Page:EB1911 - Volume 08.djvu/790

action” as a particular case. Selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy $\mathrm {H}$, so that it shall pass through the other. Hence, regarding $\mathrm {A}$ as a function of the initial and final co-ordinates and the energy, we find

$\left.{\begin{aligned}&p'_{1}={\frac {\partial \mathrm {A} }{\partial q'_{1}}},\qquad p'_{2}={\frac {\partial \mathrm {A} }{\partial q'_{2}}},\ldots ,\\&p_{1}=-{\frac {\partial \mathrm {A} }{\partial q_{1}}},\qquad p_{2}=-{\frac {\partial \mathrm {A} }{\partial q_{2}}},\ldots ,\end{aligned}}\right\}$

(10)

and

$\tau ={\frac {\partial \mathrm {A} }{\partial \mathrm {H} }}.$

(11)

$\mathrm {A}$ is called by Hamilton the characteristic function; it represents, of course, the “action” of the system in the free motion (with prescribed energy) between the two configurations. Like $\mathrm {S}$, it satisfies a partial differential equation, obtained by substitution from (10) in (7).

The preceding theorems are easily adapted to the case of cyclic systems. We have only to write

$\mathrm {S} =\int _{t}^{t'}(\mathrm {R} -\mathrm {V} )dt=\int _{t}^{t'}(\mathrm {T} -\kappa {\dot {\chi }}-\kappa '{\dot {\chi }}'-\ldots -\mathrm {V} )dt$

(12)

in place of (1), and

$\mathrm {A} =\int (2\mathrm {T} -\kappa {\dot {\chi }}-\kappa '{\dot {\chi }}'-\ldots )dt,$

(13)

in place of (8); cf. § 7 ad fin. It is understood, of course, that in (12) $\mathrm {S}$ is regarded as a function of the initial and final values of the palpable co-ordinates $q_{1},q_{2},\ldots q_{m}$, and of the time of transit $\tau \!$, the cyclic momenta being invariable. Similarly in (13), $\mathrm {A}$ is regarded as a function of the initial and final values of $q_{1},q_{2},\ldots q_{m}$, and of the total energy $\mathrm {H}$, with the cyclic momenta invariable. It will be found that the forms of (4) and (9) will be conserved, provided the variations $\delta q_{1},\delta q_{2},\ldots \!$ be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols.

9. Reciprocal Properties of Direct and Reversed Motions.

We may employ Hamilton’s principal function to prove a very remarkable formula connecting any two slightly disturbed natural motions of the system. If we use the symbols $\delta \!$ and $\Delta \!$ to denote the corresponding variations, the theorem isLagrange’s formula.

${\frac {d}{dt}}\Sigma (\delta p_{r}\cdot \Delta q_{r}-\Delta p_{r}\cdot \delta q_{r})=0;$

(1)

or integrating from $t$ to $t'\!$,

$\Sigma (\delta p'_{r}\cdot \Delta q'_{r}-\Delta q'_{r}\cdot \delta q'_{r})=\Sigma (\delta p_{r}\cdot \Delta q_{r}-\Delta p_{r}\cdot \delta q_{r}).$

(2)

If for shortness we write

$(r,s)={\frac {\partial ^{2}\mathrm {S} }{\partial q_{r}\partial q_{s}}},\qquad (r,s')={\frac {\partial ^{2}\mathrm {S} }{\partial q_{r}\partial q'_{s}}},$

(3)

we have

$\delta p_{r}=-\Sigma _{s}(r,s)\delta q_{s}-\Sigma _{s}(r,s')\delta q'_{s}$

(4)

with a similar expression for $\Delta p_{r}$. Hence the right-hand side of (2) becomes

$-\Sigma _{r}{\Sigma _{s}(r,s)\delta q_{s}+\Sigma _{s}(r,s')\delta q'_{s}}\Delta q_{r}+\Sigma _{r}{\Sigma _{s}(r,s)\Delta q_{s}+\Sigma _{s}(r,s')\Delta q'_{s}}\delta q_{r}=\Sigma _{r}\Sigma _{s}(r,s'){\delta q_{r}\cdot \Delta q'_{s}-\Delta q_{r}\cdot \delta q'_{s}}.$

(5)

The same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to Lagrange, and was employed by him as the basis of his method of treating the dynamical theory of Variation of Arbitrary Constants.

