is in one plane, but when in space it is ds2 = dr3 + dy2 + dz2 : [the element of the space divided by the element of the c being an arbitrary constant quantity introduced by integration. 77. This equation will give the velocity of the particle in any point of its path, provided its velocity in any other point be known : for if A be its velocity in that point of its trajectory whose co-ordinates are a, b, c, then and A2= c + 2f (a, b, c), v2 — A2 = 2ƒ (x, y, z) — 2f (a, b, c) ; whence v will be found when A is given, and the co-ordinates a, b, c, x, y, z, are known. It is evident, from the equation being independent of any particular curve, that if the particle begins to move from any given point with a given velocity, it will arrive at another given point with the same velocity, whatever the curve may be that it has described. 78. When the particle is not acted on by any forces, then X, Y, and Z are zero, and the equation becomes v2 = c. The velocity in this case, being occasioned by a primitive impulse, will be constant; and the particle, in moving from one given point to another, will always take the shortest path that can be traced between these points, which is a particular case of a more general law, called the principle of Least Action. B fig. 20. be obtained. Principle of Least Action. 79. Suppose a particle beginning to move from a given point A, fig. 20, to arrive at another given point B, and that its velocity at the point A is given in magnitude but not in direction. Suppose also that it is urged by accelerating forces X, Y, Z, such, that the finite value of Xdx + Ydy + Zdz can We may then determine v the velocity of the particle in terms of x, y, z, without knowing the curve described by the particle in moving from A to B. If ds be the element of the curve, the finite value of vds between A and B will depend on the nature of the path or curve in which the body moves. The principle of Least Action consists in this, that if the particle be free to move in every direction between these two points, except in so far as it obeys the action of the forces X, Y, Z, it will in virtue of this action, choose the path in which the integral fvds is a minimum; and if it be constrained to move on a given surface, it will still move in the curve in which fvds is a minimum among all those that can be traced on the surface between the given points. To demonstrate this principle, it is required to prove the variation of fvds to be zero, when A and B, the extreme points of the curve are fixed. By the method of variations fedsfd.vds: for the mark of integration being relative to the differentials, is independent of the variations. Now hence and therefore ds 8.vds dv. ds + vôds, but v = or ds = vdt; dt dv. ds = vdvdt = dtd. v2, d.vds= dt. d. v2 + v.§.ds. The values of the two last terms of this equation must be found separately. To find dt. 8.v2. It has been shown that v = c + 2/ (Xda + Ydy+Zdz), its differential is vdv (Xdx + Ydy + Zdz), and changing the differentials into variations, 8.2 If .v be substituted in the general equation of the motion of a particle on its surface, it becomes But du does not enter into this equation when the particle is free; and when it must move on the surface whose equation is u = 0, Su is also zero; hence in every case the term Adu vanishes; there A value of the first term required. second term v.8.ds must now be found. Since ds2dx dy2 + dz2, which is the value of the second term; and if the two be added, their sum is 8.vds d Sx + dy dy + dz 8 z }, dt dt dt as may easily be seen by taking the differential of the last member of this equation. Its integral is If the given points A and B be moveable in space, the last member of this equation will determine their motion; but if they be fixed points, the last member which is the variation of the co-ordinates of these points is zero: hence also fvds = 0, which indicates either a maximum or minimum, but it is evident from the nature of the problem that it can only be a minimum. If the particle be not urged by accelerating forces, the velocity is constant, and the integral is vs. Then the curve s described by the particle between the points A and B is a minimum; and since the velocity is uniform, the particle will describe that curve in a shorter time than it would have done any other curve that could be drawn between these two points. 80. The principle of least action was first discovered by Euler: it has been very elegantly applied to the reflection and refraction of light. If a ray of light IS, fig. 21, falls on any surface CD, it fig. 21. B S -D will be turned back or reflected in the direction Sr, so that ISA=rSA. But if the medium whose surface is CD be diaphanous, as glass or water, it will be broken or refracted at S, and will enter the denser medium in the direction SR, so that the sine of the angle of incidence ISA will be to the sine of the angle of refraction RSB, in a constant ratio for any one medium. Ptolemy discovered that light, when reflected from any surface, passed from one given point to another by the shortest path, and in the shortest time possible, its velocity being uniform. Fermat extended the same principle to the refraction of light; and supposing the velocity of a ray of light to be less in the denser medium, he found that the ratio of the sine of the angle of incidence to that of the angle of refraction, is constant and greater than unity. Newton however proved by the attraction of the denser medium on the ray of light, that in the corpuscular hypothesis its velocity is greater in that medium than in the rarer, which induced Maupertuis to apply the theory of maxima and minima to this problem. If IS, a ray of light moving in a rare medium, fall obliquely on CD the surface of a medium that is more dense, it moves uniformly from I to S; but at the point S both its direction and velocity are changed, so that at the instant of its passage from one to the other, it describes an indefinitely small curve, which may be omitted without sensible error: hence the whole trajectory of the light is ISR; but IS and SR are described with different velocities; and if these velocities be v and v', then the variation of IS xv+ SR × v must be zero, in order that the trajectory may be a minimum: hence the general expression dvds 0 becomes in this case 8. (IS × v + SR × v') = 0, when applied to the refraction of light; from whence it is easily found, by the ordinary analysis of maxima and minima, that v sin ISA = v sin RBS. As the ratio of these sines depends on the ratio of the velocities, it is constant for the transition out of any one medium into another, but varies with the media, on account of the velocity of light being different in different media. If the denser medium be a crystallized diaphanous substance, the velocity of light in it will depend on the direction of the luminous ray; it is constant for any one ray, but variable from one ray to another. Double refraction, as in Iceland spar and in crystallized bodies, arises from the different velocities of the rays; in these substances two images are seen instead of one. Huygens first gave a distinct account of this phenomenon, which has since been investigated by others. Motion of a Particle on a curved Surface. 81. The motion of a particle, when constrained to move on a curve or surface, is easily determined from equation (7); for if the variations be changed into differentials, and if X', Y', Z' be eliminated by their values in the end of article 69, that equation becomes dx. d'x+dy. d'y + dz.d'z dt =Xdr+Ydy+Zd: + R, {dr. cos a + dy. cos ß + dz. cos y}, R, being the reaction in the normal, and a, ß, y the angles made by the normal with the co-ordinates. But the equation of the surface being u = 0, Adu dr. cos a + dy. cos B + dz. cos y = 0; so that the pressure vanishes from the preceding equation; and when the forces are functions of the distance, the integral is and 2f (x, y, z) + c = v2, A2 — v2 = 2f (x, y, z) – 2f (a, b, c), as before. Hence, if the particle be urged by accelerating forces, the velocity is independent of the curve or surface on which the particle moves; and if it be not urged by accelerating forces, the velocity is constant. Thus the principle of Least Action not only holds with regard to the curves which a particle describes in space, but also for those it traces when constrained to move on a surface. 82. It is easy to see that the velocity must be constant, because a particle moving on a curve or surface only loses an indefinitely small part of its velocity of the second order in passing from one indefinitely small plane of a surface or side of a curve to the consecutive; for if the particle be moving on ab with b В a the velocity; then if the angle abe B, the velocity in be will be v cos B; but cos 61-62- &c.; therefore the velocity on be differs from the velocity on ab by the indefinitely small quantity v. 6. In order to determine the pressure of the particle on the surface, the analytical expression of the radius of curvature must be found. |