DYNAMICS, a branch of mechanics. The term is usually applied (as here) to the study of the motion of matter, but sometimes it includes statics, which is concerned with matter at rest.
The Fundamental Equation.—If we were to consider a material system isolated in space it would theoretically be possible (though perhaps not practicable) to describe the motion merely as related to the configuration. We more frequently consider, however, some arbitrary and not necessarily isolated portion of matter--a "dynamical system." The motion of such a system depends not merely on its configuration but also upon matter external to the system. The effect of this external matter is most simply comprehended in the notion of force. And although we could perhaps study the behaviour of a self contained system without the aid of this notion, yet when once it has been introduced there is no good reason for restricting its use.
The measurement of mass and of force, Newton's "Laws of Motion," the idea of work, and the theory of the simplest sys tems—such as a single particle or rigid body under given forces, or two particles mutually attracting—are dealt with in the article MECHANICS. One particular case of two particles subject only to their mutual attraction is of special interest and importance, namely, when the attraction varies inversely as the square of the distance between the particles. This is the famous "problem of two bodies." The methods developed in the study of this particular problem have had a profound effect on the history of the whole subject.
In the present article we consider the general theory of systems of particles and rigid bodies. A rigid body is conceived of, for our present purpose, as an aggregate of particles, finite in num ber, set in a rigid imponderable frame. (It is hardly necessary to remind the reader that this conception, adequate for the purpose of dynamics, is totally inadequate in other connections.) Thus the whole system consists of a number (perhaps large, but essentially finite) of particles, each subject to forces, including the reactions of other particles of the system; and these reactions must so adjust themselves that the necessary geometrical rela tions imposed by the constitution of the system are fulfilled. It is easy to see of what form these relations are. For if we denote by i,2, • • • , E3N the 3N co-ordinates of the N particles of the system referred to fixed rectangular axes, then we shall have m relations of the type where m < 3N. The coefficients a, depend upon the configura tion and the time. Thus for a single particle constrained to move without friction on a variable surface f (x, y, z, t) = o we have ax ay az a t In the systems usually considered we have often an important simplification, namely, that in the equations (I) there are no terms in dt, and the coefficients a„ are independent of t.
The motion of a typical particle of the system is controlled where m denotes the mass of the particle, x, y, z, its co-ordinates with respect to fixed rectangular axes, and Y+Y', Z+Z' the components parallel to these axes of the resultant force acting on the particle. The notation implies the division of the forces into two groups, and the division is to be effected thus: The "internal" forces X', Y', Z' are such that altogether they do no work in any infinitesimal displacement of the system com patible with the constraints as they exist at the instant considered. We will call such displacements "admissible." It is important to observe that if the constraints are variable the admissible displacements are those appropriate to the system as constituted at the given instant. Thus in the case of the particle constrained to move on the variable surface f(x, y, z, t) = o the admissible displacements are in the tangent plane to the surface as it is at the given instant. Explicitly the increments bx, Sy, bz must satisfy the single condition And generally the admissible displacements are subject to the equations obtained from (I) by writing for dE, and omitting the terms in dt.
where S denotes here, and throughout the present article, a sum mation over the particles of the system, and the equation holds for all sets of values of Sx, by, bz corresponding to admissible displacements. The forces which fall into this category are fa miliar from the theory of virtual work in statics. The most important are (i.) the mutual reactions between pairs of particles whose distance apart is invariable, (ii.) the reactions between perfectly smooth surfaces, and (iii.) the reactions between per fectly rough surfaces; in the last case the admissible displace ments include only a relative motion of pure rolling. From (2) and (3) we derive at once This is the fundamental equation of dynamics. It expresses the fact that the work of the "kineta" (mx, my, int) for any ad missible displacement is equal to the work of the "external" forces, X, Y, Z. It is the immediate analogue for a dynamical system of d'Alembert's principle for a single rigid body.
First Deductions from the Fundamental Equation.- It may happen that among the admissible displacements is included the displacement that the system actually suffers in a small time succeeding the instant considered. This will be the case, for example, if a particle is constrained to move on a fixed smooth surface, but not if the particle moves on a variable surface as in the example considered above. Suppose that this does hold, and consequently that we may write x, i for ox, Sy, bz in the equation (4). We have then where T = the "kinetic energy." The rate of working of the external forces is equal to the rate of increase of the kinetic energy. This is the famous "equation of energy." Again, it may happen that among the admissible displacements is included a displacement of the whole system without distor tion as if rigid, in the direction of x. Then we may write Sx = constant, Sy, Sz = o in the equation (4), whence SX = Smx.
