Mechanics

P. V. Kharlamov

TWO PARTICULAR SOLUTIONS OF THE PROBLEM OF THE MOTION OF A BODY HAVING A FIXED POINT

(Presented by Academician P. Ya. Kochina, 13 VII 1963)

Suppose that the center of gravity of the system lies on a principal axis of the gyration ellipsoid for the fixed point, and that along the same axis there is directed the vector \(\vec{\lambda}\), characterizing internal cyclic motions (for example, the motion of a fluid circulating in multiply connected cavities of the body, or the motion of a flywheel of negligible weight).

The equations of motion of such a body \(\left({}^{1}\right)\), in the usual notation \(\left({}^{2}\right)\), have the form

\[ A\frac{dp}{dt}=(B-C)qr, \]

\[ B\frac{dq}{dt}=(C-A)rp-\lambda r-\gamma_3,\qquad C\frac{dr}{dt}=(A-B)pq+\lambda q+\gamma_2, \tag{1} \]

\[ \frac{d\gamma_1}{dt}=r\gamma_2-q\gamma_3,\qquad \frac{d\gamma_2}{dt}=p\gamma_3-r\gamma_1,\qquad \frac{d\gamma_3}{dt}=q\gamma_1-p\gamma_2. \]

The known integrals are

\[ {}^{1}\!/_{2}\,(Ap^2+Bq^2+Cr^2)-\gamma_1=E,\qquad (Ap+\lambda)\gamma_1+Bq\gamma_2+Cr\gamma_3=k, \tag{2} \]

\[ \gamma_1^2+\gamma_2^2+\gamma_3^2=\Gamma^2. \]

In the solutions indicated below, \(q, r, \gamma_1, \gamma_2, \gamma_3\) are expressed as functions of \(p\). Substituting \(q=q(p)\), \(r=r(p)\) into (1), we determine, by quadrature, the dependence of \(p\) on \(t\).

First solution.

\[ \frac{C-B}{A}q^2= \frac{A-C}{2B-A}p^2+ 2\lambda\,\frac{3BC-AB-C^2}{(2B-A)^2(2C-A)}\,p+ \]

\[ +\frac{2C-A}{(A-B)(A-C)} \left\{ H-\lambda^2 \frac{3BC-AB-C^2}{(2B-A)^3(2C-A)^3}\times \right. \]

\[ \left. \times\left[A^3-2A^2(2C+B)+AC(3C+8B)-BC(5C+B)\right] \right\}, \]

\[ \frac{B-C}{A}r^2= \frac{A-B}{2C-A}p^2+ 2\lambda\,\frac{3BC-AC-B^2}{(2C-A)^2(2B-A)}\,p+ \frac{2B-A}{(A-C)(A-B)}\times \tag{3} \]

\[ \times \left\{ H-\lambda^2 \frac{3BC-AC-B^2}{(2C-A)^3(2B-A)^3} \left[A^3-2A^2(2B+C)+AB(3+8C)-BC(5B+C)\right] \right\}, \]

\[ \gamma_1 = A\frac{(A-B)(A-C)}{(2B-A)(2C-A)}p^2 -\lambda A \frac{A^2(B+C)-6ABC+2BC(B+C)} {(2B-A)^2(2C-A)^2}p+H, \]

\[ \gamma_2 = q\left\{ \frac{(A-B)(A-C)}{2C-A}p +\lambda\left[ C\frac{3BC-AC-B^2}{(2C-A)^2(2B-A)}-1 \right] \right\}, \]

\[ \gamma_3 = r\left\{ \frac{(A-C)(A-B)}{2B-A}p +\lambda\left[ B\frac{3BC-AB-C^2}{(2B-A)^2(2C-A)}-1 \right] \right\}. \]

Since \(q^2\) and \(r^2\) are polynomials of the second degree in \(p\), \(p\) is an elliptic function of time.

This solution contains 7 independent parameters: \(A, B, C, \lambda, H, p_0, \psi_0\) (\(\psi\) is the angle of precession).

On substituting (3) into (2), the constants \(E, k\), and \(\Gamma\) are determined. Taking the quantity \(\Gamma\) as prescribed, we obtain two values for the parameter \(H\):

\[ \begin{aligned} H = \pm \sqrt{\,\Gamma^2 + \lambda^4 \frac{A(B-C)^2(B+C-2A)\,[2A^2-3A(B+C)+4BC]^2} {4(A-B)^2(A-C)^2(2B-A)^4(2C-A)^4}} \\ {}+ \frac{\lambda^2 A}{(2B-A)(2C-A)} \left\{ \frac{2A^4BC+A^3(B+C)(B^2-8BC+C^2)+26A^2B^2C^2} {2(A-B)(A-C)(2B-A)^2(2C-A)^2} \right. \\ \left. {}+ \frac{-12AB^2C^2(B+C)+2B^2C^2(B+C)^2} {2(A-B)(A-C)(2B-A)^2(2C-A)^2} -1 \right\}. \end{aligned} \]

In particular, for \(\lambda=0\), the indicated solution yields Steklov’s solution (3).

