Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/624

572 forces called into play whenever the co-ordinates $$q_1, q_2\dots q_m$$ 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 above 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.

Principle of Least Action.

§ 7. 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, 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 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

$$\mathrm{A}= \sum \int mvds = \sum \int mv^2dt= 2 \int\mathrm{T}dt.$$. . (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. The proof, from first principles, has been given under Mechanics (Ency. Brit.) vol. xv. pp. 723-26), where it is also shown how the equation (2), expressed in terms of generalized co-ordinates, leads to Lagrange’s equations.

The equation (2), it is to be noticed, merely expresses that the variation of 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

$$\delta\int ds=0$$,. . . . (3)

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 P, starting from O, move along a geodesic; this geodesic will be a minimum path from O to P until P passes through a point O&prime; (if such exist), which is the intersection with a consecutive geodesic through 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 Jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:—Let O and P denote any two configurations on a natural path of the system. If this be the sole free path from O to P with the prescribed amount of energy, the action from O to P is a minimum. But if there be several distinct paths, let P vary from coincidence with O along the first-named path; the action will then cease to be a minimum when a configuration O&prime; is reached such that two of the possible paths from O to O&prime; coincide. For instance, if O and P be positions on the parabolic path of a projectile under gravity, there will be a second path (with the same energy and therefore the same velocity of projection from O), these two paths coinciding when P is at the other extremity (O&prime;, say) of the focal chord through O. The action from O to P will therefore be a minimum for all positions of P short of O&prime;. Two configurations such as O and O&prime; in the general statement are called conjugate kinetic foci. Of. Variation of an Integral, § 6.

Before leaving this topic the connexion of the principle of stationary action with a well-known theorem of optics may be noticed. For the motion of a particle in a conservative field of force the principle takes the form

$$\delta\int vds = 0.$$. . . . (4)

On the corpuscular theory of light $$v$$ is proportional to the refractive index $$\mu$$. of the medium, whence

$$\delta\int \mu ds =0.$$. . . .(5)

For the further development of this result see Optics (Ency. Brit. vol. xvii.) and Wave Theory (Ency. Brit.  vol. xxiv.).

In the formula (2) the energy in the hypothetical motion is prescribed, whilst the time of transit from the initial to the final configuration is variable.

In another theorem, due to Hamilton, the time of transit is prescribed to be the same as in the actual motion, whilst the energy may be different and need not (indeed) be constant. Under these conditions we have

$$\delta\int_{t}^{t'} \left ( \mathrm{T} - \mathrm{V} \right ) dt=0, $$. . . (6)

where $$t$$, $$t'$$ are the prescribed times of passing through the given initial and final configurations. The proof of (6) is simple; we have

$$\delta\int_{t}^{t'} \left ( \mathrm{T} - \mathrm{V} \right ) dt= \delta\int_{t}^{t'}  \left ( \delta\mathrm{T} - \delta\mathrm{V} \right ) dt = \int_{t}^{t'} \left\{ \sum m \left ( \dot{x}\delta\dot{x} + \dot{y}\delta\dot{y} + \dot{z}\delta \dot{z} \right ) - \delta\mathrm{v} \right \} dt $$

$$ = {\left [ \sum m \left ( \dot{x}\delta\dot{x} + \dot{y}\delta\dot{y} + \dot{z}\delta\dot{z} \right ) \right ]}_{t}^{t'} - \int_{t}^{t'} \left\{ \sum m \left ( \dot{x}\delta\dot{x} + \dot{y}\delta\dot{y} + \dot{z}\delta \dot{z} \right ) - \delta\mathrm{v} \right \} dt $$

. . (7)

The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by D’Alembert’s principle. The fact that in (6) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange’s equations, we have

$$\begin{align} \int_{t}^{t'}(\delta\mathrm{T} & -\delta\mathrm{V})dt = \int_{t}^{v} \left \{ \frac{\partial\mathrm{T}}{\partial\dot{q}_1} \partial\dot{q}_1 + \frac{\partial\mathrm{T}}{\partial q_1}\partial q_1 + \ldots - \frac{\partial\mathrm{V}}{\partial q_1}\partial q_1 - \ldots \right \}dt \\ & = \left [ p_1\delta q + p_2 \delta q_2 + \ldots \right ]_t^t \\ & - \int_{t}^{t'} \left \{ \left ( \dot{p}_1 - \frac{\partial\mathrm{T}}{\partial q_1} + \frac{\partial\mathrm{V}}{\partial q_1} \right )\partial q_1 + \left ( \dot{p}_2 - \frac{\partial\mathrm{T}}{\partial q_2} + \frac{\partial\mathrm{V}}{\partial q_2} \right )\partial q_2 + \ldots \right \}dt \quad \text{(8)} \end{align}$$

The integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of $$\delta q_1 \delta q_2, \dots$$ under the integral sign should vanish for all values of $$t$$, since the variations in question are independent, and subject only to the condition of vanishing at the limits of integration. We are thus led to Lagrange’s equation of motion for a conservative system. It appears that the formula (6) is a convenient as well as a compact embodiment of the whole of ordinary Dynamics.

The modification of the Hamiltonian principle appropriate to

the case of cyclic systems has been given by Larmor. If we write, as in § 1 (25),

$$\Theta = \mathrm{T} - \kappa\dot{\chi} - \kappa'\dot{\chi}' - \kappa\dot{\chi} - \ldots, \quad. \quad. \quad \text{(9)}$$

we shall have

$$\delta \int_{t}^{t'} \left ( \Theta - \mathrm{V} \right )dt = 0, \quad. \quad. \quad. \quad \text{(10)}$$

provided that the variation does not affect the cyclic moments $$ k, k', k'',\dots$$, and that the configurations at times $$t$$ and $$t'$$ are unaltered, so far as they depend on the palpable co-ordinates $$q_1, q_2, \dots q_m .$$ The initial and final values of the ignored coordinates will in general be affected.

To prove (10) we have, on the above understandings,

$$\begin{align} \delta \int_{t}^{t'} \left ( \Theta - \mathrm{V} \right )dt & = \int_{t}^{t'} \left (\delta\mathrm{T} - \kappa\delta\dot{\chi} - \ldots - \delta\mathrm{V} \right )dt \\ & = \int_{t}^{t'} \left ( \frac{\partial\mathrm{T}}{\partial\dot{q}_1}\delta\dot{q}_1 + \ldots + \frac{\partial\mathrm{T}}{\partial q_1}\delta q_1 + \ldots - \delta\mathrm{V} \right )dt, \quad. \text{(11)} \end{align}$$

where terms have been cancelled in virtue of § 5 (2). The last member of (11) represents a variation of the integral

$$\int_{t}^{t'} \left ( \mathrm{T} - \mathrm{V} \right )dt$$

on the supposition that $$\delta\chi = 0, \delta\chi'=0, \delta\chi=0,\dots$$ throughout, whilst $$\delta q_1, \delta q_2, \delta q_m$$ vanish at times $$t$$ and $$t'$$; i.e.'', it is a variation in which the initial and final configurations are absolutely unaltered. It therefore vanishes as a consequence of the Hamiltonian principle in its original form.

Larmor has also given the corresponding form of the principle of least action. He shows that if we write

$$\mathrm{A} = \int \left (2\mathrm{T} - \kappa\dot{\chi} - \kappa'\dot{\chi}' - \kappa\dot{\chi} - \ldots \right )dt,. \quad. \quad \text{(12)}$$

then

$$\delta A = 0, \quad. \quad. \quad. \quad. \quad \text{(13)}$$

provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic moments.