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General investigation

356. To investigate the effect of an infinitely small conof disturbed servative disturbance produced at any instant in the motion


of any conservative system, may be reduced to a practicable problem (however complicated the required work may be) of mathematical analysis, provided the undisturbed motion is thoroughly known.

General equation of

motion free

in two degrees.

(a) First, for a system having but two degrees of freedom to move, let

2T = P↓3 + Q¿2 + 2R¢¢


where P, Q, R are functions of the co-ordinates not depending on the actual motion,


d dT dt di





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P$ + R$,

Pÿ + R$ +





(dip + 14 ) 4 4


and the Lagrangian equations of motion [§ 318 (24)] are

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Pÿ + R$ +


Rÿ + Q$ + } { ( 2 π4 - dø
dR dP






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Q$ + R&

etc., can ever become infinite.

dR dQ

10) +3 } =



dQ dQ
dy dp

We shall suppose the system of co-ordinates so chosen that
none of the functions P, Q, R, nor their differential coefficients


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(b) To investigate the effects of an infinitely small disturbance, we may consider a motion in which, at any time t, the co-ordinates are +p and +q, p and q being infinitely small; and, by simply taking the variations of equations (3) in the usual manner, we arrive at two simultaneous differential equations of the second degree, linear with respect to

P, q, p, q, Ÿ, q,

but having variable coefficients which, when the undisturbed motion, is fully known, may be supposed to be known functions of t. In these equations obviously none of the coefficients can at any time become infinite if the data correspond to a real dynamical problem, provided the system of co-ordinates is properly chosen (a); and the coefficients of p and q are the


values, at the time t, of P, R, and R, Q, respectively, in the General inorder in which they appear in (3), P, Q, R being the coefficients of disturbed path. of a homogeneous quadratic function (1) which is essentially positive. These properties being taken into account, it may be shown that in no case can an infinitely small interval of time be the solution of the problem presented (§ 347) by the question of kinetic stability or instability, which is as follows:

(c) The component velocities &, & are at any instant changed to +a, +ẞ, subject to the condition of not changing the value of T. Then, a and ß being infinitely small, it is required to find the interval of time until q/p first becomes equal to ¿/4.

(d) The differential equations in p and q reduce this problem, and in fact the full problem of finding the disturbance in the motion when the undisturbed motion is given, to a practicable form. But, merely to prove the proposition that the disturbed course cannot meet the undisturbed course until after some finite time, and to estimate a limit which this time must exceed in any particular case, it may be simpler to proceed thus:

(e) To eliminate t from the general equations (3), let them first be transformed so as not to have t independent variable. We must put

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And by the equation of energy we have

(Pdy2+Qdp2 + 2Rdydp)
{2 (E – V)}}

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and therefore

dtd"-dod t



it being assumed that the system is conservative. Eliminating dt and det between this and the two equations (3), we find a differential equation of the second degree between and 4, which is the differential equation of the course. For simplicity, let us suppose one of the co-ordinates, & for instance, to be independent variable; that is, let d3p=0. We have, by (4),

dta dt = - $


ÿdt2 = d3& +


dy ödť2,


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d'p dp

+ L

able coefficients, none of which can become infinite as long as EV, the kinetic energy, is finite.

(f) Taking the variation of this equation on the supposition that becomes +p, where p is infinitely small, we have



where L and M denote known functions of p, neither of which has any infinitely great value. This determines the deviation, p, of the course. Inasmuch as the quadratic (1) is essentially always positive, PQ-R must be always positive. Hence, if




of the third degree, with vari

+ Mp = 0........

dp for a particular value of 4, p vanishes, and has a given value do which defines the disturbance we suppose made at any instant,

must increase by a finite amount (and therefore a finite time must elapse) before the value of p can be again zero; that is to say, before the disturbed course can again cut the undisturbed


(g) The same proposition consequently holds for a system having any number of degrees of freedom. For the preceding proof shows it to hold for the system subjected to any frictionless constraint, leaving it only two degrees of freedom; including that particular frictionless constraint which would not alter either the undisturbed or the disturbed course. The full general investigation of the disturbed motion, with more than two degrees of freedom, takes a necessarily complicated form, but the principles on which it is to be carried out are sufficiently indicated by what we have done.

(h) If for L/PQ – R3 we substitute a constant 2a, less than its least value, irrespectively of sign, and for M/PQ-R, a

constant B greater algebraically than its greatest value, we have an equation


+ 2a + B = 0.......
dp3 do


Here the value of p vanishes for values of successively exceeding one another by π/√B-a2, which is clearly less than the increase that must have in the actual problem before p vanishes a second time. Also, we see from this that if a2> -B the actual motion is unstable. It might of course be unstable even if a2 <ß; and the proper analytical methods for finding either the rigorous solution of (7), or a sufficiently near practical solution, would have to be used to close the criterion of stability or instability, and to thoroughly determine the disturbance of the course.

equation of

path of

(i) When the system is only a single particle, confined to a Differential plane, the differential equation of the deviation may be put disturbed under a remarkably simple form, useful for many practical single parproblems. Let N be the normal component of the force, per plane. unit of the mass, at any instant, v the velocity, and p the radius of curvature of the path. We have (§ 259)

ticle in a


Let, in the diagram, ON be the undisturbed, and OE the disturbed path. Let EN, cutting ON at right angles, be denoted by u, and ON by s. If further we denote by p' the radius of curvature in the disturbed path,


remembering that u is infinitely small, we easily find

SN 8






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Hence, using 8 to denote variations from N to E, we have

v2 8(v2) d'u


47) ·





1 d2u น
+ +


ρ ds* p3

General investigation of disturbed path.

Differential equation of disturbed, path of

single par ticle in a plane.

Kinetic foci.


never a minimum in a course including kinetic foci.

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or, if we denote by
tance from the point

v2 = 2 (E-V),

d'u 3u SN


p v2







the rate of variation of N, per unit of disN in the normal direction, so that dV = Lu,


This includes, as a particular case, the equation of deviation from a circular orbit, investigated above (§ 350).


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u = = 0..

357. If, from any one configuration, two courses differing infinitely little from one another have again a configuration in common, this second configuration will be called a kinetic focus relatively to the first: or (because of the reversibility of the motion) these two configurations will be called conjugate kinetic foci. Optic foci, if for a moment we adopt the corpuscular theory of light, are included as a particular case of kinetic foci in general. By § 356 (g) we see that there must be finite intervals of space and time between two conjugate foci in every motion of every kind of system, only provided the kinetic energy does not vanish.

358. Now it is obvious that, provided only a sufficiently short course is considered, the action, in any natural motion of Theorem of a system, is less than for any other course between its terminal



configurations. It will be proved presently (§ 361) that the first configuration up to which the action, reckoned from a given initial configuration, ceases to be a minimum, is the first kinetic focus; and conversely, that when the first kinetic focus is passed, the action, reckoned from the initial configuration, ceases to be a minimum; and therefore of course can never again be a minimum, because a course of shorter action, deviating infinitely little from it, can be found for a part, without altering the remainder of the whole, natural course.

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