Page:EB1911 - Volume 08.djvu/788

convolutions. Each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable.

A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates $q_{1},q_{2},\ldots q_{m}\!$ all vanish (see § 5). This has been discussed by Routh, Lord Kelvin and Tait, and Poincaré. These writers treat the question, by an extension of Lagrange’s method, as a problem of small oscillations. Whether we adopt the notion of stability which this implies, or take up the position of Klein and Sommerfeld, there is no difficulty in showing that stability is ensured if $\mathrm {V} +\mathrm {K} \!$ be a minimum as regards variations of $q_{1},q_{2},\ldots q_{m}\!$. The proof is the same as that of Dirichlet for the case of statical stability.

We can illustrate this condition from the case of the top, where, in our previous notation,

$\mathrm {V} +\mathrm {K} =\mathrm {M} gh\cos \theta +{\frac {(\mu -\nu \cos \theta )^{2}}{2\mathrm {A} \sin ^{2}\theta }}+{\frac {\nu ^{2}}{2\mathrm {C} }}.$

(1)

To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put $\mu =\nu \!$. We then find without difficulty that $\mathrm {V} +\mathrm {K} \!$ is a minimum provided $\nu ^{2}\geqq 4\mathrm {AM} gh\!$. The method of small oscillations gave us the condition $\nu ^{2}>4\mathrm {AM} gh\!$, and indicated instability in the cases $\nu ^{2}\leqq 4\mathrm {AM} gh\!$. The present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable.

The question remains, as before, whether it is essential for stability that $\mathrm {V} +\mathrm {K}$ should be a minimum. It appears that from the point of view of the theory of small oscillations it is not essential, and that there may even be stability when $\mathrm {V} +\mathrm {K}$ is a maximum. The precise conditions, which are of a somewhat elaborate character, have been formulated by Routh. An important distinction has, however, been established by Thomson and Tait, and by Poincaré, between what we may call ordinary or temporary stability (which is stability in the above sense) and permanent or secular stability, which means stability when regard is had to possible dissipative forces called into play whenever the co-ordinates $q_{1},q_{2},\ldots q_{m}\!$ vary. Since the total energy of the system at any instant is given (in the notation of § 5) by an expression of the form ${\mathfrak {T}}+\mathrm {V} +\mathrm {K}$, where ${\mathfrak {T}}$ cannot be negative, the argument of Thomson and Tait, given under Mechanics, § 23, for the statical question, shows that it is a necessary as well as a sufficient condition for secular stability that $\mathrm {V} +\mathrm {K} \!$ should be a minimum. When a system is “ordinarily” stable, but “secularly” unstable, the operation of the frictional forces is to induce a gradual increase in the amplitude of the free vibrations which are called into play by accidental disturbances.

There is a similar theory in relation to the constrained systems considered in § 3 above. The equation (21) there given leads to the conclusion that for secular stability of any type of motion in which the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{n}\!$ are zero it is necessary and sufficient that the function $\mathrm {V} -\mathrm {T} _{0}\!$ should be a minimum.

The simplest possible example of this is the case of a particle at the lowest point of a smooth spherical bowl which rotates with constant angular velocity ($\omega \!$) about the vertical diameter. This position obviously possesses “ordinary” stability. If $a\!$ be the radius of the bowl, and $\theta \!$ denote angular distance from the lowest point, we have

$\mathrm {V} -\mathrm {T} _{0}=mga(1-\cos \theta )-{\tfrac {1}{2}}m\omega ^{2}a^{2}\sin ^{2}\theta ;$

(2)

this is a minimum for $\theta =0\!$ only so long as $\omega ^{2}<g/a\!$. For greater values of $\omega \!$ the only position of “permanent” stability is that in which the particle rotates with the bowl at an angular distance $\cos ^{-1}(g/\omega ^{2}a)\!$ from the lowest point. To examine the motion in the neighbourhood of the lowest point, when frictional forces are taken into account, we may take fixed ones, in a horizontal plane, through the lowest point. Assuming that the friction varies as the relative velocity, we have

$\left.{\begin{aligned}{\ddot {x}}&=-p^{2}x-k({\dot {x}}+\omega y),\\{\ddot {y}}&=-p^{2}y-k({\dot {y}}-\omega x),\end{aligned}}\right\}$

(3)

where $p^{2}=g/a\!$. These combine into

${\ddot {z}}+k{\dot {z}}+(p^{2}-ik\omega )z=0,$

(4)

where $z=x+iy,i={\sqrt {}}-1\!$. Assuming $z=\mathrm {C} e^{\lambda t}\!$, we find

$\lambda =-{\tfrac {1}{2}}k(1\mp \omega /p)\pm ip,$

(5)

if the square of $k\!$ be neglected. The complete solution is then

$x+iy=\mathrm {C} _{1}e^{-\beta 1t}e^{ipt}+\mathrm {C} _{2}e^{-\beta 2t}e^{-ipt},$

(6)

where

$\beta _{1}={\tfrac {1}{2}}k(1-\omega /p),\qquad \beta _{2}={\tfrac {1}{2}}k(1+\omega /p).$

(7)

This represents two superposed circular vibrations, in opposite directions, of period $2\pi /p\!$. If $\omega <p\!$, the amplitude of each of these diminishes asymptotically to zero, and the position $x=0,y=0\!$ is permanently stable. But if $\omega >p\!$ the amplitude of that circular vibration which agrees in sense with the rotation $\omega \!$ will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. If the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (see Mechanics, § 13).