The formula (2) leads at once to some remarkable reciprocal relations which were first expressed, in their complete form, by Helmholtz. Consider any natural motion of a conservative system between two configurations $\mathrm {O}$ and $\mathrm {O} '\!$ through which it passes at times $t\!$ and $t'\!$ respectively, and let $t'-t=\tau \!$. Helmholtz’s reciprocal theorems.As the system is passing through $\mathrm {O} \!$ let a small impulse $\delta p_{r}\!$ be given to it, and let the consequent alteration in the co-ordinate $q_{s}\!$ after the time $\tau \!$ be $\delta q'_{s}$. Next consider the reversed motion of the system, in which it would, if undisturbed, pass from $\mathrm {O} '\!$ to $\mathrm {O}$ in the same time $\tau \!$. Let a small impulse $\delta p'_{s}\!$ be applied as the system is passing through $\mathrm {O} '\!$, and let the consequent change in the co-ordinate $q_{r}\!$ after a time $\tau \!$ be $\delta q_{r}\!$. Helmholtz’s first theorem is to the effect that

$\delta q_{r}:\delta p'_{s}=\delta q'_{s}:\delta p_{r}.$

(6)

To prove this, suppose, in (2), that all the $\delta q\!$ vanish, and likewise all the $\delta p\!$ with the exception of $\delta p_{r}\!$. Further, suppose all the $\Delta q'\!$ to vanish, and likewise all the $\Delta p'\!$ except $\Delta p'_{s}\!$, the formula then gives

$\delta p_{r}\cdot \Delta q_{r}=-\Delta p'_{s}\cdot \delta q'_{s},$

(7)

which is equivalent to Helmholtz’s result, since we may suppose the symbol $\Delta \!$ to refer to the reversed motion, provided we change the signs of the $\Delta p\!$. In the most general motion of a top (Mechanics, § 22), suppose that a small impulsive couple about the vertical produces after a time $\tau \!$ a change $\delta \theta \!$ in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of $\theta \!$ will produce after a time $\tau \!$ a change $\delta \psi \!$, in the azimuth of the axis, which is equal to $\delta \theta \!$. It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. Again, in the corpuscular theory of light let $\mathrm {O} ,\mathrm {O} '\!$ be any two points on the axis of a symmetrical optical combination, and let $\mathrm {V} ,\mathrm {V} '\!$ be the corresponding velocities of light. At $\mathrm {O} \!$ let a small impulse be applied perpendicular to the axis so as to produce an angular deflection $\delta \theta$, and let $\beta '\!$ be the corresponding lateral deviation at $\mathrm {O} '\!$. In like manner in the reversed motion, let a small deflection $\delta \theta '\!$ at $\mathrm {O} '\!$ produce a lateral deviation $\beta$ at $\mathrm {O}$. The theorem (6) asserts that

${\frac {\beta }{\mathrm {V} '\delta \theta '}}={\frac {\beta '}{\mathrm {V} \delta \theta }},$

(8)

or, in optical language, the “apparent distance” of $\mathrm {O}$ from $\mathrm {O} '\!$ is to that of $\mathrm {O} '\!$ from $\mathrm {O}$ in the ratio of the refractive indices at $\mathrm {O} '\!$ and $\mathrm {O}$ respectively.

