Work done by impact. Equations of impulsive motion. 309. It is worthy of remark, that if any number of impacts be applied to a body, their whole effect will be the same whether they be applied together or successively (provided that the whole time occupied by them be infinitely short), although the work done by each particular impact is in general different according to the order in which the several impacts are applied. The whole amount of work is the sum of the products obtained by multiplying each impact by half the sum of the components of the initial and final velocities of the point to which it is applied. 310. The effect of any stated impulses, applied to a rigid body, or to a system of material points or rigid bodies connected in any way, is to be found most readily by the aid of D'Alembert's principle; according to which the given impulses, and the impulsive reaction against the generation of motion, measured in amount by the momenta generated, are in equilibrium; and are therefore to be dealt with mathematically by applying to them the equations of equilibrium of the system. 1 Let P, Q, R, be the component impulses on the first particle, m1, and let x, y, z, be the components of the velocity instantaneously acquired by this particle. Component forces equal to (P1-m,x,), (Q1-m ̧ÿ,), must equilibrate the system, and therefore we have (§ 290) ... Σ {(P − mx) dx + (Q − mý) dy + (R − m2) dz} = 0..........................(a) where dx,, dy,, ... denote the components of any infinitely small displacements of the particles possible under the conditions of the system. Or, which amounts to the same thing, since any possible infinitely small displacements are simply proportional to any possible velocities in the same directions, Σ {(P− mx) u + (Q – mý) v + (Q − m2) w} = 0.................................. (b) where u1, v1, w, denote any possible component velocities of the Equations One particular case of this equation is of course had by suppos. This Σ (P. 1 x + Q • { ý + R . { ż) = { Σm (x2 + y2 + ż3).................(c). agrees with § 308 above. of impulsive motion. Euler, ex- impulsive 311. Euler discovered that the kinetic energy acquired from Theorem of rest by a rigid body in virtue of an impulse fulfils a maximum-tended by minimum condition. Lagrange extended this proposition to a system of bodies connected by any invariable kinematic re- Equation of lations, and struck with any impulses. Delaunay found that motion. it is really always a maximum when the impulses are given, and when different motions possible under the conditions of the system, and fulfilling the law of energy [§ 310 (c)], are considered. Farther, Bertrand shows that the energy actually acquired is not merely a "maximum," but exceeds the energy of any other motion fulfilling these conditions; and that the amount of the excess is equal to the energy of the motion which must be compounded with either to produce the other. Let x, y,... be the component velocities of any motion whatever fulfilling the equation (c), which becomes } Σ (Px' + Qÿ' + R¿′ ) = } Σm (x22 + y22 + ž2) = T′′.................(d). If, then, we take ¿‚'—¿‚ = u ̧, ÿ, −y, = etc., we have T'− T = }Σm {(2x + u) u + (2ý + v)v + (2ż + w)w} But, by (b), = · Σm (xu + ýv + żw) + } Σm (u3 + v3 +w3)................(e). .......... Σm (xu + ÿv + żw) = Σ (Pu + Qv + Rw)............................(ƒ); and, by (c) and (d), Liquid set in motion 312. The energy of the motion generated suddenly in a impulsively. mass of incompressible liquid given at rest completely filling a vessel of any shape, when the vessel is suddenly set in motion, or when it is suddenly bent out of shape in any way whatever, subject to the condition of not changing its volume, is less than the energy of any other motion it can have with the same motion of its bounding surface. The consideration of this theorem, which, so far as we know, was first published in the Cambridge and Dublin Mathematical Journal [Feb. 1849], has led us to a general minimum property regarding motion acquired by any system when any prescribed velocities are generated suddenly in any of its parts; announced in the Proceedings of the Royal Society of Edinburgh for April, 1863. It is, that provided impulsive forces are applied to the system only at places where the velocities to be produced are prescribed, the kinetic energy is less in the actual motion than in any other motion which the system can take, and which has the same values for the prescribed velocities. The excess of the energy of any possible motion above that of the actual motion is (as in Bertrand's theorem) equal to the energy of the motion which must be compounded with either to produce the other. The proof is easy:-here it is : Equations (d), (e), and (ƒ) hold as in § (311). But now each velocity component, u,, v1, w1, u,, etc. vanishes for which the component impulse P1, Q1, R1, P2, etc. does not vanish (because ¿1+u1, ÿ, + v1, etc. fulfil the prescribed velocity conditions). Hence every product Pu,, Q, etc. vanishes. Hence now instead of (g) and (h) we have and Σ (xu + ÿv+ żw) = 0......... ..