MECHANICS 679 17. In general, in any system originally with any num ber m of degrees of freedom, and subjected to a number u of degrees of constraint, the whole motion can be fully charac terized by m - n independent quantities, called generalized coordinates, and corresponding to the degrees of freedom which remain. The elegance and simplicity of a solution often depend in a marked manner upon the choice of these; and the transformation of the general equations from Cartesian to generalized coordinates forms one of the most powerful and elegant contributions to abstract dynamics which Lagrange made in the Mccanique Analytique. eedom 18. A rigid system has only six degrees of freedom: a three translations for any one of its points, and three inde pendent rotations about axes passing through that point. When one point is fixed, it loses the three translations, and has only three degrees of freedom. When a second point is fixed, it loses other two ; in fact it can no longer move except by turning round the line joining the fixed points. When a third point, not in line with the other two, is fixed, there is no degree of freedom left ; the system is fixed. inear 19. It may be well to notice here that, in all cases which we lations shall require to consider, whatever be the relations among two uong different sets of variables which we employ alternatively to deter- nall mine the relative positions of the parts of any system, the equations splace- which give the relations between corresponding small increments eats, of these variables are always linear so far as these increments are concerned. Thus, for instance, if we have as above a condition of the form A*,y, =) = {, we deduce from it at once Here the differential coefficients are partial. In such cases, any homogeneous function of the second order in S.r, 8y, 5z, &c., will be represented by a homogeneous function, also of the second order, in 8, 817, &c., however many be the coordinates in the separate systems. When, however, one or more of the equations of condition involves the clement of time explicitly, the relations among corresponding small increments of the alternative sets of coordinates, though still linear, will not be homogeneous. Thus a homogeneous function of the second order in one set will be a function of the second order in the other set, but not homo geneous, unless the increments are produced instantaneously. To give a single instance, suppose that the string of a simple pendulum (not necessarily oscillating in one plane) contracts uni formly. We shall now have x* + ?/ + z* = (a - ct) instead of the equation in the example 15, and one of our equa tions among increments is zti.v + ySy + zSz = - c(a- ct)Sf, , which, though still linear, is no longer homogeneous in the incre ments of coordinates. Kinematics of a Point. ontinu- 20. The one necessary characteristic of the path de- y of scribed by a moving point is its continuity. There can be
- j ut no break or gap in it. But, as we study kinematics, at
present, solely for its physical applications, we impose a restriction on such complete generality. The path of a moving particle must be one of continuous curvature, unless either (1) the motion ceases and commences again in a different direction (in which case we have two separate and successive states of motion to consider), or (2) an infinite force is applied to the particle (a case which we need not consider). A similar remark, we may say in passing, applies to velocity also. So that, for our purpose, we may confine ourselves to the geometrical properties of the motion of a point whose rate and direction of motion change con tinuously, if at all, and not by fits and starts. >irec- 21. If the point describe a straight line, that line gives ion of the direction of its motion at every instant. If it describe a curvej th e direction of its motion is at every instant that of the corresponding tangent to the curve. lotion. Let A, B, C, D represent four points on the path taken Change in close succession, in the order in which the moving point of direc- reaches them. From A the point moves to B, so that the tlon< line joining A and B (the tangent) is the direction of motion at A. Similarly the line joining B and C gives the direction of motion at B. The points A, B, C of course lie in one plane. This is the plane in which, for two succes sive elements of its path, the point is moving. It is there fore that in which the change of direction of motion takes place, and is called the " osculating plane." And, just as the Osculat- straight line through A and B gives the direction of motion ing at A, so the circle passing through the points A, B, C P lane - determines the " curvature " of the path at A. If we apply the same reasoning to the three successive points B, C, D, we see the difference between a " plane " and a " tortuous " curve. For, if D lie in the plane ABC, the osculating plane is the same at A and at B ; and if the same holds for other successive points the whole bending takes place in one plane. But if D be not in the plane ABC, BCD is the osculating plane at B, and we thus see that successive positions of the osculating plane of a tortuous curve are Tortuous produced by its rotation about the tangent BC to the path ; P atu - for BC is in both planes ABC and BCD. We shall not have space here to deal in detail with cases of tortuosity ; but it was necessary to point out their essential nature. 22. The curvature of ABC obviously depends upon the Curva- change of direction from AB to BC, and is directly pro- ture - porttonal to it. But it is obviously greater, for the same amount of change of direction, as ABC is less. In a circle the curvature is the same at all points, and, as the radius is everywhere perpendicular to the tangent, the change of its direction is the same as that of the tangent. Hence the curvature, being the change of direction per unit length of the arc, is measured simply by the reciprocal of the radius. Generally, if < be the angle between the tangent at A and any fixed line in the osculating plane, and if s represent the length of the curve measured from any fixed point on it to A, we have, by the fundamental property of infinitesimals, L = -r- = curvature. Ss ds (We will use, as above, the letter L for a limit, in the sense in which that term was introduced by Newton.) In a circle we have always (a being the radius) = <, and hence the curvature so that in general the measure of curvature is the reciprocal of the radius of the circle passing through three consecutive points of the path. For other analytical expressions for curvature see vol. xiii. p. 26. For a curve in space (whether tortuous or not) we have Cui vature = : while the direction cosines of the radius of curvature are 23. The chief properties connected with the curvature Evolute of a plane curve are made very clear by the artifice of *" regarding it as an " involute." This idea introduces us to the kinematics of a flexible and inextensible line. Suppose such a line, held tight, to be wrapped round a cylinder of any form, in a plane perpendicular to its axis, each point of it, when it is unwound in its own plane, will describe a curve whose form depends upon that of the transverse section of the cylinder. Let P M M (fig. 1) be such a section of the cylinder; MP, M P , two positions of the
free part of the cord; P, P , the corresponding positions of