7. Principle of Least Action.

The preceding theories give us statements applicable to the system at any one instant of its motion. We now come to a series of theorems relating to the whole motion of the system between any two configurations through which it passes, Stationary Action. viz. we consider the actual motion and compare it with other imaginable motions, differing infinitely little from it, between the same two configurations. We use the symbol $\delta \!$ to denote the transition from the actual to any one of the hypothetical motions.

The best-known theorem of this class is that of Least Action, originated by P. L. M. de Maupertuis, but first put in a definite form by Lagrange. The “action” of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of the vis viva. The action of a dynamical system is the sum of the actions of its constituent particles, and is accordingly given by the formula

$\mathrm {A} =\Sigma \int mvds=\Sigma \int mv^{2}dt=2\int \mathrm {T} dt.$

(1)

The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property

$\delta \mathrm {A} =0,$

(2)

provided the total energy have the same constant value in the varied motion as in the actual motion.

If $t,t'\!$ be the times of passing through the initial and final configurations respectively, we have

${\begin{aligned}\delta \mathrm {A} &=\delta \int _{t}^{t'}\Sigma m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})dt\\&=\int _{t}^{t'}\delta Tdt+2\mathrm {T} '\delta t'+2\mathrm {T} \delta t,\end{aligned}}$

(3)

since the upper and lower limits of the integral must both be regarded as variable. This may be written

$\delta \mathrm {A} =\int _{t}^{t'}\delta Tdt+\int _{t}^{t'}\Sigma m({\dot {x}}\delta {\dot {x}}+{\dot {y}}\delta {\dot {y}}+{\dot {z}}\delta {\dot {z}})dt+2\mathrm {T} '\delta t'-2\mathrm {T} \delta t$⁠

⁠$=\int _{t}^{t'}\delta Tdt+\left[\Sigma m({\dot {x}}\delta x+{\dot {y}}\delta y+{\dot {z}}\delta z)\right]_{t}^{t'}-\int _{t}^{t'}\Sigma m({\ddot {x}}\delta x+{\ddot {y}}\delta y+{\ddot {z}}\delta z)dt+2\mathrm {T} '\delta t'-2\mathrm {T} \delta t.$

(4)

Now, by d’Alembert’s principle,

$\Sigma m({\ddot {x}}\delta x+{\ddot {y}}\delta y+{\ddot {z}}\delta z)=-\delta \mathrm {V} ,$

(5)

and by hypothesis we have

$\delta (\mathrm {T} +\mathrm {V} )=0.$

(6)

The formula therefore reduces to

$\delta \mathrm {A} =\left[\Sigma m({\dot {x}}\delta x+{\dot {y}}\delta y+{\dot {z}}\delta z)\right]_{t}^{t'}+2\mathrm {T} '\delta t'-2\mathrm {T} \delta t.$

(7)

Since the terminal configurations are unaltered, we must have at the lower limit

$\delta x+{\dot {x}}\delta t=0,\qquad \delta y+{\dot {y}}\delta t=0,\qquad \delta z+{\dot {z}}\delta t=0,$

(8)

with similar relations at the upper limit. These reduce (7) to the form (2).

The equation (2), it is to be noticed, merely expresses that the variation of $\mathrm {A} \!$ vanishes to the first order ; the phrase stationary action has therefore been suggested as indicating more accurately what has been proved. The action in the free path between two given configurations is in fact not invariably a minimum, and even when a minimum it need not be the least possible subject to the given conditions. Simple illustrations are furnished by the case of a single particle. A particle moving on a smooth surface, and free from extraneous force, will have its velocity constant; hence the theorem in this case resolves itself into

$\delta \int ds=0,$

(9)

i.e. the path must be a geodesic line. Now a geodesic is not necessarily the shortest path between two given points on it; for example, on the sphere a great-circle arc ceases to be the shortest path between its extremities when it exceeds 180°. More generally, taking any surface, let a point $\mathrm {P} \!$, starting from $\mathrm {O}$, move along a geodesic; this geodesic will be a minimum path from $\mathrm {O}$ to $\mathrm {P}$ until $\mathrm {P}$ passes through a point $\mathrm {O} '$ (if such exist), which is the intersection with a consecutive geodesic through $\mathrm {O}$. After this point the minimum property ceases. On an anticlastic surface two geodesics cannot intersect more than once, and each geodesic is therefore a minimum path between any two of its points. These illustrations are due to K. G. J. Jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:—Let $\mathrm {O}$ and $\mathrm {P}$ denote any two configurations on a natural path of the system. If this be the sole free path from $\mathrm {O}$ to $\mathrm {P}$ with the prescribed amount of energy, the action from $\mathrm {O}$ to $\mathrm {P}$ is a minimum. But if