DYNAMICS, ANALYTICAL 569 «quations then follows exactly as before. It is to be noted that (22) the equation § 2 (17) does not as a rule now hold. The proof in- T0) — Q^i + Qtfh + • • volved the assumption that T is a homogeneous quadratic function or, in the case of conservative forces, of the velocities q2.... It has been pointed out by Hayward that Yieille’s case can be <§ +V -T =const. . . . (23) brought under Lagrange s by introducing a new co-ordinate {6) in This may be called the equation0 of relative energy. It may, of place of t, so far as it appears explicitly in the relations (1). We course, be easily established from first principles without the use have then of generalized co-ordinates. 2T=a0$2 + 2{alq1 + a^h+...)6 + kuq* + A^2 +...+ 2A12£tf2 + ... . We have still to notice the modifications which Lagrange’s (7) equations undergo when the co-ordinates gq, q2,...q„ ConThe equations of motion will be as in § 2 (10), with the additional are not all independently variable. In the first place, strained we may suppose them connected by a number m (< ft) systems. equation of relations of the type rfST_ST 8 Mt, <h, g'2,...g'„)=0, B(£, ft, ft,...g'm) = 0, &c., (24) dt dd 90“°’ • • • • () where 0 is the force corresponding to the co-ordinate 6. We may These may be interpreted as introducing partial constraints into suppose 0 to be adjusted so as to make 0 = 0, and in the remaining a previously free system. The variations 6ft, 8q2,...8qn in the exequations nothing is altered if we write t for 0 before, instead of pressions (6) and (7) of § 2 which are to be equated are no longer after, the differentiations. The reason why the equation § 2 (17) independent, but are subject to the relations no longer holds is that we should require to add a term 00 on the !*h + !3fe+... = 0, |{jl + ®J?8+...=0, fcc. (25) right-hand side; this represents the rate at which work is being done by the constraining forces required to keep 0 constant. As an example, let x, y, z be the co-ordinates of a particle relative Introducing indeterminate multipliers X, y,..., one for each of to axes fixed in a solid which is free to rotate about the axis of z. these equations, we obtain in the usual manner n equations of the If 0 be the angular co-ordinate of the solid, we find without type „ d „0T 0T _ 0A 0B difficulty 2 2 2 2 2 dtWr~dqr~Qr + r + f*dqr+-’ ‘ ’ (26) 2T = + # + z ) + 2()m{xy - yx) + {I + to(cc + y )} 0 , . (9) in place of § 2 (10). These equations, together with (24), serve where I is the moment of inertia of the solid. The equations of to determine the n co-ordinates g1( q^...qn and the m multipliers motion, viz., X, y, ... . Again, it may happen that although there are no prescribed d_ 9T _9T_x Y (10) relations between the co-ordinates ft, q2,...qn, yet from the cirdt dx dx ’ dt dy dy~ ' dt dz 0s cumstances of the problem certain geometrical conditions are im^0T_0T posed on their variations, thus and (11) dtdO 00 AiSft-fiA25g2+... =0, BjSft-fB25g2+... = 0, &c., (27) become 2 2 where the coefficients are functions of ft, q2,...qn and (possibly) m(x - 20# - a;0 - yd) = X, m(y + 20* -1/0 + £C0) = Y, ms=Z, (12) of t. It is assumed that these equations are not integrable as regards the variables ft, g^,... .qn ; otherwise, we fall back on the and ^[{(l + ™(a:2 + y2)}0 + m(a#-i/:r)] = 0. . . (13) previous conditions. Cases of the present type arise, for instance, ordinary dynamics when we have a solid rolling on a (fixed or If we suppose 0 adjusted so as to maintain 0 = 0, or (again) if in surface. The six co-ordinates which serve to specify the we suppose the moment of inertia I to be infinitely great, we moving) position of the solid at any instant are not subject to any necessary obtain the familiar equations of motion relative to moving axes, viz., relation, but the conditions to be satisfied at the point of contact impose three conditions of the form (27). The general equations m(*-2wy-w2x) = X, m(y + 2wa;-w2y)=Y, ms = Z, (14) of motion are obtained, as before, by the method of indeterminate where w has been written for 0. These are the equations which multipliers, thus we should have obtained by applying Lagrange’s rule at once to <7 0T 0T ^ A 4 the formula (28) ^0^"0^~Qr + XAr + MBr+ 1 1 1 2 l 1 2T=m{x + y + z‘ ) + 2mu{xy-yx) + mu [x + y ), . (15) The co-ordinates gq, q2,...qn, and the indeterminate multipliers which gives the kinetic energy of the particle referred to axes X, y,... are determined by these equations and by the velocityrotating with the constant angular velocity w. conditions corresponding to (27). When t does not appear exMore generally, we might apply Lagrange’s method to find the plicitly in the coefficients, these velocity-conditions take the forms Aift + A2g2 + ... = 0, Bjft + B^ +... = 0, &c. . (29) Rotating equations of motion of a system whose configuration axes. relative to axes by rotating angular velocity (w) is defined meanswith of constant generalized co-ordinates Hamiltonian Equations of Motion. 2i> to,-- 1r>, writing § 4. In the Hamiltonian form of the equations of motion of a 2 2 2 2 2 2 2T=2m(a: + y + z ) + 2wLm{xy - yx) + w 2m(x + y ) conservative system with unvarying relations, the kinetic energy is supposed expressed in terms of the momenta px,p2,... and the = 2<§ + 2w2m(a;y - ycc) + 2T0, . . . . (16) say. This problem is interesting on account of its bearing on the co-ordinates ft, g'2,..., as in § 1 (19). Since the symbol 8 now denotes a variation extending to the co-ordinates as well as to the kinetic theory of the tides. The details of the work would occupy momenta, we must add to the last member of § 1 (21) terms of the too much space, but the result may be stated. Assuming that the Cartesian co-ordinates x, y, z of any particle relative to the types moving axes are functions of g-j, q2)... qn, of the form § 1 (1), we <9 find, after cancelling a number of terms, that the typical equation is Since the variations Sjq, 5p2,... Sgq, 5g2,... may be taken to be independent, we infer the equations § 1 (23) as before, together with dt dqr “ 0^r + (r>1 + 2)?2 + • • • + (r>s)7« + • • • ~ = Qr. (17) 0T_ _0T 0T_ _0T where (r, s) = 2w. 2to 0(a;, y) (18) 0ft dqx 0ft~ 0ft’ ' ‘ ' S(5'« <lr)' Hence the Lagrangian equations § 2 (14) transform into It is to be noticed that (19) (r, r) = 0, (r, s) = - (s, r). A (3) =-i;<T'+v>The conditions of relative equilibrium are If we write H = T' + V (4) (20) ^r- cqr ’
- .
so that H denotes the total energy of the system, supposed exor, in case the forces Qr depend only on the co-ordinates q1: q2,..., pressed in terms of the new variables, we get and are conservative, 0H . 0H Vl ( } L(V-To) = 0, .... (21) ~ 0ft’ -F2_ 0ft’ If to these we join the equations i.e., the value of V - T0 must be stationary. 0H . 0H If we multiply (17) by qr, and sum the result for r=l, 2, 3,... n, (6) ft = 0p1’ we find, taking account of (19), 0/V S. III.— 72
