IM (K2+a2+h2—2 ah cos 0) 62—Mgh cos B—const. (14)
Whenever, as in the preceding examples, a body or a system of bodies, is subject to constraints which leave it virtually only one degree of freedom, the equation of energy is sufficient for the complete determination of the motion. If q be any variable coordinate defining the position or (in the case of a system of bodies) the configuration, the velocity of each particle at any instant will be proportional to 4, and the total kinetic energy may be expressed in the form 1A42, where A is in general a function of q [cf. equation (14)1. This coefficient A is called the coefficient of inertia, or the reduced inertia of the system, referred to the coordinate q.
Thus in the case of a railway truck travelling with velocity u the kinetic energy is 2(M+mK2/a2)u2, where M is the total mass, a the radius and x the radius of gyration of each wheel, and m is the sum of the masses of the wheels; the reduced inertia is therefore M+mK2/a2. Again, take the system composed of the flywheel, connecting rod, and piston of a steamengine. We have here a limiting case of three
bar motion (§ 3), and the
,11 instantaneous centre J of
i the connectingrod PQ will
have the position shown in
' I . the figure. The velocities
of P and Q will be in the
ratio of JP to JQ, or OR to
OQ; the velocity of the piston is therefore yO, where y=OR. Hence if, for simplicity, we neglect the inertia of the connectingrod, the kinetic energy will be 2(I+My2)b2, where I is
the moment of inertia of the flywheel, and M is the mass of the piston. The effect of the mass of the piston is therefore to increase the apparent moment of inertia of the flywheel by the variable amount My2. If, on the other hand, we take OP (=x) as our variable, the kinetic energy is Z (M+I/y2)x2. We may also say, therefore, that the effect of the flywheel is to increase the apparent mass of the piston by the amount I/y2; this becomes infinite at the " deadpoints " where the crank is in line with the connectingrod.
If the system be " conservative," we have
2Ag2+V =const., (15)
where V is the potential energy. If we differentiate this with respect to t, and divide out by ¢, we obtain
Aq+a, dgg2 + dq =o
as the equation of motion of the system with the unknown reactions (if any) eliminated. For equilibrium this must be
satisfied by 4=0; this requires that dV/dq=o, i.e. the potential energy must be " stationary." To examine the effect of a small disturbance from equilibrium we put V =f (q), and write q=qo+n, where qo is a root of f'(qo) =o and n is small. Neglecting terms of the second order inn we have dV/dq=f'(q)=
f"(qo).n, and the equation (16) reduces to
+f"(qo)n=o,
where A may be supposed to be constant and to have the value corresponding to q=qo. Hence if f"(qo) >o, i.e. if V is a minimum in the configuration of equilibrium, the variation of n is simpleharmonic, and the period is 2.ir‘/ {A/f"(qo)}. This depends only on the constitution of the system, whereas the amplitude and epoch will vary with the initial circumstances. If f"(qo) <0, the solution of (17) will involve real exponentials, and n will in general increase until the neglect of the terms of the second order is no longer justified. The configuration q= qo, is then unstable.
As an example of the method, we may take the problem to which equation (14) relates. If we differentiate, and divide by 6, and retain only the terms of the first order in 0, we obtain
{x2+(h—a)2J9+ghO =o, (18) as the equation of small oscillations about the position 0=o. The length of the equivalent simple pendulum is {K2+(h—a)2[/h.
The equations which express the change of motion (in two dimensions) due to an instantaneous impulse are of the forms
M (u' — u) =, M (v'  v) = n, I (ca' — w) = v. (19) Here u', v' are the values of the component velocities of G just before, and u, v their values just after, the impulse, whilst co', to denote the corresponding angular velocities. Further, i;, n are the timeintegrals of the forces parallel to the coordinate axes, and v is the timeintegral of their moment about G. Suppose, for example, that a rigid lamina at rest, but free to move, is struck by an instantaneous impulse F in a given line. Evidently G will begin to move parallel to the line of F; let its initial velocity be u', and let co' be the initial angular
velocity. Then Mu' = F, Iw' = F. GP, where GP is the perpendicular from G to the line of F. If PG be produced to any point C, the initial velocity of the point C of the lamina will be
u'—w'. GC = (F/M).(1—GC.CP/K2),
where Ice is the radius of gyration about G. The initial centre of rotation will therefore be at C, provided GC. GP= x2. If this condition be satisfied there would be no impulsive reaction at C even if this point were fixed. The point P is therefore called the centre of percussion for the axis at C. It will be noted that the relation between C and P is the same as that which connects the centres of suspension and oscillation in the compound pendulum.
