Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/625

 DYNAMICS,

ANALYTICAL

between the same two configurations, these being regarded as defined by the palpable co-ordinates alone. Hamilton's Principal and Characteristic Functions. § 8. In the investigations next to be described a more extended meaning is given to the symbol 5. We will, in the first instance, . denote by it an infinitesimal variation of the most enera function gordinates l hind, affecting notbut merely thefinal coat any instant, also the the values initial ofand configurations and the times of passing through them. If we put S=JtT-Y)dt, . . . (1) we have, then, 5S = (T' - Y')5t' -{T-Y)dt+jtdT - 8Y)dt

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from (10) in (7). Some illustrations of the theory, for the case of a single particle, are given under Mechanics (Ency. Brit. vol. xv. pp. 724-25). The preceding theorems are easily adapted to the case of cyclic systems. We have only to write jt (® - Y)dt = J^(T- kx~ k'x' - ... - Y)dt. (12) in place of (1), and A

-=f(2T-Kx-K'x-...)dt,. . . (13) in place of (8) ; cf. § 7 ad Jin. 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,...qm, 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 q^, ^qm, and of the total energy H, with the cyclic momenta invariable. It will be = (T - Y')St' — (T - V)5£ + F2m(xSx + ydi/ + z5z)~|. . (2) L " Jt found that the forms of (4) and (9) will be conserved, provided the Let us now denote by x' + 8x', y’ + 8y', z' + 8z', the final co-ordinates variations 8qv 8q2,... be understood to refer to the palpable co{i.e., at time t' + 8t') of a particle m. In the terms in (2) which ordinates alone. It follows that the equations (5), (6), and (10),. relate to the upper limit we must therefore write 8x'-x'5t', (11) will still hold under the new meanings of the symbols. Sy' - y'St', 8z' - z'8t' for Sx, oy, 5z. With a similar modification at Reciprocal Properties of Direct and Reversed Motions. the lower limit, we obtain § 9. We may employ Hamilton’s principal function to prove a SS= - H5r + Hm(x'Sx' + y'8y' + z'Sz') very remarkable formula connecting any two slightly — 'Lm{x8x + y8y + iSz),, . (3) disturbed natural motions of the system. If we use grange’s where H( =T + V) is the constant value of the energy in the free the symbols 5 and A to denote the corresponding varia- ormu^a% motion of the system, and r ( = <'-<) is the time of transit. In tions, the theorem is generalized co-ordinates this takes the form -^f<(J)pr.&qr— &pr.dqJ)=-§ . . . (1) 5S= -H8T+pq'1+p'28q'^+... or, integrating from t to f, -PMi-PiPZz(4) How if we select any two arbitrary configurations as initial and 'Z{Sp'Jkq'r-kp'r.Sq'J^^pr.Aq,.-Apr,$qr). . (2) final, it is evident that we can in general (by suitable initial If for shortness we write velocities or impulses) start the system so that it will of itself pass 32S 82S from the first to the second in any prescribed time r. On this (r, s) = (3> view of the matter, S will be a function of the initial and final dqjdq,’ (r, s') = dqrdq'J co-ordinates {qY,q2,... and q,q'2,...) and the time r, as independent we have variables. And we obtain at once from (4) 8pr= -S,(r, s)8ql - 2,(r, s')dq', . . (4), as with a similar expression for Apr. Hence the right-hand side of P-l = 3?Y P 2: (2) becomes (5) as as -2r{2s(r, s)Sqs + Zs{r, s')8q's} Aqr + 2r {2s(r, s)A?s + 2s(r, s^Aq'f^ Pl = i>2= = a?2’ ~Zqi 2r24(r, s')pqr.Aq t — Aqr,8cf , (5) tt_ as and H (6) The same value is obtained in like manner for the expression on ~ ~aY the left hand of (2); hence the theorem, which, in the form (1), S is called by Hamilton the principal function ; if its general form is due to Lagrange, and was employed by him as the basis of his for any system can be found, the preceding equations suffice to method of treating the dynamical theory of Fariation of Arbitrary determine the motion resulting from any given conditions. If we Constants. substitute the values of pY, p2,... and H from (5) and (6) in the exThe formula (2) leads at once to some remarkable reciprocal relations pression for the kinetic energy in the form T' (see § 1), the which were first expressed,_ in their complete form, by Helmholtz. equation Consider any natural motion of a conservative system He,m T' + V = H. . . . (7) between two configurations 0 and O' through which it ho,tz ' passes at times t and t' respectively, and let t'-t = r. r l rocaI ’s becomes a partial differential equation to be satisfied by S. It has As the system is passing through 0 let a small impulse ff P been shown by Jacobi that the dynamical problem resolves itself tbeorems into obtaining a “complete” solution of this equation, involving 8pr be given to it, and let the consequent alteration in n + 1 arbitrary constants. This aspect of the subject, as a problem the co-ordinate qs after the time r be 8q'a. Next consider the in partial differential equations, has received great attention at the reversed motion of the system, in which it would, if undisturbed, pass from O' to 0 in the same time r. Let a small impulse 8p's hands of mathematicians, but must be passed over here. be applied as the system is passing through O', and let the conseThere is a similar theory for the function quent change in the co-ordinate qr after a time r be bqr. HelmCharacter*A = 2 fTdt-S + Hr. . . V(8) holtz’s first theorem is to the effect that istic J ' function. Jt follows from (4) that 8qr-dp', = 8q',-.8pr. . . . (6) To prove this, suppose, in (2), that all the 8q vanish, and likewise 5A = r5H +p'18q +p'fcf^ +... all the 8p with the exception of 8pr. Further, suppose all the Af to vanish, and likewise all the Ap' except Ap’s, the formula then -PHx-PMz(9) Selecting, as before, any two arbitrary configurations, it is in gives general possible to start the system from one of these, with a 8pr.Aqr= -Ap's.8q's,. . . (7) prescribed value of the total energy H, so that it shall pass through which is equivalent to Helmholtz’s result, since we may suppose the other. Hence, regarding A as a function of the initial and the symbol A to refer to the reversed motion, provided we change final co-ordinates and the energy, we find the signs of the Ap. For example, in the most general motion or a top (§ 2)> suppose that a small impulsive couple about the , aA, 3A vertical produces after a time r a change 56 in the inclination of the axis, the theorem asserts that in the reversed motion an equal im(10) 0A pulsive couple in the plane of 6 will produce after a time r a change Pl ?>2_ 8f/, m the azimuth of the axis, which is equal to 86. It is under- ~ a9l’ a?2’ ••• ’ stood, of course, that the couples have no components (in the aA and T— (ii) generalized sense) except of the types indicated ; for instance, they SH" consist in each case of a force applied to the top at a point of A is called by Hamilton the characteristic function ; it represents, may the axis, and of the accompanying reaction at the pivot. Again, in of course, the “action” of the system in the free motion (with the corpuscular theory of light let 0, O' be any two points on the axis prescribed energy) between the two configurations. Like S, it of a symmetrical optical combination, and let V, V' be the correspondsatisfies a partial differential equation, obtained by substitution ing velocities of light. At 0 let a small impulse be applied perpen-