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Originally appearing in Volume V08, Page 764 of the 1911 Encyclopedia Britannica.
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CALCULUS OF. 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 S f vds=0. . (to) On the corpuscular theory of light v is proportional to the refractive indexµ of the medium, whence S f Ads =0. . . . (II) In the formula (2) the energy in the hypothetical motion is pre-scribed, whilst the time of transit from the initial to the final con- figuration is variable. In another and generally more Hamilton- convenient theorem, due to Hamilton, the time of transit Ian prin- is prescribed to be the same as in the actual motion, whilst ciple. the energy may be different and need not (indeed) be constant. Under these conditions we have s f t'(T —V)dt =0, . . (12) c where t, t' are the prescribed times of passing through the given initial and final configurations. The proof of (12) is simple; we have SJ t(T—V)dt= f: (ST—SV)dt= f i (Mm(±U ySj/+ibi)—SV}dt = [Em(isx+ioy+zoz)] " t — f:' 12m (lax+pay+2Sz) +SV } dt. 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 (12) 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 f (ST—SV)dt= f /aq Sq+a4 aq Sql—... dt = [p1Sq+p2Sg2+...] o' _it" 1 (P1_aq +aq) Eqi+ (1~2—aQz+aQ2 Eq2+... dt.(14) 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 SQ1, Sq2,... 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 (12) 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 J. Larmor. Extension If we write, as in § 1 (25), to cyclic systems. R=T—KX–K'X'–K"X"—..., . (15) we shall have of (R—V)dt=0, . c provided that the variation does not affect the cyclic momenta s, K', K",..., and that the configurations at times t and t' are unaltered, so far as they depend on the palpable co-ordinates qi, gz,• The initial and final values of the ignored co-ordinates will in general be affected. To prove (16) we have, on the above understandings, S f: (R —V)dt= f (ST—KEX—...—SV)dt t' aT OT = f (a-Eq~+...+ag-Oa'+... –EV) dl, . (17)where terms have been cancelled in virtue of § 5 (2). The last member of (17) represents a variation of the integral f t'(T—V)dt on the supposition that SX=o, 3X' =o, SX"=o,... throughout, whilst Sqi, Sq2, Sqm 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 A= f (2T—KX—KX'——...)dt, . (18) then 6A =0, . . (19) provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these .being regarded as defined by the palpable co-ordinates alone. § 8. Hamilton's Principal and Characteristic Functions. In the investigations next to be described a more extended meaning is given to the symbol S. We will, in the first instance, denote by it an infinitesimal variation of the most Principal general kind, affecting not merely the values of the co- function. ordinates at any instant, but also the initial and final con-figurations and the times of passing through them. If we put s= f t'(T—V)dt, . (1) we have, then, SS=(T'—V')St'—(T—V)St+ f ` (ST—SV)dt _ (T'—V') at' — (T —V) St+ [Em (AEx+DEy+zOz) ] . (2) Let us now denote by x'+Sx', y'+Sy', z'+Ez', the final co-ordinates (i.e. at time t'+St') of a particle m. In the terms in (2) which relate to the upper limit we must therefore write Ox'—x'St', 6z'—2'St' for Sx, Sy, Sz. With a similar modification at the lower limit, we obtain SS =—HSr+gym (x'Sx'+y'Sy'+t'Sz') Em(tOx+iOy+zEz), . . (3) where H(=T+V) is the constant value of the energy in the free motion of the system, and r(=t'—t) is the time of transit. In generalized co-ordinates this takes the form SS = —HSr+p'lOq'i+p'23q'2+... piEgi—p2Eg2—.... . (4) Now if we select any two arbitrary configurations as initial and final, it is evident that we can in general (by suitable initial velocities or impulses) start the system so that it will of itself pass from the first to the second in any prescribed time r. On this view of the matter, S will be a function of the initial and final co-ordinates (qi, q2,... and q'l, q'2,...) and the time r, as independent variables. And we obtain at once from (4) , aS aS P _ , —aq, y'2 =—+ a2, ... , OS aS Pi = –agl,p2=–age, ... , and H= Or S is called by Hamilton the principal function; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. If we substitute the values of p,, p2,... and H from (5) and (6) in the expression for the kinetic energy in the form T' (see § I), the equation TI+V=H (7) becomes a partial differential equation to be satisfied by S. It has been shown by Jacobi that the dynamical problem resolves itself into obtaining a " complete " solution of this equation, involving n+I arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here. There is a similar theory for the function (8) Characteristic It follows from (4) that function. SA = rSH+p'iOq'i+p'2Sq'z+... — piOgi — p2E42 — (9) This formula (it may be remarked) contains the principle of " least . (13) . (16) • (5) . (6) A =2 fTdt=S+Hr. . . 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 H, so that it shall pass through the other. Hence, regarding A as a function of the initial and final co-ordinates and the energy, we find aA aA pi=aq,r p2=aq,,, ... , aA aA Pi = - dql; Pi = - h, .. . 