(g'), ..(h'). Impulsive motion referred to generalized co-ordinates. We return to the subject in §§ 316, 317 as an illustration of the use of Lagrange's generalized co-ordinates; to the introduction of which into Dynamics we now proceed. 313. The method of generalized co-ordinates explained above (§ 204) is extremely useful in its application to the dynamics of a system; whether for expressing and working out the details of any particular case in which there is any motion re generalized nates. finite number of degrees of freedom, or for proving general Impulsive principles applicable even to cases, such as that of a liquid, as ferred to described in the preceding section, in which there may be an co-ordiinfinite number of degrees of freedom. It leads us to generalize the measure of inertia, and the resolution and composition of forces, impulses, and momenta, on dynamical principles corresponding with the kinematical principles explained in § 204, which gave us generalized component velocities: and, as we shall see later, the generalized equations of continuous motion are not only very convenient for the solution of problems, but most instructive as to the nature of relations, however complicated, between the motions of different parts of a system. In the meantime we shall consider the generalized expressions for the impulsive generation of motion. We have seen above (308) that the kinetic energy acquired by a system given at rest and struck with any given impulses, is equal to half the sum of the products of the component forces multiplied each into the corresponding component of the velocity acquired by its point of application, when the ordinary system of rectangular co-ordinates is used. Precisely the same statement holds on the generalized system, and if stated as the convention agreed upon, it suffices to define the generalized components of im- Generalized pulse, those of velocity having been fixed on kinematical of impulse principles (§ 204). Generalized components of momentum mentum. of any specified motion are, of course, equal to the generalized components of the impulse by which it could be generated from rest. ... (a) Let 4, 4, 0, be the generalized co-ordinates of a material system at any time; and let 4, 4, 0, ... be the corresponding generalized velocity-components, that is to say, the rates at which 4, 4, 0, ... increase per unit of time, at any instant, in the actual motion. If x, y, z, denote the common rectangular co-ordinates of one particle of the system, and 1, ý,, ż, its component velocities, we have components or mo Generalized expression for kinetic energy. Generalized components of force, Hence the kinetic energy, which is Σm (x2+y+*), in terms of rectangular co-ordinates, becomes a quadratic function of 4, 4, etc., when expressed in terms of generalized co-ordinates, so that if we denote it by T we have $$ T = } {(4, v) * + (, ) $* + ... +2 (, ) + ...}......(2), where (4, 4), (4, 4), (4, 6), etc., denote various functions of the co-ordinates, determinable according to the conditions of the system. The only condition essentially fulfilled by these coefficients is, that they must give a finite positive value to T for all values of the variables. (b) Again let (X, Y1, Z1), (X2, Y, Z), etc., denote component forces on the particles (x,, y, z,), (x, y, z), etc., respectively; and let (dx,, dy,, d,), etc., denote the components of any infinitely small motions possible without breaking the conditions of the system. The work done by those forces, upon the system when so displaced, will be To transform this into an expression in terms of generalized coordinates, we have and it becomes where dx dx, 84 + do аф dy бу = 84 + etc. dy, 84 + do dy etc. etc. dy dz +2 dy dy dy These quantities, V, P, etc., are clearly the generalized components of the force on the system. Let !, !, etc. denote component impulses, generalized on the same principle; that is to say, let pdt, etc., |