This is the equation of the "conservation of linear momentum." And in the same way, if a rotation without distortion of the system about the axis of x is admissible, and if further the "conservation of angular momentum." Transformation of the Fundamental Equation.—The equation (4) is the starting point for a number of lines of de velopment. Thus (a) for systems of a suitably restricted type we can derive from it a new variational equation (see equation [8] below), the variables whose differentials appear being new generalized co-ordinates and their derivatives. The compact and elegant form of such variational equations—equivalent, of course, to a set of differential equations—is of peculiar value in dynamics. Or (b) we can exhibit the equation as the condition that some function is to be stationary under given conditions. In Gauss's principle we have to minimize an algebraic function of the accelerations (the co-ordinates and velocities, which appear in the function, being treated as mere constants while the equa tions are being formed). In Hamilton's principle and in the "Principle of Least Action" of Maupertuis it is an integral that has to be minimized by the usual methods of the calculus of variations. Or (c) we can transform to differential equations— equations associated with the names of Appell, Lagrange, Hamilton, Routh. The various lines of development are of course intimately connected. Thus the minimizing equations for Gauss' function are Appell's equations, and the minimizing equations obtained from Hamilton's principle are Lagrange's equations.
The methods just enumerated are not applicable over identical ranges. Thus Appell's equations of motion have the advantage that they may be applied at once to systems containing rolling surfaces, e.g., a rough sphere, to which given forces are applied, rolling on a fixed or moving surface. Lagrange's equations may not be applied directly to such problems—though it is easy to modify them (by the introduction of multipliers associated with the reaction at the point of rolling contact) to include problems of this type. Again, Hamilton's principle has a wide range of applicability. It may be applied, by a natural extension, even to continuous systems, such as a vibrating string or membrane; though in practice the differential equations can usually be ob tained more simply in other ways—either by considering the motion of a small element treated as a particle, or directly from a suitably modified form of the fundamental equation.
In a short article it is of course impossible to consider all these methods in detail. We shall therefore restrict our further enquiry to systems of a particular type—the type with which we are in fact most often concerned—leaving the reader to consult the books mentioned at the end of this article for information as to systems of more general types. To begin with, we suppose that the co-ordinates x, y, z, of every particle of the system can be expressed as functions, not involving t, of n generalized co ordinates ql, q2, • • - , qn, capable of continuous variation in a certain domain; and that the admissible displacements of the system are represented by arbitrary infinitesimal increments bq,, bq2, . • • , This is a very sweeping simplification. The class of admissible displacements is identical with the class of possible displacements (there are no dt terms in the equations [i]), and these displacements are represented in a particularly simple way—by arbitrary increments in the q's. The restriction excludes the possibility of a motion inexorably imposed on the system from without. It also excludes the possibility of rolling surfaces; for although in this case all possible displacements may be admissible, it is not possible, in general, to represent the dis placements by arbitrary increments of a set of parameters. Thus, as a simple concrete example, for a rough sphere rolling on a plane we can express the co-ordinates of any particle as functions of five parameters; but the increments of these para meters representing the displacements are subject to two non integrable relations.
We suppose further that the external forces are "conservative," i.e., that the sum S(Xbx+Yby-}-Zbz) is the perfect differential of a function, — V, of the q's. The outstanding example of con servative forces is the mutual attraction F between a pair of particles, when F depends only on the distance r: for these forces With these limitations the equation (4) takes a very simple form. Consider separately the term a:bx appropriate to one particular particle. We have where repeated suffixes imply summation (this convention is to be understood throughout), and the three similar terms for each particle can be transformed in the same way. The kinetic energy T is now a quadratic form in the q's with coefficients depending on the q's. We write This is the form of the equations of motion given by Joseph Louis Lagrange, the author of the great Mecanique Analytique (1788). It is usually the most convenient form to apply to con crete problems.