§ 18. Equations of Motion in Three Dimensions.—It was proved in § 7 that a body moving about a fixed point 0 can be brought from its position at time t to its position at time t+St by an infinitesimal rotation e about some axis through 0; and the limiting position of this axis, when St is infinitely small, was called the " instantaneous axis." The limiting value of the ratio a/St is called the angular velocity of the body; we denote it by w. If t, n, are the components of e about rectangular coordinate axes through 0, the limiting values of /81, n/St, ]'/St are called the component angular velocities; we denote them by p, q, r. If 1, m, n be the directioncosines of the instantaneous axis we have
p=lw, q=mw, r=nw, (1)
p2+q2+r2 =w2• (2)
If we draw a vector OJ to represent the angular velocity, then J traces out a certain curve in the body, called the polhode, and a certain curve in space, called the her polhode. The cones generated by the instantaneous axis in the body and in space are called the polhode and herpolhode cones, respectively; in the actual motion the former cone rolls on the latter (§7).
(13)
(16)
(17)
The special case where both cones are right circular and w is constant is important in astronomy and also in mechanism (theory of bevel wheels). The " precession of the equinoxes " is due to the fact that the earth performs a motion of this kind about its centre, and the whole class of such motions has therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the
0 FIG. 78.
axis of the fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode, which are of course both circles. if a be the semiangle of the rolling cone, l3 the constant inclination of OC to OZ, and the angular velocity with which the plane ZOC revolves about OZ, then, considering the velocity of a point in OC at unit distance from 0, we have
w sin a= d sin 1, (3) where the lower sign belongs to the third case. The earth's precessional motion is of this latter type, the angles being a=•oo87", $=23° 28'.
If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by
2T =E{m(w. PN)2} =w2. (m.PN2) =Iw2, (4) where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have
I =Al2+Bm2+Cn2—2Fmn—2Gnl 2Hlm,
and therefore by (I),
2T=Ap2+Bq2+Cr2—2Fgr—2Grp2Hpq• (5) Again, if x, y, z be the coordinates of P, the component velocities of m are
qz—ry, rx—pz, py—qx, (6) by § 7 (5); hence, if JI,µ, v be now used to denote the component angular momenta about the coordinate axes, we have X=tin( py—qx)y—m(rx—pz)z}, with two similar formulae, or
X= Ap—Hq—Gr=a
8T 1
T
=—Hp+Bq—Fr=aq,  (7) v=—Gp—Fq+Cr=a1. j
If the coordinate axes be taken to coincide with the principal axes of inertia at 0, at the instant under consideration, we have the simpler formulae
2T =Ap2+Bq2+Cr2, (8)
X=Ap, µ=Bq, v=Cr. (9) It is to be carefully noticed that .the axis of resultant angular momentum about 0 does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momenta] ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § Ir (41), the coordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the directioncosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, µ, v. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if F be the resultant angular momentum, so that
a2 +µ2 +v2 =r2, (to) the length of the perpendicular OH on the tangent plane at J is
OH = . pP+r ..P+ Cr . wr P =2P . W,
r ,T where p=OJ. This relation will be of use to us presently (§ 19).
The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point 0 of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose coordinates relative to 0 are x, y, z are then
u+qz—ry, v+rx—pz, w+py—qx (12) by § 7 (6). It is usually convenient to take as our basepoint the masscentre of the body. In this case the kinetic energy is given by
2T=Mo(u2+v2+w2)+Ap2+Bge+Cr2—2Fgr—2Grp—2Hpq, (13) where M„ is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the masscentre; cf. § 15 (9).
