/A \ dp dr A h3
(25) (-- + z^2) -- - xz -- -- -pq cot[theta] + q --
\M / dt dt M M
- pqz(x + z cot[theta] - [rho] sin[theta]) + qrz[rho] cos[theta] = 0,
dr /A \ dp A h3
(B) -xz -------- + (-- + z^2) -------- + -- p cot[theta] - --
d[theta] \M / d[theta] M M
+ pz(x + z cot[theta] - [rho] sin[theta]) + rz[rho] cos[theta] = 0.
In the special case of a gyrostat rolling on the sharp edge of a
circle passing through G, z = 0, [rho] = 0, (A) and (B) reduce to
/ C \ dr / 1 1 \ dh3
(26) p = ( --- + 1 ) -------- = ( ---- + -- ) --------,
\Mx^2 / d[theta] \Mx^2 C / d[theta]
dp h3 d.p sin[theta] h3sin [theta]
(27) -------- + p cot[theta] = --, -------------- = -------------;
d[theta] A d[theta] A
d^2h3 dh3 CMx^2
(28) ---------- + -------- cot[theta] = -----------,
d[theta]^2 d[theta] A(Mx^2 + C)
a differential equation of a hypergeometric series, of the form of
Legendre's zonal harmonic of fractional order n, given by
(29) n(n + 1) = CMx^2/A(Mx^2 + C).
For a sharp point, x = 0, [rho] = 0, and the previous equations are
obtained of a spinning top.
The elimination of X and Z between (18) (20) (22), expressed
symbolically as
(30) (22) - z(18) + x(20) = 0,
gives
/A \ dq h3 /A \
(C) ( -- + x^2 + z^2 ) -- - p -- + ( -- + z^2 ) p^2 cot [theta] + p^2xz
\M / dt M \M /
+ q^2[rho](x cos[theta] - z sin[theta]) - prx(x + z cot[theta])
- g(x cos[theta] - z sin[theta]) = 0,
and this combined with (A) and (B) will lead to an equation the
integral of which is the equation of energy.
13. The equations (A) (B) (C) are intractable in this general form;
but the restricted case may be considered when the axis moves in
steady motion at a constant inclination [alpha] to the vertical; and
the stability is secured if a small nutation of the axis can be
superposed.
It is convenient to put p = [Omega] sin [theta], so that [Omega] is
the angular velocity of the plane Gzx about the vertical; (A) (B) (C)
become
/C \ dr d[Omega]
(A*) ( -- + x^2) -------- - xz sin[theta] --------
\M / d[theta] d[theta]
- [Omega]x(x sin[theta] - 2z cos[theta]
- [rho] sin^2[theta]) + rx[rho] cos[theta] = 0,
dr /A \ d[Omega] h3 /A \
(B*) -xz -------- + ( -- + z^2) sin[theta] -------- - -- + 2[Omega]( -- + z^2) cos [theta]
d[theta] \M / d[theta] M \M /
+ [Omega]z sin[theta](x - [rho] sin[theta]) - rz[rho] cos[theta] = 0,
/A \ dq h3
(C*) ( -- + x^2 + z^2) -- + q^2p(x cos[theta] - z sin[theta]) - [Omega] -- sin [theta]
\M / dt M
/A \
+ [Omega]^2 ( -- + z^2) sin [theta] cos [theta] + [Omega]^2xz sin^2[theta]
\M /
- [Omega]rx(x sin [theta] + z cos [theta]) - g(x cos [theta] - z sin [theta]) = 0.