector in the direction perpendicular to the plane of a and b, drawn in the sense which makes the triple of vectors a, b, e a positive or a right-handed one.

From the definition it follows that the vector product is not commutative. Rather

(6.8)

The distributive law holds:

(6.9)

The associative law does not hold, for in general

a x (b x c) ≠ (a x b) x c

In terms of cartesian coordinates

(6.10)

and the other components are obtained by cyclically permuting the indices 1, 2, 3.

Two kinds of triple products occur. The first is

(6.11)

This is a scalar, and it gives the volume of the parallelepiped with a, b, c as edges. It is clearly unchanged by a cyclic permutation of a, b, c. It is positive if the vectors taken in this order form a right-handed triple, and negative if they form a left-handed one.

The second kind of triple product is

(6.12)

The result is