As a simple example, consider a particle moving in a plane under given conservative forces, the position of the particle at any time being given by polar co-ordinates (r, 0). Here In particular for a central attraction for which V is a function only of r we have from 04) and express H as a function of q's and p's, suppressing the q's in favour of the p's by means of the equations (7). Then (q) may be written 1• These are the equations of motion given by Sir William Rowan Hamilton in 1834. We have here 2n equations of the first order as contrasted with the n equations of the second order of La grange. Hamilton's equations, though usually less convenient than Lagrange's for concrete problems, are of great importance in theoretical work.
In connection with Lagrange's equations we can picture the motion of the system as the motion of a representative point (qi, q2, • • • , in an n-dimensional manifold, the q-space. The co ordinates and velocities of the representative point at a given instant suffice to determine the motion. In the same way, in connection with Hamilton's equations, we can consider the mo tion of a representative point Q2, • • • , pi, P2, • • • , pn) in a 2n-dimensional manifold, the (q, p) space. The co-ordinates alone of the representative point at a given instant suffice to determine the motion. The path of the representative point, either in the q-space or the (q, p) space, is called an "orbit" of the system.
In the q-space, but not in the (q, p) space, any path is geo metrically possible. When, as in Hamilton's principle, we con sider variations to neighbouring paths which are not necessarily natural orbits of the system, we think in terms of the q-space. But when we consider variations to neighbouring natural orbits, i.e., orbits satisfying the equations of motion, we may use either representation.
It is possible to produce a function which plays the part of a Lagrangian function for some freedoms, and of a Hamiltonian function for the rest. For the equation (9) may be written Thus the equations corresponding to the first m co-ordinates are of the form (i7), and the rest of the form (io). The method was devised by Edward John Routh (1876) for the special case in which qi, q2, • • • , qm do not appear in L, nor, consequently, in R; so that pi, • • • , pm remain constant during the motion. Such co-ordinates appear conspicuously in the study of gyro scopes, and Routh's method is commonly applied to such problems.
As we have already hinted above the principle holds under less restricted conditions than those imposed in the proof just given. But in some cases (e.g., that of the rolling sphere) there is a new feature which must be carefully borne in mind. The variation to the contemporaneous point on the neighbouring orbit must always be an admissible variation; but the neigh bouring orbit itself may be, not only not a natural orbit, but not even an orbit which is geometrically possible. (In the case of the sphere rolling on a rough plane the motion in the varied orbit would, in general, involve sliding as well as rolling on the plane.) The Equation of Energy.—Consider now variations to con temporaneous points on a neighbouring orbit which is a natural orbit for the system. And suppose first that the neighbouring orbit is the same as the original orbit, but not quite synchronized with it; so that the system passes through the same sequence of configurations, but reaches any configuration a small time •r earlier than it did in the original orbit. Then This is the equation of energy already established in a different form and under less restricted conditions, above.
As an example of the use of this equation, consider again the problem of a particle moving in a plane under a central attraction derived from a potential function V = V(r). The equation of energy is Z -}- V = constant = E, say. (2I ) We have also the equation (r 5) of angular momentum From (2 I) and (22) we have at once say, where the positive value of the radical is to be taken so long as r increases with t. If r lies originally between two consecutive real positive zeros of p, then r lies always in the range From (23) we have This completes the solution; t = and 0= 00 when r = (25) is the equation of the orbit, and (24) gives the position of the par ticle in its orbit at any time.
If the attraction at distance r is we have V = — yMm/r. For E < o the orbit (25) is an ellipse having the centre of attrac tion in one focus.
A transformation from to where are functions of (qi, q2, • • • , qn, p1, P2, • • • , pn, t) is called a contact transformation if = (33) where R, W can be expressed at will in various forms—for example, as functions of the q's and p's and 1, or as functions of the a's and /3's and 1. In particular the equation (3 2) is of the form (33); the solution of a dynamical problem—the trans formation from to a contact transformation.
Suppose now we have a dynamical problem for which the variables are the equation (32)—and we transform to new variables by means of a contact trans formation satisfying (33). We have then = (34) so that satisfy new Hamiltonian equations, the new Hamiltonian function being (H+R). This important theorem is due in substance to Carl Gustav Jacob Jacobi (1837).
As a simple example of the use of this theorem consider again the problem of a particle moving in a plane under a central at traction derived from a potential function V = V(r). The equa tion (36) is These are the integrals of the equations, of motion, obtained previously by another method. The compactness of the present method is very striking.