The components , rl, of linear momentum are
t=Mou=du' n=Mov av, 3'=Mow=aw, (14)
whilst those of the relative angular momentum are given by (7). The preceding formulae are sufficient for the treatment of instantaneous impulses. Thus if an impulse (, n, X, v) change the motion from (u, v, w, p, q, r) to (u', v', w', p', q', r') we have
Mo(u'—u) =E, Mo(v'—v) =n, Mo(w'—w) =(5)
A(p'—p)=X, B(q'—q)=a, C(r'—r) =v, (15)
where, for simplicity, the coordinate axes are supposed to coincide with the principal axes at the masscentre. Hence the change of kinetic energy is
T'—T= .i(u+u')+n.1(v+v')+i'•1(w+w'),
+ A . i(p+p') + i.'a(q+q')+v.1(r+r'). (16) The factors of , rl, r, X, v on the righthand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the work done by the given system of impulsive forces in such a displacement, As in § 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse. We are thus led to the following statement: the change of kinetic energy due to any system of impulsive forces is equal to the sum of the products of the several forces into the semisum of the initial and final velocities of their respective points of application, resolved in the directions of the forces. Thus in the problem of fig. 77 the kinetic energy generated is zM(K21Cg2)w'2, if C be the instantaneous centre; this is seen to be equal to F. w'. CP, where w'. CP represents the initial velocity of P.
The equations of continuous motion of a solid are obtained by substituting the values of E, rl, , X, µ, v from (14) and (7) in the general equations
d_ dn— —=Z
dt—X' dt Y' dt '
d =L' dt =M' dt =N'
where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the masscentre as base, the coordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the coordinate axes.
An exception occurs, however, in the case of a solid which is kinetically symmetrical (§ I I) about the masscentre, e.g. a uniform sphere. The equations then take the forms
Molt=X, Mov=Y, Mow=Z, CP=L, Cq=M, Cs=N,
where C is the constant moment of inertia about any axis through
(II)
(17)
(18)
the masscentre. Take, for example, the case of a sphere rolling on a plane; and let the axes Ox, Oy be drawn through the centre parallel to the plane, so that the equation of the latter is z=a. We will suppose that the extraneous forces consist of a known force (X, Y, Z) at the centre, and of the reactions (Fl, F2, R) at the point of contact. Hence
Moir=X+Fi, Moii=Y+F2, o=Z+R, ? C¢=Fla, Col= —Fla, Cf =o.
(19)
The last equation shows that the angular velocity about the normal to the plane is constant. Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the geometrical relations
u+qa=o, v+pa=o, w=o, (20) by (12). Eliminating p, q, we get
(Mo+Ca2A =X, (Mo+Ca2)t =Y. (21) The acceleration of the centre is therefore the same as if the plane were smooth and the mass of the sphere were increased by C/a2. Thus the centre of a sphere rolling under gravity on a plane of inclination a describes a parabola with an acceleration
g sin a/(I+C/Ma2)
parallel to the lines of greatest slope.
Take next the case of a sphere rolling on a fixed spherical surface. Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, z be the coordinates of this centre relative to axes through 0, the centre of the fixed sphere. If the only extraneous forces are the reactions (P, Q, R) at the point of contact, we have
Mox=P, Moy=Q, Mor=R,
Cp = 2c (yR—zQ), Cq = —d(zP—xR), Cr = —¢(xQ—yP),
the standard case being that where the rolling sphere is outside the fixed surface. The opposite case is obtained by reversing the sign of a. We have also the geometrical relations
x=(a/c)(qz—ry), y=(a/c)(rx—pz), =(a/c)(py—gx)• (23) If we eliminate P, Q, R from (22), the resulting equations are integrable with respect to t; thus
p=— °(yzzy)+a, q = — C (zx+0,
r = — C° (xy—yx) +y, (24) where a, S, y are arbitrary constants. Substituting in (23) we find
( :
1+c—)x ' 2=`cr(Rz—yy), 1+ y=¢&x—az),
( z
I+MCa )i=(ay—Qx)• (25)
Hence az+s3y+yz=o, or
ax+toy+7z = const. ; (26) which shows that the centre of the rolling sphere describes a circle. If the axis of z be taken normal to the plane of this circle we have a=o, $=o, and
( z
1+ Moe) x=—7ccy, (I+MC )y cx.
