Constraint expressed analytically.

the component angular velocity round an axis through the origin and in the direction whose, direction-cosines are proportional to A', B', C',

a = √ 12 + B22 + C22

A+B+C

It might be supposed that by altering the origin of co-ordinates we could do away with the angular velocities, and leave only a linear equation among the components of velocity of translation. It is not so; for let the origin be shifted to a point whose coordinates are §, 7, . The angular velocities about the new axes, parallel to the old, will be unchanged; but the linear velocities which, in composition with these angular velocities about the new axes, give w1, w2, w3, u, v, w, with reference to the old, are ($ 89)

η,

Generalized co-ordinates.

Of a point.

Hence the equation of constraint becomes

Au'+Bv'+Cw'+(A'+B§— Cn)w1+etc.=0.

Now we cannot generally determine έ, n, so as to make 1, etc., disappear, because this would require three conditions, whereas their co-efficients, as functions of §, 7, , are not independent, since there exists the relation

A(B§— Cn)+B(C§ — A¿)+C(Aŋ — B§)=0.

The simplest form we can reduce to is

lu+mv'+nw'+a(lw,+mw,+nw1)=0,

that is to say, every longitudinal motion of a certain axis must be accompanied by a definite proportion of rotation about it.

202. These principles constitute in reality part of the general theory of "co-ordinates" in geometry. The three co-ordinates of either of the ordinary systems, rectangular or polar, required to specify the position of a point, correspond to the three degrees of freedom enjoyed by an unconstrained point. The most general system of co-ordinates of a point consists of three sets of surfaces, on one of each of which it lies. When one of these surfaces only is given, the point may be anywhere on it, or, in the language we have been using above, it enjoys two degrees of freedom. If a second and a third surface, on each of which also it must lie, it has, as we have seen, no freedom left: in other words, its position is completely

co-ordinates.

specified, being the point in which the three surfaces meet. Generalized The analytical ambiguities, and their interpretation, in cases in which the specifying surfaces meet in more than one point, need not occupy us here.

To express this analytically, let y=a, p=3, 0=y where of a point. ,, are functions of the position of the point, and a, B, y constants, be the equations of the three sets of surfaces, different values of each constant giving the different surfaces of the corresponding set. Any one value, for instance, of a, will determine one surface of the first set, and so for the others and three particular values of the three constants, specify a particular point, P, being the intersection of the three surfaces which they determine. Thus a, B, y are the "co-ordinates" of P; which may be referred to as "the point (a, B, y)." The form of the co-ordinate surfaces of the (4, 4, 0) system is defined in terms of co-ordinates (x, y, z) on any other system, plane rectangular co-ordinates for instance, if y, p, 0 are given each as a function of (x, y, z).

differential

203. Component velocities of a moving point, parallel to the Origin of the three axes of co-ordinates of the ordinary rectangular system, are, calculus. as we have seen, the rates of augmentation of the corresponding co-ordinates. These, according to the Newtonian fluxional notation, are written x, y, ż; or, according to Leibnitz's notation, dx dy dz

which we have used above,

Lagrange has com

bined the two notations with admirable skill and taste in his Mécanique Analytique, as we shall see in Chap. II. In specifying the motion of a point according to the generalized system of co-ordinates, 4, 6, 8 must be considered as varying with the dy do do

time:,, 0, or

components of velocity

d3 d2 d20

dr2 dtdt

tion.

will then be the generalized

will be the generalized components of accelera

of any

204. On precisely the same principles we may arrange sets of Co-ordinates co-ordinates for specifying the position and motion of a material system. system consisting of any finite number of rigid bodies, or

of any system.

