UDC 531.38

Mechanics

A. I. DOKSHEVICH

ON A PARTICULAR SOLUTION OF THE PROBLEM OF THE ROTATION OF A HEAVY RIGID BODY ABOUT A FIXED POINT

(Presented by Academician P. Ya. Kochina on 19 VII 1965)

The equations of the problem of the rotation of a heavy rigid body about a fixed point, in the notation of [1], have the form

\[ \begin{aligned} \dot{x} &= (a_2-a_1)yz+(b_2y-b_1z)x,\\ \dot{y} &= (a-a_2)zx+(b_1y+b_2z)z-b_2x^2-\gamma_3,\\ \dot{z} &= (a_1-a)xy-(b_1y+b_2z)y+b_1x^2+\gamma_2,\\ \dot{\gamma}_1 &= \gamma_2\omega_3-\gamma_3\omega_2,\qquad \dot{\gamma}_2=\gamma_3\omega_1-\gamma_1\omega_3,\qquad \dot{\gamma}_3=\gamma_1\omega_2-\gamma_2\omega_1, \end{aligned} \tag{1} \]

where

\[ \omega_1=ax+b_1y+b_2z,\qquad \omega_2=a_1y+b_1x,\qquad \omega_3=a_2z+b_2x. \]

The known integrals of these equations are

\[ \begin{gathered} \tfrac12(ax^2+a_1y^2+a_2z^2)+(b_1y+b_2z)x-\gamma_1=h,\\ \gamma_1x+\gamma_2y+\gamma_3z=m,\\ \gamma_1^2+\gamma_2^2+\gamma_3^2=e^2. \end{gathered} \tag{2} \]

Let the center of gravity of the body lie on the perpendicular to the circular section of the gyration ellipsoid:

\[ a_2-a_1=0,\qquad b_2=0. \]

We assume that the initial value of the variable \(x\) is different from zero and that \(b_1\ne0\), as a result of which the conditions under which the Hess and Lagrange solutions occur are not fulfilled. Under these assumptions, equations (1) and the integrals (2) can be written as follows:

\[ \begin{aligned} dx/d\tau &= -zx,\\ dy/d\tau &= (a_0-b_0)zx+zy-\gamma,\\ dz/d\tau &= (b_0-a_0)xy-y^2+x^2+\beta,\\ d\alpha/d\tau &= -\gamma(b_0y+x)+b_0\beta z,\\ d\beta/d\tau &= \gamma(a_0x+y)-b_0\alpha z,\\ d\gamma/d\tau &= \alpha(b_0y+x)-\beta(a_0x+y); \end{aligned} \tag{3} \]

\[ \begin{gathered} \tfrac12(ax^2+a_1y^2+a_2z^2)+(b_1y+b_2z)x-\gamma_1=h_0,\\ \alpha x+\beta y+\gamma z=m_0,\qquad \alpha^2+\beta^2+\gamma^2=l_0^2, \end{gathered} \tag{4} \]

where

\[ \tau=b_1t,\qquad \alpha=\gamma_1/b_1,\qquad \beta=\gamma_2/b_1,\qquad \gamma=\gamma_3/b_1,\qquad a_0=a/b_1, \]

\[ b_0=a_1/b_1=a_2/b_1. \]

The system (3) admits the particular solution

\[ y=y_1x+y_2x^{-1}, \qquad z^2=r_1x^2+r_2x^{-2}+r_0, \]

\[ \alpha=\alpha_0+\alpha_1x^2, \qquad \beta=\beta_0+\beta_1x^2, \qquad \gamma=\gamma_0xz. \]

The dependence of the variables of the problem on time is determined by means of the equation

\[ dx/dt=-zx, \]

which can be transformed to the form

\[ (dx/dt)^2=-r_1b_1^2(x_1^2-x^2)(x^2-x_2^2). \]

The constants \(y_1, y_2, r_1, r_2, r_0, \alpha_0, \alpha_1, \beta_0, \beta_1, \gamma_0, x_1^2, x_2^2\) are expressed in terms of the parameters \(a_0, b_0\) as follows:

\[ 3y_1=b_0-2a_0+\delta, \qquad \delta=\pm\sqrt{a_0^2-a_0b_0+b_0^2+3}, \]

\[ 3\gamma_0=-(a_0+b_0)+2\delta, \qquad \alpha_0=\pm l_0, \qquad \beta_0=b_0\alpha_0, \]

\[ (4+b_0^2)\alpha_1=\gamma_0(3b_0y_1+a_0b_0+2), \]

\[ (4+b_0^2)\beta_1=\gamma_0[(b_0^2-2)y_1+b_0-2a_0], \]

\[ y_2=\frac{\beta_0}{\gamma_0}, \qquad r_1=-\frac{1}{\gamma_0^2}(\alpha_1^2+\beta_1^2), \qquad r_0=-\frac{2}{\gamma_0^2}(\alpha_0\alpha_1+\beta_0\beta_1), \qquad r_2=-\frac{\beta_0^2}{\gamma_0^2}, \]

\[ x_1^2=\frac{1}{2r_1}\left(-r_0+\sqrt{r_0^2-4r_1r_2}\right), \qquad x_2^2=\frac{1}{2r_1}\left(-r_0-\sqrt{r_0^2-4r_1r_2}\right). \]

The signs of these constants, except for \(y_1\), can be determined from the inequalities

\[ \delta b_0<0, \qquad \delta\gamma_0>0, \qquad b_0\alpha_1>0, \qquad b_0\alpha_0<0, \]

\[ \beta_0<0, \qquad \beta_1>0, \qquad r_1<0, \qquad r_2<0, \qquad r_0>0, \qquad b_0y_2>0. \]

Institute of Mechanics and Computing Center
of the Academy of Sciences of the Uzbek SSR

Received
5 VII 1965

REFERENCES

1. P. V. Kharlamov, Prikl. matem. i mekhanika, 27, 4 (1963).