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Slightly disturbed equilibrium.

of the transformation of quadratic functions, that we may, by a determinate linear transformation of the co-ordinates, reduce the

ous trans

of two

to sums of

Simultane. number) we have ia quantities A, A', A", ... B, B', B", ... etc., to be determined formation by i2 equations expressing that in 2T the coefficients of 1,0?, etc. are each

equal to unity, and of yo, etc. each vanish, and that in V the coefficients of quadratic functions 4.9., etc. each vanish. But, particularly in respect to our dynamical problem,

the following process in two steps is instructive:squares.

(1) Let the quadratic expression for T in terms of 4s, 5?, jo, etc., be reduced to the form 4,2 +0+... by proper assignment of values to A, B, etc. This may be done arbitrarily, in an infinite number of ways, without the solution of any algebraic equation of degree higher than the first; as we may easily see by working out a synthetical process algebraically according to the analogy of finding first the conjugate diametral plane to any chosen diameter of an ellipsoid, and then the diameter of its elliptic section, conjugate to any chosen diameter of this ellipse. Thus, of the

ili - 1)

equations expressing that

2 the coefficients of the products 4,0, , 0, 0, 0, etc. vanish in T, take first the one expressing that the coefficient of 4.°, vanishes, and by it find the value of one of the B's, supposing all the A's and all the B's but one to be known. Then take the two equations expressing that the coefficients of 1,0, and 0,0 vanish, and by them find two of the C's supposing all the C's but two to be known, as are now all the A's and all the B's: and so on. Thus, in terms of all the A's, all the B's but one, all the C's but two, all the D's but three, and so on, supposed known, we find by the solution of linear equations the remaining B's, C's, D's, etc. Lastly, using the values thus found for the unassumed quantities, B, C, D, etc., and equating to unity the coefficients of ?, 0,, 09, etc. in the transformed expression for 2T, we have i equations among the squares

(i+1) i and products of the assumed quantities, (i) A's, (i-1) B’s, (i – 2) C's,

2 etc., by which any one of the A's, any one of the B's, any one of the C's, and so

i(i-1) on, are given immediately in terms of the ratios of the others to them.

2

i (i - 1) Thus the thing is done, and disposable ratios are left undetermined.

ili - 1 (2) These quantities may be determined by the

2

. ing that also in the transformed quadratic V the coefficients of ., 4.0,, 0.9., etc. vanish. Or, having made the first transformation as in (1) above, with assumed values

i (i-1) for disposable ratios, make a second transformation determinately thus:

2

2

Generalized -Let

orthogonal
transforma.
tion of co-
ordinates.

v=ly. + m¢.. +
0,=l'y,+ m'on +...

etc., etc.,
where the i? quantities l, m, ..., l', m', satisfy the fi (i+1) equations

U' + mm' + =0, I'l" + m'm" + ... =0, etc.,

energies.

or of

expression for 27, which is essentially positive, to a sum of Simplified

expressions squares of generalized component velocities, and at the same for the

kinetic and time V to a sum of the squares of the corresponding co-ordi- potential nates, each multiplied by a constant, which may be either positive or negative, but is essentially real. [In the case of an equality

any number of equalities among the values of these constants (a, b, etc. in the notation below), roots as they are of a determinantal equation, the linear transformation ceases to be wholly determinate; but the degree or degrees of indeterminacy which supervene is the reverse of embarrassing in respect to either the process of obtaining the solution, or the interpretation and use of it when obtained.] Hence y, ¢, may be so chosen that T= 1 (4 + 0 + etc.)

:(4), and

V = } (aya + B8% + etc.) ............ .(5), a, B, etc., being real positive or negative constants. Hence Lagrange's equations become j=-ay, =-Bø, etc..............

...(6). The solutions of these equations are

Integrated y = A cos (t Ja - e), ø= A' cos (tJB - e'), etc. ...(7), equations

of motion, A, e, A', e, etc., being the arbitrary constants of integration. the fundas Hence we conclude the motion consists of a simple harmonic variation of each co-ordinate, provided that a, b, etc., are all vibration. positive. This condition is satisfied when V is a true minimum at the configuration of equilibrium; which, as we have seen (5 292), is necessarily the case when the equilibrium is stable. If any one or more of a, b,... vanishes, the equilibrium might

mental modes of

ous trans.

of two

and
12 + m2 + ... =1, 142 + m'? + ...=1, etc.,

Simultane. leaving } i (i – 1) disposables.

formation We shall still have, obviously, the same form for 27, that is :

quadratic

functions 2T=4,2 +0,2+...

to sums of And, according to the known theory of the transformation of quadratic functions, squares. we may determine the 1 i (i – 1) disposables of l, m, ..., l', m', ... so as to make the ļi(i-1) products of the co-ordinates Yu, ,,, etc. disappear from the expression for V, and give

2 V=ay..? +Bot..., where a, b, y, etc., are the roots, necessarily real, of an equation of the ith degree of which the coefficients depend on the coefficients of the squares and products in the expression for V in terms of t., 0, etc. Later [(7'), (8) and (9) of g 343 f], a single process for carrying out this investigation will be worked out.

Integrated equations of motion, expressing the fundamental modes of vibration;

or of falling & way from contiguration of unstabie equilibrium.

be either stable or unstable, or neutral; but terms of higher
orders in the expansion of V in ascending powers and products
of the co-ordinates would have to be examined to test it; and if
it were stable, the period of an infinitely small oscillation in the
value of the corresponding co-ordinate or co-ordinates would be
infinitely great. If any or all of a, b, y, ... are negative, Vis
not a minimum, and the equilibrium is (S292) essentially un-
stable. The form (7) for the solution, for each co-ordinate for
which this is the case, becomes imaginary, and is to be changed
into the exponential form, thus; for instance, let - a=p, a positive
quantity. Thus
v=CE+IVP + Ke-tvp ...

