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(J. J. H.)

Dynamics, Analytical.—The fundamental principles of Dynamics, and their application to special problems, are explained under Mechanics (Ency. Brit. vol. xv.), 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 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. This article is devoted to an outline of such portions of general dynamical theory as seem to be most important from this latter point of view.

§ 1. 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 $$q_1,q_2,\dots q_n$$ In, 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

where the functions $$f$$ differ (of course) from particle to particle. In modern language, the variables $$q_1, q_2,\dots q_n$$ In are generalized co- ordinates serving to specify the configuration of the system; their derivatives with respect to the time are denoted by $$\dot{q}_1, \dot{q}_2,\dots \dot{q}_n,$$ 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. 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 $$m$$,

where $$\mathrm{X}', \mathrm{Y}', \mathrm{Z}'$$ are the components of the impulse on $$m$$. Now let $$\delta x, \delta y, \delta z$$ be any infinitesimal variations of $$x, y, z$$ which are consistent with the connexions of the system, and let us form the equation

where the sign $$\Sigma$$ indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations oq, qg,... of the generalized Co-ordinates, we have 2=1+2+..., &c., &c. (4) ax axc 8x= -og₁+ 692+ &c., &c., (5) 812 and therefore Em(idx+ydy+62)=(An91+A1242+...)691 +(A1+A2+...)2+(6) where Ar=2 Ση 2 + 2 อน + dar dgr dx dx dy dy, dz dz Ara=Σm { car + or 64 + or - =Ar⋅ (7) If we form the expression for the kinetic energy T of the system, we find 2T=2m(+y+2)=A1+A2292²+...+2A12192+... (8) The coefficients A, A, ... A12 are by an obvious analogy called the coefficients of inertia of the system; they are in general func- tions of the co-ordinates 1, 2,.... The equation (6) may now be written эт эт Zm(idx+yoy+dz)=1+ 892+... 842 (9) This may be regarded as the cardinal formula in Lagrange's method. For the right-hand side of (3) we may write 2(X'8x+Y'dy+Z'82)=Q'101+Q'2092 + ..., (10) ⋅ where Q'=2(x" Y Ах dz Car (11) The quantities Q1 Q2: are called the generalized components of impulse. Comparing (9) and (10), we have, since the variations 81, 8q... are independent, ат эт =Q'v =Q'2, ... These are the general equations of impulsive motion. It is now usual to write ат = Progr (12) . . (13) The quantities P1, P... represent the effects of the several com- ponent impulses on the system, and are therefore called the generalized components of momentum. In terms of them we have Zm(idx+joy+282)=p1091+P2092+... . (14) Also, since T is a homogeneous quadratic function of the velocities 91, 92..., 2T P+Pal2+.... (15) This follows independently from (14), assuming the special varia- tions dx=dt, &c., and therefore dq=dt, q=dt,.... Again, if the values of the velocities and the momenta in any other motion of the system through the same con- Reciprocal figuration be distinguished by accents, we have the Theorems. identity ני: P+P 2+...=P'₁₁+P'A₂+.... each side being equal to the symmetrical expression . . (16) A1+A2222+...+A12(414'2+'12)+..... (17) The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta Pi, Pa... all vanish with the exception of P1, and similarly that the momenta p', P',... all vanish except p'. We have then po' p', or move. 92: P1=9'1: P'2 • (18) The interpretation is simplest when the co-ordinates 91, q2 are both of the same kind, e.g., both lines, or both angles. We may then conveniently put p₁=p', and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. As an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to A blow at right angles to the chain, at any point P, will produce a certain velocity at any other point Q; the theorem asserts that an equal velocity will be produced at P by an equal blow at Q. Again, an impulsive couple acting on any link A will produce a certain angular velocity in any other link B; an equal couple applied to B will produce an equal angular velocity in A. Also if an impulse F applied at P produce an angular velocity w in a link A, a couple Fa applied to A will produce a linear velocity wa at P. Historically, we may note that reciprocal relations in