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 S, it satisfies a partial differential equation, obtained by substitution from (I o) in (7). The preceding theorems are easily adapted to the case of cyclic systems. We have only to write S= f: (R-V)dt= f: (T-KX-K',y''-...-V)dt . (12) in place of (I), and A= f (2T-KX—K'X'-...)dt, . . . (3) in place of (8); cf. § 7 ad fin. It is understood, of course, that in (12) S is regarded as a function of the initial and final values of the palpable co-ordinates q2,...q,,,, and of the time of transit r, the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of qi, Q2,, and of the total energy H, with the cyclic momenta invariable. It will be found that the forms of (4) and (9) will be conserved, provided the variations 6q1, Sq2,... be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (Io), (II) 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 La- remarkable formula connecting any two slightly disturbed grange's natural motions of the system. If we use the symbols forma/a. S and i to denote the corresponding variations, the theorem is dtZ(bpr.Agr opr.bgr) =0; . or. integrating from t to t', Aq'r-Aq''r) =Z(Spr.Agr-Apr•Sq,). If for shortness we write a'S a~. (r, s) = s') = aa qr g,e, agraq, we have bQr=s)Sq.-E,(r, s')5g'. . . (4) with a similar expression for Apr. Hence the right-hand side of (2) becomes s)Sq,+E,(r, s')Sq',}~qr Er}£. r, s).q.,+E,(r, s')Oq',15gr =2r2,(r, s')(Sgr..7q',',} . . (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 (I), 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 0 and 0' through which it passes at times t and t' respectively, and let I'-t=r. As the system is passing through 0 let a small impulse bp, be given to it, and let the consequent alteration in the co-ordinate q, after the time r be Sq',. Next consider the reversed motion of the system, in which it would, if undisturbed, pass from 0' to 0 in the same time r. Let a small impulse Sp', be applied as the system is passing through 0', and let the consequent change in the co-ordinate qr after a time r be Sqr. Helmholtz's first theorem is to the effect that Sqr: bp', =bq',: Spr• (6) To prove this, suppose, in (2), that all the Sq vanish, and likewise all the Sp with the exception of bp,. Further, suppose all the Aq' to vanish, and likewise all the op' except op'„ the formula then gives bpr•Ogr = -Ap',-Sq'„ (7) which is equivalent to Helmholtz's result, since we may suppose the symbol A to refer to the reversed motion, provided we 763 change the signs of the op. In the most general motion of a top (MECHANICS, § 22), suppose that a small impulsive couple about the vertical produces after a time r a change S8 in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of 8 will produce after a time r a change Sip, in the azimuth of the axis, which is equal to M. It is under-stood, 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 0, Of be any two points on the axis of a symmetrical optical combination, and let V, V' be the corresponding velocities of light. At 0 let a small impulse be applied perpendicular to the axis so as to produce an angular deflection SO, and let be the corresponding lateral deviation at 0'. In like manner in the reversed motion, let a small deflection SO' at 0' produce a lateral deviation (3 at O. The theorem (6) asserts that R Of V~-=Vbe or, in optical language, the " apparent distance " of 0 from 0' is to that of 0' from 0 in the ratio of the refractive indices at 0' and 0 respectively. In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change Sq,- in one of the co- Helm-ordinates, the momenta being all unaltered, and Sq'. is ho/tz's the consequent variation in one of the momenta after second time T. Similarly in the reversed' motion a change Sp', reciprocal produces after time r a change of momentum Spr. The theorem. theorem asserts that Sp',:Sgr=Spr:Sq'. . (9) This follows at once from (2) if we imagine all the Sp to vanish, and likewise all the Sq save Sqr, and if (further) we imagine all the op' to vanish, and all the Oq' save Aq',. Reverting to the optical illustration, if F, F', be principal foci, we can infer that the convergence at F' of a parallel beam from F is to the convergence at F of a parallel beam from F' in the inverse ratio of the refractive indices at F' and 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, 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. Uber 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. Poincare, " Sur 1'equilibre d'une masse fluide animee d'un mouvement de rotation " Acta math. vol. vii. (1885) ; F. Klein and A. Sommerfeld, Theorie des Kreisels, pts. I, 2 (Leipzig, 1897-1898); A. Lioupanoff and J. Hadamard, Liouville, 5me serge, vol. iii. (1897); T. J. I. Bromwich, Proc. Land. 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. Ls.) and aA r-aH . (Io) . (II) Helmholtz's reciprocal theorems. (8)
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