Equations a system of bodies connected by any invariable kinematic re- of impulsive Let x, y... be the component velocities of any motion whatever fulfilling the equation (c), which becomes }Σ(Px' + Qÿ' +Rż' )=T'={Σm(¿'2 +ÿ'2+ż’2) If, then, we take i1— x=u1, ÿ1—ÿí v1, etc., we have But, by (b), (d). =Σm(żu+ÿv+żw) — §Σm(u2+v2+w3) (e). motion impulsively. 312. The energy of the motion generated suddenly in a Liquid set in 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 the general minimum property proved below regarding motion acquired by any system when any given velocities are generated suddenly in any of its parts. motion re 313. The method of generalized co-ordinates explained Impulsive above (§ 204) is extremely useful in its application to the ferred to dynamics of a system; whether for expressing and working co-ordinates. out the details of any particular case in which there is any generalized Impulsive generation of motion re ferred to co-ordinates. finite number of degrees of freedom, or for proving general principles applicable even to cases, such as that of a liquid, as generalized described in the preceding section, in which there may be an infinite 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 Generalized the generalized system, and if stated as the convention agreed upon, it suffices to define the generalized components of impulse, those of velocity having been fixed on kinematical principles (§ 204). Generalized components of momentum of any specified motion are, of course, equal to the generalized components of the impulse by which it could be generated from components of impulse or mo mentum. rest. (a) Let 4, 4, 0, ... be the generalized co-ordinates of a material system at any time; and let ỳ, ¿, 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, ý1, 2, its component velocities, we have Hence the kinetic energy, which is Em(x2++), in terms of rectangular co-ordinates, becomes a quadratic function of ,, etc., when expressed in terms of generalized co-ordinates, so that if we denote it by 7' we have T=}{(,)+(, ) *+...+2(, )+...} (2), Generalized expression energy. where (4, 4), (4, 4), (4, 4), etc., denote various functions of the for kinetic 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 (X1, Y1, Z1), (X2, Y2, Z2), etc., denote component forces on the particles (x1, 1, 1), (X2, Y2, Z2), etc., respectively; and let (or, dy1, dz,), 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 (3). To transform this into an expression in terms of generalized coordinates, we have dx184+ do аф These quantities, Y, Þ, etc., are clearly the generalized components of the force on the system. Let Y, P, etc., denote component impulses, generalized on the same principle; that is to say, let where Y,, ... denote generalized components of the continuous force acting at any instant of the infinitely short time 7, within which the impulse is completed. If this impulse is applied to the system, previously in motion Generalized components, of force, of impulse. Impulsive generation of motion referred to generalized co-ordinates. in the manner specified above, and if dỷ, 84, denote the resulting augmentations of the components of velocity, the means of the component velocities before and after the impulse will be .... Hence, according to the general principle explained above for calculating the work done by an impulse, the whole work done in this case is ¥(†+¿8†)+ḍ($+¿86)+etc. To avoid unnecessary complications, let us suppose dv, dp, etc., to be each infinitely small. The preceding expression for the work done becomes ¥Ý+Þ¿+etc.; and, as the effect produced by this work is augmentation of kinetic energy from T to T+ ST, we must have dT Momenta in terins of velocities. to the coefficients of 4, 4, ... respectively in (c) From this we see, further, that the impulse required to produce the component velocity from rest, or to generate it in the system moving with any other possible velocity, has for its components Hence we conclude that to generate the whole resultant velocity n=(4,6)4+(4, 6)$+(0, 4)0+.. etc. (7), where it must be remembered that, as seen in the original expression for T, from which they are derived, (4, 4) means the same thing as (4, 4), and so on. The preceding expressions are the differential coefficients of T with reference to the velocities; Momenta in terms of velocities. (8). (d) The second members of these equations, being linear functions of y,,..., we may, by ordinary elimination, find ỳ, ò̟, etc., in terms of έ, 7. etc., and the expressions so obtained are of course linear functions of the last-named elements. And, since T is a quadratic function of y, p, etc., we have Now the algebraic process by which 4, 4, etc., are obtained in terms of έ, 7, etc., shows that, inasmuch as the coefficient of in the expression, (7), for έ, is equal to the coefficient of , in the expression for , and so on; the coefficient of 7 in the expression for must be equal to the coefficient of § in the expression for 4, and so on; that is to say, These expressions solve the direct problem,-to find the velocity produced by a given impulse (§, 7,...), when we have the kinetic energy, T, expressed as a quadratic function of the components of the impulse. (e' If we consider the motion simply, without reference to the impulse required to generate it from rest, or to stop it, the quantities έ, 7, ... are clearly to be regarded as the components of the momentum of the motion, according to the system of generalized co-ordinates. |