The solution of these equations is of the type
x = b cos (ot+e), y = b sin (et+e), where b, a are arbitrary, and
ya/c
=1+Moat/C
The circle is described with the constant angular velocity a.
When the gravity of the rolling sphere is to be taken into account the preceding method is not in general convenient, unless the whole motion of G is small. As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory. If Oz be drawn vertically upwards, then in the beginning of the disturbed motion the quantities x, y, p, q, P, Q will all be small. Hence omitting terms of the second order, we find
Moz=P, MoY=Q, R=Mog,
Cj = — (Moga/c)y+aQ, Cq = (Moga/c)x —aP, Cf =o.
(30)
The last equation shows that the component r of the angular velocity retains (to the first order) the constant value n. The geometrical relations reduce to
x=aq—(na/c)y, y=—ap+(na/c)x.
Eliminating p, q, P, Q, we obtain the equations
(C+Moa2)z+(Cna/c)y—(Moga2/c)x=o, (C +Moa2)Y— (Cna/c)x— (Moga2/c)y =o, which are both contained in
(C+Moa2)dt,—iCna~—Moga2 (x+iy) =o.This has two solutions of the type x+iy=ae'(a:+e), where a, e are arbitrary, and a is a root of the quadratic
(C+Moa2)a2— (Cna/c)a+Moga2/c =o. If
n'> (4Mgc/C) (1+ Moa'/C),
both roots are real, and have the same sign as n. The motion of G then consists of two superposed circular vibrations of the type
x=a cos (at+e), y=a sin (et+e), (36) in each of which the direction of revolution is the same as that of the initial spin of the sphere. It follows therefore that the original position is stable provided the spin n exceed the limit defined by (35). The case of a sphere spinning about a vertical axis at the lowest point of a spherical bowl is obtained by reversing the signs of a and c. It appears that this position is always stable.
It is to be remarked, however, that in the first form of the problem the stability above investigated is practically of a limited or temporary kind. The slightest frictional forces—such as the resistance of the air—even if they act in lines through the centre of the rolling sphere, and so do not directly affect its angular momentum, will cause the centre gradually to descend in an everwidening spiral path.
§ 19. Free Motion of a Solid.—Before proceeding to further problems of motion under extraneous forces it is convenient to investigate the free motion of a solid relative to its masscentre 0, in the most general case. This is the same as the motion about a fixed point under the action of extraneous forces which have zero moment about that point. The question was first discussed by Euler (1750); the geometrical representation to be given is due to Poinsot (1851).
The kinetic energy T of the motion relative to 0 will be constant. Now T= ZIw2, where w is the angular velocity and I is the moment of inertia about the instantaneous axis. If p be the radiusvector OJ of the momental ellipsoid
Axe+By2+Cz2 = Me4 (1)
drawn in the direction of the instantaneous axis, we have I=Me4/p2(§ II); hence w varies as p. The locus of J may therefore be taken as the " polhode " (§ 18). Again, the vector which represents the angular momentum with respect to 0 will be constant in every respect. We have seen (§ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that
OH=r • w
(22)
(27)
(29)
(31)
(32)
(33)
(34) (35)
(2)
(28) space, with an angular velocity proportional at each instant to
where I' is the resultant angular momentum about O. Since w varies as p, it follows that OH is constant, and the tangent plane at J is therefore fixed in space. The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in
the radiusvector of the point of contact. The fixed plane is parallel to the invariable plane at 0, and the line OH is called the invariable line. The trace of the point of contact J on the fixed plane is the " herpolhode."