In the second reciprocal theorem of Helmholtz the configuration $\mathrm {O} \!$ is slightly varied by a change $\delta q_{r}\!$ in one of the co-ordinates, the momenta being all unaltered, and $\delta q'_{s}\!$ is the consequent variation in one of the momenta after time $\tau \!$. Similarly in the reversed motion a change $\delta p'_{s}\!$ produces after time $\tau \!$ a Helmholtz’s second reciprocal theorem.change of momentum $\delta p_{r}\!$. The theorem asserts that

$\delta p'_{s}:\delta q_{r}=\delta p_{r}:\delta q'_{s}$

(9)

This follows at once from (2) if we imagine all the $\delta p\!$ to vanish, and likewise all the $\delta q\!$ save $\delta q_{r}\!$, and if (further) we imagine all the $\Delta p'\!$ to vanish, and all the $\Delta q'\!$ save $\Delta q'_{s}\!$. Reverting to the optical illustration, if $\mathrm {F} ,\mathrm {F} '\!$, be principal foci, we can infer that the convergence at $\mathrm {F} '\!$ of a parallel beam from $\mathrm {F}$ is to the convergence at $\mathrm {F}$ of a parallel beam from $\mathrm {F} '\!$ in the inverse ratio of the refractive indices at $\mathrm {F} '\!$ and $\mathrm {F}$. This is equivalent to Gauss’s relation between the two principal focal lengths of an optical instrument. It may be obtained otherwise as a particular case of (8).

We have by no means exhausted the inferences to be drawn from Lagrange’s formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by R. J. E. Clausius, Helmholtz, Thomson (Lord Kelvin) and Tait, and Lord Rayleigh. In applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium.

It may be worth while to point out, however, that there is no such limitation to the use of Lagrange’s formula (1). In applying it to cyclic systems, it is convenient to introduce conditions already laid down, viz. that the co-ordinates $q_{r}$ are the palpable co-ordinates and that the cyclic momenta are invariable. Special inference can then be drawn as before, but the interpretation cannot be expressed so neatly owing to the non-reversibility of the motion.

Authorities.—The most important and most accessible early authorities are J. L. Lagrange, Mécanique analytique (1st ed. Paris, 1788, 2nd ed. Paris, 1811; reprinted in Œuvres, vols. xi., xii., Paris, 1888–89); Hamilton, “On a General Method in Dynamics,” Phil. Trans. 1834 and 1835; C. G. J. Jacobi, Vorlesungen über Dynamik (Berlin, 1866, reprinted in Werke, Supp.-Bd., Berlin, 1884). An account of the extensive literature on the differential equations of dynamics and on the theory of variation of parameters is given by A. Cayley, “Report on Theoretical Dynamics,” Brit. Assn. Rep. (1857), Mathematical Papers, vol. iii. (Cambridge, 1890). For the modern developments reference may be made to Thomson and Tait, Natural Philosophy (1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord Rayleigh, Theory of Sound, vol. i. (1st ed. London, 1877; 2nd ed. London, 1894); E. J. Routh, Stability of Motion (London, 1877), and Rigid Dynamics (4th ed. London, 1884); H. Helmholtz, “Über die physikalische Bedeutung des Prinzips der kleinsten Action,” Crelle, vol. c., 1886, reprinted (with other cognate papers) in Wiss. Abh. vol. iii. (Leipzig, 1895); J. Larmor, “On Least Action,” Proc. Lond. Math. Soc. vol. xv. (1884); E. T. Whittaker, Analytical Dynamics (Cambridge, 1904). As to the question of stability, reference may be made to H. Poincaré, “Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation” Acta math. vol. vii. (1885); F. Klein and A. Sommerfeld, Theorie des Kreisels, pts. 1, 2 (Leipzig, 1897–1898); A. Lioupanoff and J. Hadamard, Liouville, 5me série, vol. iii. (1897); T. J. I. Bromwich, Proc. Lond. Math. Soc. vol. xxxiii. (1901). A remarkable interpretation of various dynamical principles is given by H. Hertz in his posthumous work Die Prinzipien der Mechanik (Leipzig, 1894), of which an English translation appeared in 1900.  (H. Lb.)