DYNAMICS (from Gr. δύναμις, strength), the name of a branch of the science of Mechanics (q.v.). The term was at one time restricted to the treatment of motion as affected by force, being thus opposed to Statics, which investigated equilibrium or conditions of rest. In more recent times the word has been applied comprehensively to the action of force on bodies either at rest or in motion, thus including “dynamics” (now termed kinetics) in the restricted sense and “statics.”
Analytical Dynamics.—The fundamental principles of dynamics, and their application to special problems, are explained in the articles [Mechanics] and [Motion, Laws of], where brief indications are also given of the more general methods of investigating the properties of a dynamical system, independently of the accidents of its particular constitution, which were inaugurated by J.L. Lagrange. These methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. In determining the properties of such systems the methods of analytical geometry and of the infinitesimal calculus (or, more generally, of mathematical analysis) are necessarily employed; for this reason the subject has been named Analytical Dynamics. The following article is devoted to an outline of such portions of general dynamical theory as seem to be most important from the physical point of view.
1. General Equations of Impulsive Motion.
The systems contemplated by Lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables q1, q2, ... qn, each of which admits of continuous variation over a certain range, so that if x, y, z be the Cartesian co-ordinates of any one particle, we have for example
x = ƒ(q1, q2, ... qn), y = &c., z = &c.,
(1)
where the functions ƒ differ (of course) from particle to particle. In modern language, the variables q1, q2, ... qn are generalized co-ordinates serving to specify the configuration of the system; their derivatives with respect to the time are denoted by q˙1, q˙2, ... q˙n, and are called the generalized components of velocity. The continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called the path.
For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration of Impulsive motion. impulsive motion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular co-ordinates of any particle m,
mẋ = X′, mẏ = Y′, mz˙ = Z′,