....(8), which (unless the disturbance is so adjusted as to make the arbitrary constant C vanish) indicates an unlimited increase in the deviation. This form of solution expresses the approximate law of falling away from a configuration of unstable equilibrium. In general, of course, the approximation becomes less and less accurate as the deviation increases.

Infinitely small disa turbance from un stable equi. librium.

Potential and Kinetic enerkies expressed as functions of time.

|-ete... (9),

or

or

We have, by (5), (4), (7) and (8),

V = a1[1 + cos 2 (tJa – e)] + etc.

V=-1p [2CK + C?e20VP + K*c-2VP] – etc. and

T= A' [1 - cos 2 (t/a – e)] + etc.

T = P(-2CK+C'<240P + K'c-2VP] +etc.) and, verifying the constancy of the sum of potential and kinetic energies, T + V = } (aA + BA" + etc.)

(11). T + V =-2 (pCK+qC'K' + etc.) One example for the present will suffice. Let a solid, immersed in an infinite liquid (8 320), be prevented from any motion of rotation, and left only freedom to move parallel to a certain fixed plane, and let it be influenced by forces subject to the conservative law, which vanish in a particular position of equilibrium. Taking any point of reference in the body, choosing its position when the body is in equilibrium, as origin of rectangular co-ordinates 0X, OY, and reckoning the potential energy from it, we shall have, as in general,

2T = Acc® + Bjø + 2Ccy; 2V = ax + by° + 2cxy,

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or

Example of fundaniental modes.

fundamen

the principles stated in $ 320 above, allowing us to regard the Example of co-ordinates x and y as fully specifying the system, provided tal modes. always, that if the body is given at rest, or is brought to rest, the whole liquid is at rest (S 320) at the same time. By solving the obviously determinate problem of finding that pair of conjugate diameters which are in the same directions for the ellipse

Amc* + By' + 2Cay = const., and the ellipse or hyperbola,

ax® + by° + 2cxy =const.,
and choosing these as oblique axes of co-ordinates (x, y), we
shall have

2T = A,2,' + B,y', and 2V = a;X;'+by;":
And, as d,, B, are essentially positive, we may, to shorten our
expressions, take x, 7/A, = 4, y, VB, = $; so that we shall have

2T = 42 + $, 2V = aye + B,
the normal expressions, according to the general forms shown
above in (4) and (5).

The interpretation of the general solution is as follows:338. If a conservative system is infinitely little displaced General from a configuration of stable equilibrium, it will ever after fundamenvibrate about this configuration, remaining infinitely near it ; infinitely each particle of the system performing a motion which is com- motion posed of simple harmonic vibrations. If there are i degrees of figuration freedom to move, and we consider any system (S202) of gene- librium. ralized co-ordinates specifying its position at any time, the deviation of any one of these co-ordinates from its value for the configuration of equilibrium will vary according to a complex harmonic function ($ 68), composed of i simple harmonics generally of incommensurable periods, and therefore (8 67) the whole motion of the system will not in general recur periodically through the same series of configurations. There are, however, i distinct displacements, generally quite determinate, which we shall call the normal displacements, fulfilling the condition, that Normal disif any one of them be produced alone, and the system then left from equito itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic func

tal modes of

about a con

librium.

Theorem of kinetic energy;

energy.

Fundamen., tion of the time, and consequently every particle of the system tal modes of vibration. will execute a simple harmonic movement in the same period.

This result, we shall see later (Vol. 11.), includes cases in which there are an infinite number of degrees of freedom; as for instance a stretched cord; a mass of air in a closed vessel; waves in water, or oscillations of water in a vessel of limited extent, or of an elastic solid; and in these applications it gives the theory of the so-called "fundamental vibration," and successive “harmonics” of a cord or organ-pipe, and of all the different possible simple modes of vibration in the other cases. In all these cases it is convenient to give the name "fundamental mode” to any one of the possible simple harmonic vibrations, and not to restrict it to the gravest simple barmonic mode, as has been hitherto usual in respect to vibrating cords and organ-pipes.

The whole kinetic energy of any complex motion of the sys

tem is [S 337 (4)] equal to the sum of the kinetic energies of of potential the fundamental constituents; and (S 337 (5)] the potential

energy of any displacement is equal to the sum of the potential energies of its normal components.

Corresponding theorems of normal constituents and funda

mental modes of motion, and the summation of their kinetic couligurn. and potential energies in complex motions and displacements, stable equi- hold for motion in the neighbourhood of a configuration of unlibrium.

stable equilibrium. In this case, some or all of the constituent
motions are fallings away from the position of equilibrium
(according as the potential energies of the constituent normal
vibrations are negative).

339. If, as may be in particular cases, the periods of the equality

vibrations for two or more of the normal displacements are equal, any displacement compounded of them will also fulfil the condition of being a normal displacement. And if the system be displaced according to any one such normal displacement, and projected with velocity corresponding to another, it will execute

a movement, the resultant of two simple harmonic movements Graphic in equal periods. The graphic representation of the variation

of the corresponding co-ordinates of the system, laid down as two rectangular co-ordinates in a plane diagram, will consequently (8 65) be a circle or an ellipse ; which will therefore,

Infinitesi. mal motions in neighbourhood of

Case of

among periods.

representation.

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