of the of motion of the concise language of mathematical analysis, constitutes Deduction Lagrange's application of the "principle of virtual velocities equations to express the conditions of D'Alembert's equilibrium between any system. the forces acting, and the resistances of the masses to acceleration. It comprehends, as we have seen, every possible condition of every case of motion. The "equations of motion" in any particular case are, as Lagrange has shown, deduced from it with great ease. Let m be the mass of any one of the material points of the system; x, y, z its rectangular co-ordinates at time t, relatively to axes fixed in direction (§ 249) through a point reckoned as fixed (§ 245); and X, Y, Z the components, parallel to the same axes, of the whole force acting on it. Thus d2z d2x m m 27 dt2 m are the components of the reaction against acceleration. And these, with X, Y, Z, for the whole system, must fulfil the where Σ denotes summation to include all the particles of the any system. spaces. If the moving bodies constitute a conservative system, and if V denote its potential energy in the configuration specified by (x, y, z, etc.), we have of course (§§ 241, 273) Of conserva tive system. Equation of energy. and therefore the indeterminate equation of motion becomes .(4), where & denotes the excess of the potential energy in the configuration (x+dx, y + dy, z+dz, etc.) above that in the configuration (x, y, z, etc.). One immediate particular result must of course be the common equation of energy, which must be obtained by supposing dx, dy, Sz, etc., to be the actual variations of the co-ordinates in an infinitely small time St. Thus if we take dx = xồt, etc., and divide both members by St, we have Σ (Xx + Yỷ + Zê) = Σm (ë¿ + ÿÿj + ïž) ..................................(5). Here the first member is composed of Newton's Actiones Agentium; with his Reactiones Resistentium so far as friction, gravity, and molecular forces are concerned, subtracted: and the second consists of the portion of the Reactiones due to acceleration. As we have seen above (§ 214), the second member is the rate of increase of Σ1⁄2m (x2 + ÿ3 + 2) per unit of time. Hence, denoting by v the velocity of one of the particles, and by W the integral of the first member multiplied by dt, that is to say, the integral work done by the working and resisting forces in any time, we have E being the initial kinetic energy. This is the integral equation of energy. In the particular case of a conservative system, W is a function of the co-ordinates, irrespectively of the time, or of the paths which have been followed. According to the previous notation, with besides V to denote the potential energy of the system in its initial configuration, we have WV - V, and the integral equation of energy becomes 0 Σmv2 = V2- V+ E ̧, or, if Ę denote the sum of the potential and kinetic energies, a Σžmv2 = E – V .... The general indeterminate equation gives immediately, for the motion of a system of free particles, m ̧ï1 = X1, m1ÿ1 = Y1, m ̧ï1 = Z1, m ̧ï ̧ = X2, etc. m.x 1 1 mỗi 1 Of these equations the three for each particle may of course be treated separately if there is no mutual influence between the particles: but when they exert force on one another, X,, Y1, etc., will each in general be a function of all the co-ordinates. introduced determinate From the indeterminate equation (1) Lagrange, by his method Constraint of multipliers, deduces the requisite number of equations for into the indetermining the motion of a rigid body, or of any system of con- equation. nected particles or rigid bodies, thus:-Let the number of the particles be i, and let the connexions between them be expressed by n equations, being the kinematical equations of the system. By taking the variations of these we find that every possible infinitely small displacement dx,, dy,, dz,, dx,, ... must satisfy the n linear equations Multiplying the first of these by A, the second by X,, etc., adding to the indeterminate equation, and then equating the coefficients of dx,, dy,, etc., each to zero, we have nate equa motion These are in all 3i equations to determine the n unknown Determiquantities A, A,, ..., and the 3i-n independent variables to tions of which x1, y1, ... are reduced by the kinematical equations (8). deduced. The same equations may be found synthetically in the following manner, by which also we are helped to understand the precise meaning of the terms containing the multipliers A, A, etc. First let the particles be free from constraint, but acted on both by the given forces X, Y,, etc., and by forces depending on mutual distances between the particles and upon their positions relatively to fixed objects subject to the law of conservation, and having for their potential energy − 1⁄2 (kF2 + k‚F,2 + etc.), so that components of the forces actually experienced by the different particles shall be Now suppose k, k, etc. to be infinitely great:-in order that the forces on the particles may not be infinitely great, we must have F=0, F=0, etc., that is to say, the equations of condition (8) must be fulfilled; and the last groups of terms in the second members of (11) now disappear because they contain the squares of the infinitely small quantities F, F,, etc. Put now kF=λ, k,F,=λ,, etc., and we have equations (10). This second mode of proving Lagrange's equations of motion of a constrained system corresponds precisely to the imperfect approach to the ideal case which can be made by real mechanism. The levers and bars and guidesurfaces cannot be infinitely rigid. Suppose then k, k,, etc. to be finite but very great quantities, and to be some functions of the co-ordinates depending on the elastic qualities of the materials of which the guiding mechanism is composed:-equations (11) will express the motion, and by supposing k, k,, etc. to be greater and greater we approach more and more nearly to the ideal case of absolutely rigid mechanism constraining the precise fulfilment of equations (8). The problem of finding the motion of a system subject to any unvarying kinematical conditions whatever, under the action of any given forces, is thus reduced to a question of pure analysis. In the still more general problem of determining the motion when certain parts of the system are constrained to move in a specified manner, the equations of condition (8) involve not only the co-ordinates, but also t, the time. It is easily seen however that the equations (10) still hold, and with (8) fully determine the motion. For:-consider the equations of equilibrium of the particles acted on by any forces X, Y,', etc., and constrained by nate equa proper mechanism to fulfil the equations of condition (8) with Determithe actual values of the parameters for any particular value tions of of t. The equations of equilibrium will be uninfluenced deduced. by the fact that some of the parameters of the conditions (8) have different values at different times. Hence, with to D'Alembert's principle, the equations of motion will still be (8), (9), and (10) quite independently of whether the parameters of (8) are all constant, or have values varying in any arbitrary manner with the time. motion energy. To find the equation of energy multiply the first of equations Equation of (10) by,, the second by y,, etc., and add. Then remarking that in virtue of (8) we have partial differential coefficients of F, F,, etc. with reference to t being denoted by (d), dt dt When the kinematic conditions are unvarying," that is to say, when the equations of condition are equations among the co-ordinates with constant parameters, we have showing that in this case the fulfilment of the equations of condition involves neither gain nor loss of energy. On the other hand, equation (12) shows how to find the work performed or consumed in the fulfilment of the kinematical conditions when they are not unvarying. VOL. I. 18 |