If p, q, r be the component angular velocities about the principal axes at 0, we have
(A2p2+B2g2+C2r2)/r2=(Ap2+Bq2+Cr2)/2T, (3)
each side being in fact equal to unity. At a point on the polhode cone x: y:z=p: q: r, and the equation of this cone is therefore / z
A2 (1—2T) x2+B2I I2BT) y2+C2 (I—2CT) z2=O. (4)
Since 2AT—PZ=B (A—B)g2+C(A—C)r2, it appears that if A> B > C the coefficient of x' in (4) is positive, that of z is negative, whilst that of y2 is positive or negative according as 2BT < P. Hence the polhode cone surrounds the axis of greatest or least moment according as 2BT P. In the critical case of 2BT= P it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a reentrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion. The invariable line OH describes another cone in the
body, called the invariable cone. At any point of this we have x : y : z =((Ap. Bq : Cr, and the 11 equation is therefore
I2 (I+ r2
2T/ y2+ (II 2T)z2=o. (5)
The signs of the coefficients follow the same rule as in the case of (4). The possible forms of the invariable cone are indicated in fig. 8o by means of the intersections with a concentric spherical
surface. In the critical case of 2 BT = r2 the cone degenerates into two planes. It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place.
Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. The question is greatly simplified when two of the principal moments are equal, say A=B. The polhode and herpolhode cones are then right circular, and the motion is " precessional " according to the definition of § 18. If a be the inclination of the instantaneous axis to the axis of symmetry, (3 the inclination of the latter axis to the invariable line, we have
I' cos fl= Cis cos a, r sin 0 = Aw sin a, (6)
tan 0 =A
c tan a.
Hence a, and the circumstances are therefore those of the
first or second case in fig. 78, according as A C. If ik be the
C,
rate at which the plane HOJ revolves about OH, we have
sina C cos a )
sin 13 Acos/3w' (8)
by § IS (3). Also if g be the rate at which J describes the polhode, we have it' sin ((3 a) =)'c sin (3, whence sin (a—0)
X = w.
sin a
If the instantaneous axis only deviate slightly from the axis of symmetry the angles a, are small, and = (AC) A .w; the instantaneous axistherefore completes its revolution in the body in the period
2w A—C x = A w.
In the case of the earth it is inferred from the independent phenomenon of lunisolar precession that (C—A)/A=•oo313. Hence if the earth's axis of rotation deviates slightly from the axis of figure, it should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in the latitude of any place on the earth's surface, as determined by astronomical methods. There appears to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as the Eulerian nutalion, since it is supposed to come under the free rotations first discussed by Euler.
§ 2o. Motion of a Solid of Revolution.—In the case of a solid of revolution, or (more generally) whenever there is kinetic symmetry about an aY:s through the masscentre, or through a fixedpoint 0, a number of interesting problems can be treated almost directly from first principles. It frequently happens that the extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant. For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.
Suppose, for example, that a flywheel is rotating with angular velocity n about its axis, which is (say) horizontal, and that this axis is made to rotate with the angular velocity in the horizontal plane. The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A 4, respectively, where A is the moment of inertia about any axis through the masscentre (or through the fixed point 0) perpendicular to that of sym
metry. If OK be the vector representing the former component at time t, the vector which represents it at time t+St will be OK', equal to OK in magnitude and making with it an angle 4. Hence
KK' (=COO will represent the change in this component due to the extraneous forces. Hence, so far as this component is concerned, the extraneous forces must supply a couple of moment Cm/. in a vertical plane through the axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such a sense that the direction of the spin n approximates to that of the azimuthal rotation i/'. The remaining constituent of the extraneous forces is a couple All; 0 about the vertical; this vanishes if
is constant. If the axis of the flywheel
make an angle 0 with the vertical, it is seen in like manner that the required couple in the vertical plane through the axis is Cn sin 0 1(.. This matter can be strikingly illustrated with an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical diameter.