Second solution.

\[ \begin{aligned} \frac{(10B-9C)^2}{36C^2}\,q^2 + \left[ p-\frac{\lambda}{2}\frac{10B-9C}{BC} -\frac{3}{2}\frac{(3C-2B)^2(3C-4B)} {bB(10B-9C)} \right]^2 &= \\ \frac{3}{2}\frac{\lambda}{b} \frac{(3C-2B)(B-C)(10B-9C)}{B^2C} &+ \\ \frac{9}{4} \frac{(3C-2B)^3(3C-4B)} {b^2B(10B-9C)^2} (18C^2-36BC+17B^2), \end{aligned} \]

\[ \begin{aligned} r^2={}& \frac{36b}{10B-9C}\,p^3 -18\left[ 3\frac{b\lambda}{BC} + \frac{(3C-2B)(81C^2-156BC+61C^2)} {B(10B-9C)^2} \right]p^2 \\ &+9\left[ 3b\lambda^2\frac{10B-9C}{B^2C^2} + \frac{\lambda}{2B^2C(10B-9C)} (729C^3-1917BC^2+1520B^2C-388B^3) \right. \\ &\left.\qquad\quad +\frac{3}{2} \frac{(3C-2B)^2(3C-4B)} {bB^2(10B-9C)^3} (243C^3-648BC^2+495B^2C-122B^3) \right]p \\ &-\frac{9}{2}b\lambda^3 \frac{(10B-9C)^2}{B^3C^3} -\frac{3\lambda^2}{4B^3C^2(3C-2B)} (2187C^4-7533BC^3-9234B^2C^2 \\ &\qquad\qquad -5036B^3C+1064B^4) \\ &+9\frac{\lambda}{b} \frac{3C-2B}{B^2C(10B-9C)^2} (729C^4-2754BC^3+3951B^2C+632B^4) \\ &-\frac{27}{4} \frac{(B-C)(3C-2B)^3(3C-4B)} {b^2B^3(10B-9C)^4} (2187C^4-5832BC^3+4131B^2C^2 \\ &\qquad\qquad -30B^3C-488B^4), \end{aligned} \]

\[ \begin{aligned} \gamma_1={}& \frac{18bC}{10B-9C}\,p^3 -27\left[ \frac{b\lambda}{B} + \frac{C(3C-2B)(27C^2-53BC+22B^2)} {B(10B-9C)^2} \right]p^2 \\ &+\left[ \frac{27}{2}b\lambda^2\frac{10B-9C}{B^2C} + \frac{9\lambda}{4B^2(10B-9C)} (729C^3-1917BC^2+1528B^2C-388B^3) \right. \\ &\left.\qquad\quad +\frac{27}{4} \frac{C(3C-2B)^2(3C-4B)} {bB^2(10B-9C)^3} (243C^2-648BC^2+503B^2C-122B^3) \right]p \\ &-\frac{9}{4}b\lambda^3 \frac{(10B-9C)^2}{B^3C^2} -\frac{\lambda^2}{8B^3C} (2187C^3-5589BC^2+4104B^2C-916B^3) \\ &+\frac{9}{8}\frac{\lambda}{b} \frac{(3C-2B)^2} {B^2(10B-9C)^2} (-243C^3+1161BC^2-1526B^2C+568B^3) \\ &+\frac{9}{8} \frac{C(3C-2B)^4(3C-4B)} {b^2B^3(10B-9C)^4} (2187C^4-8748BC^3+12879B^2C^2 \\ &\qquad\qquad -8238B^3C+1888B^4), \end{aligned} \]

\[ \gamma_2 = q \left\{ 3bp^2 - 3 \frac{10B - 9C}{BC} \left[ b\lambda + \frac{C(3C - 2B)(27C^2 - 54BC + 22B^2)}{(10B - 9C)^2} \right] p \right. \]
\[ \left. + \frac{3}{4} b\lambda^2 \frac{10B - 9C}{B^2C^2} + \frac{\lambda}{8B^2C}(729C^3 - 1917BC^2 + 1512B^2C - 388B^3) \right. \]
\[ \left. + \frac{3}{8}\, \frac{(3C - 2B)^2(3C - 4B)}{bB^2(10B - 9C)^2} (243C^3 - 648BC^2 + 495B^2C - 122B^3) \right\}, \]
\[ \gamma_3 = r \left[ 3C \frac{3C - 2B}{10B - 9C} p - 2\lambda - 3 \frac{C(3C - 2B)^2(3C - 4B)}{b(10B - 9C)^2} \right]. \]

This solution applies if the principal moments of inertia are subject to the condition

\[ A = 18C \frac{B - C}{10B - 9C}. \]

There are 6 independent parameters: \(B, C, \lambda, b, p_0, \psi_0\).
The constants \(E, k, \Gamma\) are determined from the integrals (2).
For \(\lambda = 0\), from the second solution there follows the solution of N. Kowalewski \(^{4}\).

Institute of Hydromechanics
Siberian Branch of the Academy of Sciences

Received
9 VII 1963

REFERENCES

\(^{1}\) N. E. Zhukovskii, On the motion of a rigid body having cavities filled with a homogeneous capillary liquid, Collected Works, Moscow—Leningrad, 1936.
\(^{2}\) P. V. Kharlamov, DAN, 150, No. 4 (1963).
\(^{3}\) V. A. Steklov, Proceedings of the Department of Physical Sciences of the Society of Devotees of Natural Science, 10, issue 1 (1899).
\(^{4}\) N. Kowalewski, Math. Ann., 65 (1908).