components

Co-ordinates material points, connected together in any way. Thus if ↓, 6, 8, etc., denote any number of elements, independently variable, which, when all given, fully specify its position and configuration, being of course equal in number to the degrees of freedom to move enjoyed by the system, these elements are its co-ordinates. When it is actually moving, their rates of variation per Generalized unit of time, or 4, 4, etc., express what we shall call its generof velocity. alized component velocities; and the rates at which 4, 4, etc., augment per unit of time, or Ÿ, $, etc., its component accelerations. Thus, for example, if the system consists of a single Examples. rigid body quite free, 4, 4, etc., in number six, may be three common co-ordinates of one point of the body, and three angular co-ordinates (§ 100, above) fixing its position relatively to axes in a given direction through this point. Then &, &, etc., will be the three components of the velocity of this point, and the velocities of the three angular motions explained in § 100, as corresponding to variations in the angular co-ordinates. Or, again, the system may consist of one rigid body supported on a fixed axis; a second, on an axis fixed relatively to the first; a third, on an axis fixed relatively to the second, and so on. There will be in this case only as many co-ordinates as there are of rigid bodies. These co-ordinates might be, for instance, the angle between a plane of the first body and a fixed plane, through the first axis; the angle between planes through the second axis, fixed relatively to the first and second bodies, and so on; and the component velocities, , p, etc. would then be the angular velocity of the first body relatively to directions fixed in space; the angular velocity of the second body relatively to the first; of the third relatively to the second, and so on. Or if the system be a set, i in number, of material points perfectly free, one of its 3i co-ordinates may be the sum of the squares of their distances from a certain point, either fixed or moving in any way relatively to the system, and the remaining 32-1 may be angles, or may be mere ratios of distances between individual points of the system. But it is needless to multiply examples here. We shall have illustra tions enough of the principle of generalized co-ordinates, by actual use of it in Chap. II., and other parts of this book.

APPENDIX TO CHAPTER I.

A. EXTENSION OF GREEN'S THEOREM.

IT is convenient that we should here give the demonstration of a few theorems of pure analysis, of which we shall have many and most important applications, not only in the subject of spherical harmonics, which follows immediately, but in the general theories of attraction, of fluid motion, and of the conduction of heat, and in the most practical investigations regarding electricity, and magnetic and electro-magnetic force.

(a) Let U and U denote two functions of three independent variables, x, y, z, which we may conveniently regard as rectangular co-ordinates of a point P, and let a denote a quantity which may be either constant, or any arbitrary function of the variables. Let /dxdydz denote integration throughout a limited space bounded by a closed surface S; let //dS denote integration over the whole surface S; and let d, prefixed to any function, denote its rate of variation at any point of S, per unit of length in the direction perpendicular to S outwards.

For, taking one term of the first member alone, and integrating

by parts," we have

dx dx

dU
dx

dx dx

dxdydz,

dxdyd:= U'u2 dydz-U

the first integral being between limits corresponding to the sur-
face S; that is to say, being from the negative to the positive
end of the portion within S, or of each portion within S, of the
line through the point (0, y, z). Now if A, and A, denote the
inclination of the outward normal of the surface to this line, at
points where it enters and emerges from S respectively, and if

a constant

gives a

theorem of Green's.

Equation of the conduction of heat.

ds, and dS, denote the elements of the surface in which it is cut at these points by the rectangular prism standing on dydz, we have

dydz=-cos AdS1=cos A1dS1.

Thus the first integral, between the proper limits, involves the elements U ́acos AdS1, and— Uad cos AdS; the latter

dU

dx

dx

of which, as corresponding to the lower limit, is subtracted. Hence, there being in the whole of S an element dS, for each element dS1, the first integral is simply

cos A dS, dx

for the whole surface. Adding the corresponding terms for y and z, and remarking that

where B and C denote the inclinations of the outward normal through ds to lines drawn through dS in the positive directions parallel to y and z respectively, we perceive the truth of (1).

(b) Again, let U and U' denote two functions of x, y, z, which have equal values at every point of S, and of which the first fulfils the equation

2

+//S{ (a du )*+ (a du)*+ (a du ) * } dædydz

dx

dz

(3)

For the first number is equal identically to the second member with the addition of

of which each term vanishes; the first, or the double integral, be

cause, by hypothesis, u is equal to nothing at every point of S, and the second, or the triple integral, because of (2).