If the direction of the axis of kinetic symmetry be specified by means of the angular coordinates 0,>G
of § 7, then considering the component Z velocities of the point C in fig. 83, which are 0 and sin 0>l' along and perpendicular to the meridian ZC, we see that the component angular velocities about the lines OA', OB' are sin 0 t(i and 0 respectively. Hence if the principal moments of inertia at 0 be A, A, C, and if n be the constant angular velocity about the axis OC, the kinetic energy is given by
2T = A (62 + sin' O ¢2)+Cn2. (I)
Again, the components of angular momentum about OC, OA' are Cn,A sin 0 ', and therefore the angular momentum (µ, say) about OZ is
µ=A sin' 0%bICncos O. (2)
We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point 0 on its axis of symmetry, its masscentre G being in this axis at a distance It from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction 0G. If 0 is constant the points C, A' will in time at come to positions C", A" such that CC" = sin 0 a¢, A'A" = cos 0 a¢, and the angular momentum about OB' will become Cn sin 0 &P A sin 0 i%i. cos 0 SIP. Equating this to Mgh sin 0 at, and dividing out by sin 0, we obtain
A cos 0 L2—Cn,i+Mgh=o, (3) as the condition in question. For given values of n and 0 we have two possible values of provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with ij (4AMgh cos 0), one value of 11' is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos 0 approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very
whence
(7)
(9)
(to)
nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.
Again, take the case of a circular disk rolling in steady motion
on a horizontal plane. The centre 0 of the disk is supposed to
describe a horizontal circle of
radius c with the constant angular
velocity L, whilst its plane pre
serves a constant inclination 6 to
the horizontal. The components
of the reaction of the horizontal
lane will be Mc+I'2 at right angles
to the tangent line at the point
of contact and Mg vertically up
wards, and the moment of these
the disk, which corresponds to 013' in fig. 83, is Mc>'2. a sin 0  Mga cos 0, where a is the radius of the disk. Equating this to the rate of increase of the angular momentum about OB', investigated as above, we find
(z
C+Ma2+AQ cos e)=Mga cot 0, (4)
where use has been made of the obvious relation na=c'I. If c and 8 be given this formula determines the value of for which the motion will be steady.
In the case of the top, the equation of energy and the condition of constant angular momentum (µ) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have
A(92+sine 81%.2) + ICn2 + Mgh cos 0 = const., (5)
A sin2 0 ' + v cos O = µ, (6)
where v is written for Cn. From these II/ may be eliminated, and on differentiating the resulting equation with respect to t we obtain
Ao (µ  v cos0)(ocos0v)Mghsin0=
A sin' B O. (7)
If we put 8=o we get the condition of steady precessional motion in a form equivalent to (3). To find the small oscillation about a state of steady precession in which the axis makes a constant angle a with the vertical, we write 6=a+x, and neglect terms of the second order in x. The result is of the form
z+a2x =o, (8)
where
a2 = { (µ — v cos a)2 + 2 (g — v cos a) (µ cos a — v) COS a +
(µ cos av)2{/A2 sin, a. (9) When v is large we have, for the " slow " precession a=v/A, and for the " rapid " precession a=A/v cos a=i/i, approximately. Further, on examining the small variation in i(/, it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be easily observed. In a slightly disturbed rapid precession the superposed vibration is ellipticharmonic, with a period equal to that of the precession itself. The ratio of the axes of the ellipse is sec a, the longer axis being in the plane of B. The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical. This is, in fact, the " invariable line " of the free Eulerian rotation with which (as already remarked) we are here virtually concerned. For the more general discussion of the motion of a top see GYROSCOPE.
§ 21. Moving Axes of Reference.—For the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.
We will assume for the present that the origin 0 is fixed. The moving axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The components of angular momentum about Ox, Oy, Oz will be denoted as usual by X, µ, v. Now consider a system of fixed axes Ox', Oy', Oz' chosen so as to coincide at the instant t with the moving system Ox, Oy, Oz. At the instant t+St, Ox, Oy, Oz will no longer coincide with Ox', Oy', Oz';
'Mc 12
A d(BC)qr= L,
B d4(CA)rp=M,
C di(AB)pq=N.
If we multiply these by p, q, r and add, we get
d
dt z(Ap2+B42+Cr2)=Lp+M4+Nr, (3)
which is (virtually) the equation of energy.
As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity w about a fixed axis (1, rn, n). Since p, q, r are then constant, the requisite constraining couple is
L=(CB)mnw2, M=(AC)nlw2, N(BA)lmw2. (4) If we reverse the signs, we get the " centrifugal couple " exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at 0, and in no other case (cf. § 17).
If in (2) we put, L, M, N=0 we get the case of free rotation; thus
A2= (B C)qr,
B2 =(CA)rp, F (5)
C dt = (AB)pq.
