DYNAMICS, ANALYTICAL 567 dynamics were first recognized by Helmholtz in the domain of and it is evident that the terms in 0 which are bilinear in respect acoustics ; their use has been greatly extended by Lord Rayleigh. of the two sets of variables q^ q<i,...qm and x, x', x",...will disThe equations (13) determine the momenta yq, y?2,... as linear appear from the right-hand side. functions of the velocities yy, y2,... . Solving these, we can express It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin Velocities e( ua 0Ilsas ^venear us functions ofyq, y?2,.... The resulting la terms ot s is ern ^ S^ velocities produced by any given respectively. Let us suppose, in the first place, that momenta. we ^ ^can express impulses. Further, by substitution in (8), the system is started by given impulses of certain aa the kinetic energy as a homogeneous types, but is otherwise free. Bertrand’s theorem is to . . quadratic function of the momentayq, y?2,... . The kinetic energy, the effect that the kinetic energy is greater than if by m n mum energ as so expressed, will be denoted by T'; thus impulses of the remaining types the system were conxstrained to take any other course. We may suppose the co2T' = A'1Iy312 + A'22p22+... + 2A'12p-y?2+... . (19) to be so chosen that the constraint is expressed by the where A jj, A^,... A'12, ••• are certain coefficients depending on the ordinates vanishing of the velocities qx, <)2, ...qm, whilst the given impulses configuration. They have been called by Maxwell the coefficients are x, x', x",.... Hence the energy in the actual motion is greater of mobility of the system. When the form (19) is given, the values in the constrained motion by the amount (s>. of the velocities in terms of the momenta can be expressed in a re- than Again, suppose that the system is started with prescribed markable form due to Hamilton. The formula (15) may be written velocity components q^ g2,... qm, by means of proper impulses of the corresponding types, but is otherwise free, so that in the Mi+M2 + -=T + T', . . . (20) where T is supposed expressed as in (8), and T' as in (19). Hence motion actually generated we have x = 0, k' — 0, x" = 0,... and thereif, for the moment, we denote by 5 a variation affecting the velocities, fore K=:0. The kinetic energy is therefore less than in any other and therefore the momenta, but not the configuration, we have ' motion consistent with the prescribed velocity-conditions by the value which K assumes when k, k', k", ... represent the impulses due PMi + ffi5y? +p^q2 + q2bp2 + ... = 5T + 5T' to the constraints. Simple illustrations of these theorems are afforded by the chain _0T 0T , 0T' , 0T' (21) of straight links already employed. Thus if a point of the chain ?1 + 0^ 22+- + ^1 + ^2+" be held fixed, or if one or more of the joints be made rigid, the In virtue of (13) this reduces to energy generated by any given impulses is less than if the chain 0T' RT' had possessed its former freedom. Wi>1+jA+... = ¥;5fl+^a (22) Continuous Motion of a System. Since 5yq, dp2,... may be taken to be independent, we infer that § 2. We may proceed to the continuous motion of a system. . 0T' . 3T' The equations of motion of any particle of the system are of the qi (23) form c>Pi q*~dpz mx=X, my—Y, mz-Z. . . (1) In the very remarkable exposition of the matter given by Maxwell in his Electricity^ and Magnetism, the Hamiltonian expressions Now let x + 5x, y + by, z -f 5^ be the co - ordinates of m in any (23) for the velocities in terms of the impulses are obtained directly arbitrary motion of the system differing infinitely La . , from first principles, and the formulae (13) are then deduced by an little from the actual motion, and let u>- form the gran%e s inversion of the above argument. equation equations. An important modification of the above process has been intro'Zm{x8x + yby + zSz) = 'Z{X5x + Yby + 7ibz). . (2) duced in recent times by Routh and Thomson and Tait. Instead Lagrange’s investigation consists in the transformation of (2) into 0 ex ress n Routh’s .^es aPon i org the kinetic energy in terms of the veloci- an equation involving the independent variations Sqi, Sq2,... Sq„. modiflca- ^ ^ p> . iu terms of the momenta alone, we may It is important to notice that the symbols S and djdt are comtlon. express it inco-ordinates, terms of thesay velocities corresponding to mutative, since some of the q2, ... qm, and of the .. d. , . . dx momenta corresponding to the remaining co-ordinates, which (for Sz=^(* + te)-s=#',4c. • • (3) the sake of distinction) we may denote by x i x',--- • Thus, Hence T being expressed as a homogeneous quadratic function of qx, q2, ••• x %">•••, the momenta corresponding to the co-ordinates 2m(xdx + ydy + zdz) = ~2m(xSx + ydy + z5z) X> x'j xV- may be written - 2m(xdx + y5y + zdz) 0T , 0T _ 0T (24) K K K = j (Pifyi+P2Sq2+...)-5T,. . . (4) ~ 0x’ ~ dx” t “0X"’ These equations, when written out in full, determine %, ... by § 1 (14). The last member may be written as linear functions of yp fe... qm, k, k' , k" We now consider the function PlSSl +PlSil +P‘25P2 +P2Si-2 + — K 0T 0T 0T 0W 0 = T - xx - 'x' ~ k"x!' (25) supposed expressed, by means of the above relations, in terms of iii 9l 02' qi 0225^2 cq2 q2 ' 5 in ?2)-” im, x, x', k",... . Performing the operation 5 on both sides Hence, omitting the terms which cancel in virtue of § 1 (13), we of (25), we have find 2m(*5ce + yby + z8z) = ^A~5^ + ^2-8q2+... . (6) dx For the right-hand side of (2) we have — x5x-xSx—..., . . (26) where, for brevity, only one term of each type has been exhibited. 2(X5a; + Y52/ + Z5z) = Q15g1 + Q25g'2+..., . (7) Omitting the terms which cancel in virtue of '24), we have x|^+y|^+z|^ (8) ^I+...+^x+... = 0^1 + ...—x5x— (27) •=< Ctyr Utyr CQr )■ The quantities Qj, Q2,... are called the generalized components Since the variations Sq^ dq2,... Sqm, 5x, 8k', 5k",... may be taken to force acting on the system. be independent, we have Comparing (6) and (7) we find 0T 00 _ 0T_ 00 9T P2 (28) Pl= a-?=q1 P2• 0?2_n (%i '02i’ '~dq2~dq2’ ~Q2 (9) 00 00 00 and or, restoring the values of yq, p ,..., = (29) 2
- -0? * = “ 0x” X = - W’
d/0T 0T d 0T An important property of the present transformation is that, (10) — Q2 dt V Cip 02i — Qi> dt when expressed in terms of the new variables, the kinetic energy is 072 the sum of two homogeneous quadratic functions, thus These are Lagrange’s general equations of motion. Their number is of course equal to that of the co-ordinates qv gq,... to be deterT = (5 + K, .... (30) where © involves the velocities q^ q2,... qm alone, and K thi mined. the above proof is that given by Lagrange, but momenta x, x', x",... alone. For in virtue of (29) we have, fron theAnalytically, terminology employed is of much more recent date, having (25), been first introduced by Thomson and Tait; it has greatly pro. ..,00 00 , moted the physical application of the subject. Another proof of T = 0 - . SO o5- , x„ 0^( (31) dx + X dx the equations (10), by direct transformation of co-ordinates, has
