Stable equilibrium.

Unstable equilibrium.

Test of the nature of

But if, when displaced infinitely little in any direction from a particular position of equilibrium, and left to itself, it commences and continues vibrating, without ever experiencing more than infinitely small deviation in any of its parts, from the position of equilibrium, the equilibrium in this position is said to be stable. A weight suspended by a string, a uniform sphere in a hollow bowl, a loaded sphere resting on a horizontal plane with the loaded side lowest, an oblate body resting with one end of its shortest diameter on a horizontal plane, a plank, whose thickness is small compared with its length and breadth, floating on water, etc. etc., are all cases of stable equilibrium; if we neglect the motions of rotation about a vertical axis in the second, third, and fourth cases, and horizontal motion in general, in the fifth, for all of which the equilibrium is neutral.

If, on the other hand, the system can be displaced in any way from a position of equilibrium, so that when left to itself it will not vibrate within infinitely small limits about the position of equilibrium, but will move farther and farther away from it, the equilibrium in this position is said to be unstable. Thus a loaded sphere resting on a horizontal plane with its load as high as possible, an egg-shaped body standing on one end, a board floating edgeways in water, etc. etc., would present, if they could be realized in practice, cases of unstable equilibrium.

When, as in many cases, the nature of the equilibrium varies with the direction of displacement, if unstable for any possible displacement it is practically unstable on the whole. Thus a coin standing on its edge, though in neutral equilibrium for displacements in its plane, yet being in unstable equilibrium for those perpendicular to its plane, is practically unstable. A sphere resting in equilibrium on a saddle presents a case in which there is stable, neutral, or unstable equilibrium, according to the direction in which it may be displaced by rolling, but, practically, it would be unstable.

292. The theory of energy shows a very clear and simple equilibrium. test for discriminating these characters, or determining whether the equilibrium is neutral, stable, or unstable, in any case. If there is just as much work resisted as performed by the applied and internal forces in any possible displacement the equilibrium

nature of

is neutral, but not unless. If in every possible infinitely small Test of the displacement from a position of equilibrium they do less work equilibrium. among them than they resist, the equilibrium is thoroughly stable, and not unless. If in any or in every infinitely small displacement from a position of equilibrium they do more work than they resist, the equilibrium is unstable. It follows that if the system is influenced only by internal forces, or if the applied forces follow the law of doing always the same amount of work upon the system passing from one configuration to another by all possible paths, the whole potential energy must be constant, in all positions, for neutral equilibrium; must be a minimum for positions of thoroughly stable equilibrium; must be either an absolute maximum, or a maximum for some displacements and a minimum for others when there is unstable equilibrium.

of the
equations
of motion of

293. We have seen that, according to D'Alembert's prin- Deduction ciple, as explained above (§ 264), forces acting on the different points of a material system, and their reactions against the any system. accelerations which they actually experience in any case of motion, are in equilibrium with one another. Hence in any actual case of motion, not only is the actual work done by the forces equal to the kinetic energy produced in any infinitely small time, in virtue of the actual accelerations; but so also is the work which would be done by the forces, in any infinitely small time, if the velocities of the points constituting the system, were at any instant changed to any possible infinitely small velocities, and the accelerations unchanged. This statement, when put in the concise language of mathematical analysis, constitutes Lagrange's application of the "principle of virtual velocities" to express the conditions of D'Alembert's equilibrium between 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

Indetermin

ate equation of

motion of any system.

Of conservative system.

Equation of energy.

axes, of the whole force acting on it. Thus

d2z

m are the components of the reaction against acceleration. dt2

And these, with X, Y, Z, for the whole system, must fulfil the conditions of equilibrium. Hence if &r, dy, dz denote any arbitrary variations of x, y, z consistent with the conditions of the system, we have

Σ { ( X −m x ) dx + ( Y — m312)8y+(Z—m2)ôz} = 0 (1),

ጎ

dex
dt2

dt2

dt2

where denotes summation to include all the particles of the system. This may be called the indeterminate, or the variational, equation of motion. Lagrange used it as the foundation of his whole kinetic system, deriving from it all the common equations of motion, and his own remarkable equations in generalized co-ordinates (presently to be given). We may write it otherwise as follows:

Em cô+jôy+8z)=2(X&r+Yông+Zô:)

(2,, where the first member denotes the work done by forces equal to those required to produce the real accelerations, acting through the spaces of the arbitrary displacements; and the second member the work done by the actual forces through these imagined

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)

where & denotes the excess of the potential energy in the configuration (x+x, 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 &r, dy, Sz, etc., to be the actual variations of the co-ordinates in an infinitely small time St. Thus if we take Ex = ¿ôt, etc., and divide both members by St, we have

Σ(X¿+Yỷ+Z) =Σm(xx+ÿÿ+żż)

(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

of energy.

seen above (§ 214), the second member is the rate of increase of Equation Σ {m(x2+ÿ2+¿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

Σmv2=W+E.

(6), 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 W=V-V, and the integral equation of energy becomes,

or, if E denote the sum of the potential and kinetic energies, a
constant,
Σmv2=E-V
(7).

The general indeterminate equation gives immediately, for the motion of a system of free particles,

m ̧ï ̧=X1, m‚ÿ1=Y1, m1ë1=Z1, mq=X2, etc.

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, X1, 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 &x, dy1, 821, 8x,, ... must satisfy the n linear equations.

Multiplying the first of these by A, the second by A, etc., adding to the indeterminate equation, and then equating the coefficients of dx,, dy1, etc., each to zero, we have

Gauss's principle of least constraint.

Impact.

...

etc.

These are in all 3 equations to determine the n unknown
quantities A,,,,,..., and the 3i-n independent variables to
which X1, Y1,
are reduced by the kinematical 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.

When there are connexions between any parts of a system, the motion is in general not the same as if all were free. If we consider any particle during any infinitely small time of the motion, and call the product of its mass into the square of the distance between its positions at the end of this time, on the two suppositions, the constraint: the sum of the constraints is a minimum. This follows easily from (1).

294. When two bodies, in relative motion, come into contact, pressure begins to act between them to prevent any parts of them from jointly occupying the same space. This force commences from nothing at the first point of collision, and gradually increases per unit of area on a gradually increasing surface of contact. If, as is always the case in nature, each body possesses some degree of elasticity, and if they are not kept together after the impact by cohesion, or by some artificial appliance, the mutual pressure between them will reach a maximum, will begin to diminish, and in the end will come to nothing, by gradually diminishing in amount per unit of area on a gradually diminishing surface of contact. The whole process would occupy not greatly more or less than an hour if the bodies were of such dimensions as the earth, and such degrees of rigidity as copper, steel, or glass. It is finished, probably, within a thousandth of a second if they are globes of any of these substances not exceeding a yard in diameter.