dP dl dm dn
-- = --x1 + --x2 + --x3
dt dt dt dt
dx1 dx2 dx3
+ l--- + m--- + n---,
dt dt dt
/ dx1 \
= l ( --- - x2R + x3Q )
\ dt /
/ dx2 \
+ m ( --- - x3P + x1R )
\ dt /
/ dx3 \
+ n ( --- - x1Q + x2P )
\ dt /
= lX + mY + nZ, (7)
for all values of l, m, n.
Next, taking a fixed origin [Omega] and axes parallel to Ox, Oy, Oz
through O, and denoting by x, y, z the coordinates of O, and by G the
component angular momentum about [Omega] in the direction (l, m, n)
G = l(y1 - x2z + x3y)
+ m(y2 - x3x + x1z)
+ n(y3 - x1y + x2x). (8)
Differentiating with respect to t, and afterwards moving the fixed
origin up to the moving origin O, so that
dx dy dz
x = y = z = 0, but -- = U, -- = V, -- = W,
dt dt dt
dG / dy1 \
-- = l ( --- - y2R + y3Q - x2W + x3V )
dt \ dt /
/ dy2 \
+ m ( --- - y3P + y1R - x3U + x1W )
\ dt /
/ dy3 \
+ n ( --- - y1Q + y2P - x1V + x2U )
\ dt /
= lL + mM + nN, (9)
for all values of l, m, n.
When no external force acts, the case which we shall consider, there
are three integrals of the equations of motion
(i.) T = constant,
(ii.) x1² + x2² + x3² = F², a constant,
(iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;
and the dynamical equations in (3) express the fact that x1, x2, x3
are the components of a constant vector having a fixed direction;
while (4) shows that the vector resultant of y1, y2, y3 moves as if
subject to a couple of components
x2W - x3V, x3U - x1W, x1V - x2U, (10)
and the resultant couple is therefore perpendicular to F, the
resultant of x1, x2, x3, so that the component along OF is constant,
as expressed by (iii).
If a fourth integral is obtainable, the solution is reducible to a
quadrature, but this is not possible except in a limited series of
cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary,
Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian
mathematicians; and the general solution requires the double-theta
hyperelliptic function.
49. In the motion which can be solved by the elliptic function, the
most general expression of the kinetic energy was shown by A. Clebsch
to take the form
T = ½p(x1² + x2²) + ½p´x3²
+ q(x1y1 + x2y2) + q´x3y3
+ ½r(y1² + y2²) + ½r´y3² (1)
so that a fourth integral is given by
dy3/dt = 0, y3 = constant; (2)
dx3
--- = x1(qx2 + ry2) - x2(qx1 + ry1) = r(x1y2 - x2y1), (3)
dt
1 / dx3 \²
-- ( --- ) = (x1² + x2²)(y1² + y2²) - (x1y1 + x2y2)²
r² \ dt /
= (x1² + x2²)(y1² + y2²) - (FG - x3y3)²
= (x1² + x2²)(y1² + y2² + y3² - G²) - (Gx3 - Fy3)², (4)
in which
x1² + x2² = F² - x3², x1y1 + x2y2 = FG - x3y3, (5)
r(y1² + y2²) = 2T - p(x1² + x2²) - p´x3²
- 2q(x1y1 + x2y2) - 2q´x3y3 - r´y3²
= (p - p´)x3² + 2(q - q´)x3y3 + m1, (6)
m1 - 2T - pF² - 2qFG - r1y3² (7)
so that
1 / dx3 \²
-- ( --- ) = X3 (8)
r² \ dt /
where X3 is a quartic function of x3, and thus t is given by an
elliptic integral of the first kind; and by inversion x3 is in
elliptic function of the time t. Now
(x1 - x2i)(y1 + y2i) = x1y1 + x2y2 + i(x1y2 - x2y1)
= FG - xy3y3 + i[V-]X3, (9)
y1 + y2i FG - x3y3 + i[root]X3
-------- = --------------------- , (10)
x1 + x2i x1² + x2²
d
-- (x1 + x2i) = -i[(q´ - q)x3 + r´y3] + irx3(y1 + y2i), (11)
dt
d FG - x3y3 + i[root]X3
--- log (x1 + x2i) = dti -(q´ - q)x - r´y + rx ---------------------, (12)
dti F² - x3²
d /x1 + x2i Fy3 - Gx3
--- log \ / -------- = -(q´ - q)x3 - (r´ - r)y3 - Fr---------, (13)
dti \/ x1 - x2i F² - x3²
requiring the elliptic integral of the third kind; thence the
expression of x1 + x2i and y1 + y2i.
Introducing Euler's angles [theta], [phi], [psi],
x1 = F sin [theta] sin [phi], x2 = F sin [theta] cos [phi],
x1 + x2i = iF sin [theta][epsilon]^(-[psi]i), x3 = F cos [theta]; (14)
d[psi]
sin [theta] ------ = P sin [phi] + Q cos[phi], (15)
dt
d[psi] dT dT
F sin²[theta] ------ = --- x1 + --- x2
dt dy1 dy2
= (qx1 + ry1)x1 + (qx2 + ry2)x2
= qx1² + x2²) + r (x1y1 + x2y2)
= gF² sin² [theta] + r(FG - x3y3), (16)
_
/ FG - x3y3 Fr dx3
[psi] - qFt = | --------- --------, (17)
_/ F² - x3² [root]X3
elliptic integrals of the third kind.
Employing G. Kirchhoff's expressions for X, Y, Z, the coordinates of
the centre of the body,
__ __ __
FX = y1 cos xY + y2 cos yY + y3 cos zY, (18)
__ __ __
FY = -y1 cos xX + y2 cos yX + y3 cos zX, (19)
__ __ __
G = y1 cos xZ + y2 cos yZ + y3 cos zZ, (20)