These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 + Bq2 + Cr2 and A2p2 + B2g2 + C2r2 are both constant. The former is, in fact, equal to 2T, and the latter to P2, where T is the kinetic energy and P the resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875). If we write
u= J y d^:^5 = (1k2 sin' ‘/o)
we have, in the notation of elliptic functions, 0= am u. If we assume
p=po cos am (at+e), q =qo sin am (at+e), r=ro0 am (atle), (7)
These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by Po, qo, ro, k2, a, e. We will suppose that A> B >C. From the form of the polhode curves referred to in §19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have a=o, and Po, ro will become identical with the initial values of p, r. The conditions (9) then lead to
2 A(AC) 2 02=(AC)(BC)roz k2=A(AB). (IO)
Qo B(BC) p°' AB C(BC) ro
(2)
we find z (8)
p=Q oqr, 4=yrp, =p4opg.
Hence (5) will be satisfied, provided (9)
=apoB C aqo C  A
 k2aro A  B
'
goro A ' ropo B pogo C
in particular they will make with Ox' angles whose cosines are, to the first order, 1,rat, qSt, respectively. Hence the altered angular momentum about Ox' will be X +5X + (µ}8µ) ( rot) + (v+Sv)gSt. If L, M, N be the moments of the extraneous forces about Ox, Oy, Oz this must be equal to XlLSt. Hence, and bar symmetry, we obtain
dt  rp + qv = L, di — pv +rX =M,
dv
at  qX +pa = N. J
These equations are applicable to any dynamical system whatever. If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, p., v=Ap, Bq, Cr, whence
(I)
For a real solution we must have k2 < I, which is equivalent to 2BT > 1'2. If the initial conditions are such as to make 2BT < r2, we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever 4' increases by 2r, i.e. whenever t increases by 4K/0, when K is the " complete " elliptic integral of the first kind with respect to the modulus k.
The elliptic functions degenerate into simpler forms when k2=o or k2 = I. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in §19. If k2=1, we must have 2BT = r2. We have seen that the alternative 2BT j 1'2 determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT=1'2, exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.
When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are
d1 —rn+qf=X, dt —Pi+rE=Y, TIT —q +p+I =Z, (II)
dt —r, +qv—wo+v3'=L, di—pv+ra—of+w1=M,
dt —qx+pµ—vi;+un=N. J
To prove these, we may take fixed axes O'x', O'y', O'z' coincident with the moving axes at time t, and compare the linear and angular momenta E+5E, rt+Sn, +5~ , ?+5X, µ+Sµ, v+Sv relative to the new position of the axes, Ox, Oy, Oz at time t+St with the original momenta , n, , k, µ, v relative to O'x', O'y', O'z' at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the masscentre, we have , j, =Mou, Mov, Mow, where Mo is the total mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point 0; the angular coordinates 0, 4), >G of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are
p=d sin ¢— sin 9 cos 4i IA geed cos 41+ sins sin ,¢ , (13)
r=~+cos Bf,
by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by
2T=A(d sin 41—sin 0 cos 4;b)2+B (d cos ¢+sin 0 sin 0)2
+C (4) +cos 0 O. (14)
If A =B this reduces to
2T =A(d2+sine 0 1%2) +C(¢+cos 0 0; (15)
cf. § 20 (I).
§ 22. Equations of Motion in Generalized Coordinates.—Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized coordinates of the system. These coordinates may be chosen in an endless variety of ways,'but their number is determinate, and expresses the number of degrees of freedom of the system. We denote these coordinates by gi,g2, ...q,.. It is implied in the above description of the system that the Cartesian coordinates x, y, z of any particle of the system are known functions of the q's, varying in form (of course) from particle to particle. Hence the kin et;c energy T is given by
2T = E(m (t2+Y2+z2) } =a11412+a22422+... +2a124112+... , where
ar=tm (aqr) 2+ (9) 2+ (aQr) 2 ~' (2)
Ox ax ay ay az az
a,s = E m (a4, aq.+aqr aq.+aqr aq.) =air.
