ANALYTICAL 2 been given by Hamilton and other writers, but the variational Hence A0 - A sin 0 cos 0i/' (24) 2 + v sin 6^ = Mg* sin 0, method of Lagrange is that which stands in closest relation to the A sin 0i/' + v cos 9 — p, / subsequent developments of the subject. The chapter of Maxwell, or, eliminating yp, already referred to, is a most instructive commentary on the sub(y-v cos 0)(|U3 cos 0 -v) = Mg* sin 0. (25) ject from the physical point of view, although the proof there A0A sin 0 attempted of the equations (10) is fallacious. In a “ conservative system ” the work which would have to be An immediate application of (24) is to find the condition for a done by extraneous forces to bring the system from rest in some steady precessional motion, i.e., one in which the axis of the top standard configuration to rest in the configuration (qx, q^,- ■ ■ qn) describes a right circular cone about the vertical. Putting 0 = 0, is independent of the path, and may therefore be regarded as a 0 = a, in (24), we obtain definite function of ft, ... ?»• Denoting this function (the A cos a^2-^-!-Mg* = 0. . . • (26) potential energy) by V, we have, if there be no extraneous force For given values of v and a we have two possible values of p, proon the system, 2(X5a; + YSy + Z5z) = - 5V, . . (11) vided v exceed a certain limit. With very rapid rotation, or more precisely, with v large in comparison with N/(4AMyA cos a), one and therefore value of Y is small and the other large, viz., the two values are av av Qs Qi—, pr“, — • • (-^ Mg*/i' and vjK cos a, approximately. To find the small Small 8ft 8ft oscillation about the steady motion we put 0 = a + % m oscillaHence the typical Lagrange’s equation may be now written in (25), and neglect terms of the second order in %. The tions. the form usual method leads to an equation of the type 2 ^/ar _ar = _av X+^ X=0> • ... (27) (13) dtcqr) dir 8ft’ where or. again, „ {p-v cos a)2 + 2{p-v cos a)(y. cos a - p) cos a + {p cos a - r)2^^ _ P ~~ A2 sin4a (14) • • When v is large we find that for the slow precession p = v/A, It has been proposed by Helmholtz to give the name kinetic whilst for the rapid precession p = v/Acosa = ip, approximately. potential to the combination Y - T. . Further, on examining the small variation in p, it appears that m To calculate the rate at which the kinetic energy varies m the a slightly disturbed slow precession the motion of any point of the actual motion we have axis consists of a rapid circular motion superposed on the slow precession, so that the resultant path has a trochoidal character. dT d. . , . ^ 2 This is a type of motion commonly observed in a top spun in the sr=<«<Ml+Ml+' ) = Mi + Mi +Ma + M2 + ' ordinary way, although the successive undulations oi the trochoids 0T.. 0TI.. , ST 2 may be too small to be directly observed. In a slightly disturbed (15) i+ +Q ^ +--- +^0'l + gT?2+•• Oft cq (s 2 8ft rapid precession the added vibration is elliptic-harmonic, with a period equal to that of the precession itself. The ratio of the axes by (9). This may be written of the ellipse is sec a, the major axis being in the plane of 0. The ^ ./h+ n2dT dT+qiqi+Q (16) result is that the axis describes a circular cone about a fixed line dt=dr ‘ making a small angle with the vertical. Another view of the matter is appropriate when we study the small oscillations about (17) —Qkii + Q2Y2 + ••• the vertical position a = 0. The motion of any point of the axis may then be described as an elliptic-harmonic motion superposed This expresses that the kinetic energy is increasing at a rate equal on a uniform rotation with angular velocity v/2A about the vertical to that at which work is being done by the forces. In the case of OZ. The period of revolution in the ellipse is 2irlp, where a conservative system free from extraneous force, we have, sub, v2 - 4 AMg* (29) stituting the values of Q2,••• from (12), (18)
- (T + V) = 0 or T-fV = const.,
This would indicate that the upright position of a top (with the 2 centre of gravity above the pivot) is stable if, and only if, which is the equation of energy. ,• • >> >-4AMgA. See, however, § 6. A classical example of the application of Lagrange s equations is Constrained Systems. to the motion of a top. A rigid body, symmetrical about an axis, is supposed free to turn 3»bout 3, fixed point O in tins r § 3. It has so far been assumed that the geometrical relations, if Applica- axig at a distance h (say) from the centre of mass. any, which exist between the parts of the system case 0f tlon to the Let ’9 be the angle which OG makes with the vertical are of the type § 1 (1), andvarious so do not contain t ex- varying top ‘ OZ, p the azimuth of the plane ZOG relative to a fixed plicitly. The extension of Lagrange’s equations to the reiations. vertical plane ZOX, and <t> the angle which a meridian plane fixed case of “varying relations” of the type x=f(t, qi, q2>... qn), y = &c.,z=kc. . . (1) in the solid makes with the plane ZOG, the notation being m fact that of Mechanics, § 81. Denoting the principal moments was made by Vieille. We now have of inertia by A, A,C, we have . dx dx.dx &c., &c., • (2) 2T = A(w2 + w2) + Cm2 si?5+' 2 2 2 2 = A(# + sin 0^ ) + C{4> + P cos 0) ... . (19) • (3) s*=^s?1 + |!8?2+..., Sc., 4c., If X, p, v be the corresponding components of momentum, we have so that the expression § 1 (8) for the kinetic energy is to be 2 cos cos d =^ > 00 = A0 u=S' Cl/' = A sin 0d' + C(0 + ’A '(20). replaced by 2T = a0 + 2a1ft + 2a.2ft + ... + Auft2 + A^ft2 + ... + A12ftft +..., (4) r = ^ = C(</> + yp cos 0).... where a 2 The geometrical meaning of these quantities is recognized at once 0 = 2m{(|) +(|) +(!) }’] from the expression for the virtual moment of the impulses, (5) (dx dx dy dy dz dz [ S9 + p5p + vd<p, . . . (21) ft—2m|^ dq+ dt dq+ dt dqj y J viz., X is the angular momentum about an axis normal to the plane of 6 p is the angular momentum about the vertical, and v that and the forms of Am Ars are as given by § 1 (7). It is to be about the axis OG. If M be the total mass, the potential energy remembered that the coefficients a0, a1( a2, ... An, A^,... A12... ivni in general involve t explicitly as well as implicitly through the V = Mg*cos0. . . . (22) co-ordinates ft, ft, ••• • Again, we find 2to(cc5x + ydy + z5z) = (aj + Auft -1- A12ft + • • ■ )^ft Hence the equations (10) become + (ft + A21ft + A22y2 -(-.. .)^ft + • • • A0 - A sin 0 cos 0t/'2 + C(</> + ’A cos 9)yp sin 0 = Mg* sin 0,' 2 a.}, i rVA_l_ A r*n« d { t O.CS 9 =0, , 0T. ^ _ 3T, - jA sin 9p + C(j> + yp cos 0) cos 0} (23) _ dtx 0ft z=p.lSq1+p2Sq2+ Ci^ + yp cos 0) = O, where pr is defined as in § 1 (13). The derivation of Lagrange’s of which the last two express the constancy of the momenta p, v.
568
DYNAMICS,
