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CALCULUS OF

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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 See also:

action with a well-known theorem of See also:optics may be noticed. For the See also:motion of a particle in a conservative See also:field of force the principle takes the See also:form S f vds=0. . (to) On the corpuscular theory of See also:light v is proportional to the refractive indexµ of the See also:medium, whence S f Ads =0. . . . (II) In the See also:formula (2) the See also:energy in the hypothetical motion is pre-scribed, whilst the See also:time of transit from the initial to the final See also:con- figuration is variable. In another and generally more See also: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 See also: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 See also:proof of (12) is See also: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 See also:hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d'See also: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 See also:system of co-ordinates.

Thus, to deduce See also:

Lagrange's equations, we have f (ST—SV)dt= f /aq Sq+See also: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 See also:order that the See also:remainder of the right-See also: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 See also:variations in question are See also:independent,. and subject only to the See also:condition of vanishing at the limits of integration. We are thus led to Lagrange's See also: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 See also:ordinary See also:dynamics. The modification of the Hamiltonian principle appropriate to the See also:case of cyclic systems has been given by J. Larmor. See also: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,...gm• The initial and final values of the ignored co-ordinates will in See also: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 See also:

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 See also: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 See also:Principal and Characteristic Functions. In the investigations next to be described a more extended meaning is given to the See also:symbol S. We will, in the first instance, denote by it an infinitesimal variation of the most Principal general See also:kind, affecting not merely the values of the co- See also: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+See also: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 See also:

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 See also: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 See also: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 =—+ See also: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 See also:differential equation to be satisfied by S. It has been shown by See also:Jacobi that the dynamical problem resolves itself into obtaining a " See also:complete " See also:solution of this equation, involving n+I arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received See also:great See also: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 See also:

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,...gm, 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 See also:

Direct and Reversed Motions. We may employ Hamilton's principal function to prove a very La- remarkable formula connecting any two slightly disturbed See also: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 See also:dtZ(bpr.Agr opr.bgr) =0; . or. integrating from t to t', Aq'r-Aq'r.bq'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 See also: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',-ogr.bq',} . . (5) The •same value is obtained in like manner for the expression on the See also: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 See also: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-See also:

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 See also: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 See also: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 See also:top (See also:MECHANICS, § 22), suppose that a small impulsive couple about the See also:vertical produces after a time r a change S8 in the inclination of the See also:axis, the theorem asserts that in the reversed motion an equal impulsive couple in the See also:plane of 8 will produce after a time r a change Sip, in the See also: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 See also:pivot. Again, in the corpuscular theory of light let 0, Of be any two points on the axis of a symmetrical See also:optical See also: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 See also: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 See also:

save Sqr, and if (further) we imagine all the op' to vanish, and all the Oq' save Aq',. Reverting to the optical See also:illustration, if F, F', be principal foci, we can infer that the convergence at F' of a parallel See also: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 See also:Gauss's relation between the two principal See also:focal lengths of an optical See also:instrument. It may be obtained otherwise as a particular case of (8). We have by no means exhausted the inferences to be See also:drawn from Lagrange's formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and See also:acoustics formulated by R. J. E. See also:Clausius, Helmholtz, See also:Thomson (See also:Lord See also:Kelvin) and See also:Tait, and Lord See also: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 See also:rest.

Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the See also:

propagation of See also:sound in a See also:wind and the propagation of light in a magnetic medium. It may be See also:worth while to point out, however, that there is no such See also: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. See also:Special inference can then be drawn as before, but the See also:interpretation cannot be expressed so neatly owing to the non-reversibility of the motion. Uber See also:die physikalische Bedeutung See also:des Prinzips der kleinsten Action," Crelle, vol. c., 1886, reprinted (with other cognate papers) in Wiss. Abh. vol. iii. (See also:Leipzig, 1895); J. Larmor, " On Least Action," Proc. Lond. Math. See also:Soc. vol. xv. (1884) ; E.

T. Whittaker, See also:

Analytical Dynamics (See also:Cambridge, 1904). As to the question of stability, reference may be made to H. See also:Poincare, " Sur 1'equilibre d'une masse fluide animee d'un mouvement de rotation " Acta math. vol. vii. (1885) ; F. See also:Klein and A. See also: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. See also:

Land. Math. Soc. vol. xxxiii. (1901). A remarkable interpretation of various dynamical principles is given by H. See also:Hertz in his See also:posthumous See also:work Die Prinzipien der Mechanik (Leipzig, 1894), of which an See also:English See also:translation appeared in 1900. (H. Ls.) and aA r-aH . (Io) . (II) Helmholtz's reciprocal theorems.

End of Article: CALCULUS OF

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