Thus T is expressed as a homogeneous quadratic function of the quantities 41, 42, . . . 4n, which are called the generalized
components of velocity. The coefficients arr, a„ are called the coefjecients of inertia; they are not in general constants, being functions of the q's and so variable with the configuration. Again, if (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is
E(XSx+YIy+ZIz) =QiSgi+Q2ag2+... +Q.agn, (3) where
Qr= (X— qr+Yaq +Zaq,. . (4)
The quantities Q, are called the generalized components of force.
The equations of motion of m being
mg=X, my=Y, mff=Z, (5)
l m (xf~gr+ q + Oa,)  " (6)
ax ax aqr = aqr
d lax a2x a2x a2x ax dt \aqr/ agiaq,i+aQ2aq42+...+aq aq,gaqr
tax =d (lax) (ax) =d (xaz l —x q (Io)
OD. dt aq, dt aqr dt ` aq /
By these and the similar transformations relating to y and z the equation (6) takes the form
d raT aT
dt aqr) aqr—Qr.
If we put r= 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange's own proof will be found under DYNAMICS, § Analytical. In a conservative system free from extraneous force we have
E(Xax+YSy+Zaz) = —SV, where V is the potential energy. Hence
aV Qr=—aqr
and d OT \ dt aqr) aqr —eV
If we imagine any given state of motion (41, 42, . . . 40 through
the configuration (al, q2, . . . q,.) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (II) with respect tot over the infinitely short duration of the impulse
aT ' a1 = Qr'
where Qr' is the time integral of Qr and so represents a generalized component of impulse. By an obvious analogy, the expressions aT/a4r may be called the generalized components of momentum; they are usually denoted by pr, thus
pr=aT/a4r=alr41+a2r42+... +a,.,4,.. (16)
Since T is a homogeneous quadratic function of the velocities 4% 42, . . . 4,., we have
2T=aq 4i+a4 42+... +4,. =p142+p242+... +pngn.
Hence
2 =x14+p242+• • • +pngn +pigs+p242+..  +pngn
_ (8Q+0 41+ (aQ2+Q2) 42+.. + (aq +Qn) 4° , (x8)
+aq qi+aQ 42+... +~Q 4
dT
= dt +Qi4i+Q242+   • +Qn4,,,
= Qi4i+Q242 + ... +Qn4n. (19)
(I2)
(1)
we have Now whence Also Hence
x = aqg1 +aQg2 + ... +agngn,
ax .
ax . ax . (7)
(8) (9)
(12)
(13) (14)
(15)
(17)
or
This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy
d
(Ti (T+V) =o, or T+V=const., (20)
in virtue of (13).
As a first application of Lagrange's formula (11) we may form the equations of motion of a particle in spherical polar coordinates. Let r be the distance of a point P from a fixed origin 0, 0 the angle which OP makes with a fixed direction OZ, the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these coordinates are Sr along OP, rOO perpendicular to OP in the plane ZOP, and r sin B SIP perpendicular to this plane. The component velocities in these directions are therefore r, r6, r sin and if m be the mass of a moving particle at P we have
2T = m(rz + r?62 + r2 sin2 0 P). (21)
Hence the formula (u) gives
m(rre'—rsin' 0 2)=R,
d
d (mr26) —mr2sinBcosB,G2=e,
dt (mt.' sin2 0 t) =
~.
The quantities R, e, * are the coefficients in the expression Rbr+ebo++4 for the work done in an infinitely small displacement ; viz. R is the radial component of force, ® is the moment about a line through 0 perpendicular to the plane ZOP, and s is the moment about OZ. In the case of the spherical pendulum we have r=l, e= —mgl sin 0, '=o, if OZ be drawn vertically downwards, and therefore
6— sinocoso1G2=—f sin 0,
sin2 of= h,
where h is a constant. The latter equation expresses that the angular momentum mP sin2 opt about the vertical OZ is constant. By elimination of )G we obtain
8h2 cos' o/sin3o= —Fin 0. (24)
If the particle describes a horizontal circle of angular radius a with constant angular velocity 12, we have 0=o, h=S22 sin a, and therefore
End of Article: IM (K2+a2+h2—2 ah cos 0) 

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