Abstract Generated abstract
This paper studies particular periodic solutions of the Euler-Poisson equations for the motion of a heavy rigid body about a fixed point, in the limiting case where the initial rotation about a principal inertia axis is large and the parameter satisfies omega equals one half. It derives additional existence conditions for periodic solutions, including cases with nonzero initial transverse angular velocity components and cases where these components vanish, and gives asymptotic formulas for the period and corresponding Euler angles. The results are summarized in theorems specifying when such solutions exist, the number of arbitrary constants in the Euler-angle representations, and a necessary condition for further new periodic solutions. A special consequence is obtained for the Kovalevskaya case: any periodic solution under the stated rapid-rotation assumptions must satisfy p(0,0)q(0,0) equals zero and can involve only four arbitrary constants.
Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 159, No. 1
MECHANICS
Yu. A. ARKHANGELSKII
NEW PARTICULAR SOLUTIONS OF THE PROBLEM OF THE MOTION OF A HEAVY RIGID BODY ABOUT A FIXED POINT
(Presented by Academician A. Yu. Ishlinskii, 11 V 1964)
In paper (¹) new particular solutions were found of the Euler–Poisson equations
\[ A\frac{dp}{dt}+(C-B)qr=Mg(y_0\gamma''-z_0\gamma'),\qquad \frac{d\gamma}{dt}=r\gamma'-q\gamma'' \]
\[ \binom{ABC,\ pqr}{\gamma\gamma'\gamma'',\ x_0y_0z_0}, \tag{1} \]
under the conditions
\[ r_0\ \text{large},\qquad \gamma_0''\ne 0,\ \pm 1;\qquad \lim_{r_0\to\infty}(p_0^2+q_0^2)<\infty \qquad (u_0=u(t)_{t=0}) \tag{2} \]
and two theorems were formulated concerning the motion about a fixed point of a heavy rigid body set into rapid rotation about the greater or the smaller axis of its ellipsoid of inertia, under the assumption
\[ z_0\ne 0,\qquad \omega\ne \frac12 \left(\omega^2=\frac{(A-C)(B-C)}{AB}\right). \]
In paper (²), for \(\omega=1/2\), periodic solutions of system (1) were found under conditions (2), satisfying the additional relations
\[ A\geq B>C,\qquad p(t,\mu)_{\mu=0}=q(t,\mu)_{\mu=0}=0 \qquad \left(\mu^2=\frac{Mg\sqrt{x_0^2+y_0^2+z_0^2}}{r_0^2C}\right). \]
- It can be shown that, for \(\omega=1/2\), when the relations
\[ s^2=z_0(C-A)R_1^2 \pm \frac{\sqrt{1-\gamma_0''{}^2}}{\gamma_0''} \sqrt{x_0^2L_2^2+y_0^2L_3^2}>0; \tag{3} \]
\[ (x_0^2+y_0^2)(3B-2C)\ne 0; \tag{4} \]
\[ R_1^2=\frac{3(A+B)C-4AB-2C^2}{AB(C-B)},\qquad L_2=\frac{3B-2C}{4B},\qquad L_3=\frac{3B-2C}{8(B-C)} \]
are fulfilled, system (1) under conditions (2) has periodic solutions of period \(T\)
\[ T=\frac{4\pi}{r_0} -\frac{2\pi}{r_0^3} \left[ Ap^2(0,0)+Bq^2(0,0)+2Mg\left(z_0\gamma_0''-x_0\sqrt{1-\gamma_0''{}^2}\right) \right] +\frac{1}{r_0^4}(\cdots), \]
for which
\[ p^2(0,0)+q^2(0,0)\ne 0, \tag{5} \]
\[ p(0,0)=\pm R_2s\sqrt{1\pm s_1},\qquad q(0,0)=\pm \frac{2CR_2s}{3B-4C}\sqrt{1\mp s_1}, \]
\[ R_2^2=\frac{MgB\gamma_0''}{(A-C)(A+B-2C)},\qquad s_1^2=\frac{x_0^2l_2^2}{x_0^2L_2^2+y_0^2L_0^2}. \]
In this case, as the \(z\)-axis of the moving coordinate system one chooses an axis for which the inequality \(\gamma_0''>0\) is satisfied.
The expressions for the Euler angles \(\theta,\varphi,\psi\) corresponding to these solutions will be
\[ \begin{gathered} r_0(\theta-\theta_0)= \frac{s_2}{6(C-B)}[\theta_1(t+h)-\theta_1(h)] +\frac{1}{r_0}(\ldots),\\ r_0(\psi-\psi_0)= -\frac{Mgz_0}{C}t+ \frac{s_2}{6(C-B)\sqrt{1-\gamma_0''^{\,2}}} [\psi_1(t+h)-\psi_1(h)] +\frac{1}{r_0}(\ldots),\\ \varphi-\varphi_0=r_0t+\frac{1}{r_0}(\ldots), \end{gathered} \tag{6} \]
\[ \begin{gathered} \theta_1(t)=(A-2B+2C)\cos\left({}^{3}/_{2}\,r_0t-\varepsilon\right) -3(A+2B-2C)\cos\left({}^{1}/_{2}\,r_0t+\varepsilon\right),\\ \psi_1(t)=(A-2B+2C)\sin\left({}^{3}/_{2}\,r_0t-\varepsilon\right) -3(A+2B-2C)\sin\left({}^{1}/_{2}\,r_0t+\varepsilon\right),\\ \operatorname{tg}\varepsilon=\pm\sqrt{\frac{1\mp s_1}{1\pm s_1}},\qquad s_2^2=p^2(0,0)+\frac{(3B-4C)^2}{4C^2}q^2(0,0),\\ r_0h=\varphi_0+\frac{\pi}{2}+\frac{1}{r_0}(\ldots). \end{gathered} \]
Moreover, when the relation
\[ z_0(C-A)R_1^2 \pm \frac{\sqrt{1-\gamma_0''^{\,2}}}{\gamma_0''} \sqrt{x_0^2L_2^2+y_0^2L_3^2} \ne 0 \tag{7} \]
is fulfilled, for the cases \(A\ge B>C\) and \(C>B\ge A\) \((\omega={}^{1}/_{2})\) the system (1) under conditions (2) has periodic solutions of period \(T\)
\[ T=\frac{4\pi}{r_0} -\frac{4\pi}{r_0^3}Mg\left(z_0\gamma_0''-x_0\sqrt{1-\gamma_0''^{\,2}}\right) +\frac{1}{r_0^4}(\ldots), \]
for which
\[ p(0,0)=q(0,0)=0. \tag{8} \]
The expressions for the Euler angles corresponding to these solutions are determined from formulas (7.4) of paper \((^2)\).
Let us note that condition (7), for any fixed values of
\(x_0,y_0,z_0\ (x_0^2+y_0^2+z_0^2\ne0);\ A,B,C\)
\((\omega={}^{1}/_{2},\ A\ge B>C,\ C>B\ge A)\)
may hold for arbitrary values of \(\theta_0\) \((0<\theta_0<\pi/2)\), with the exception of one value \(\theta_0=\theta_0^*\),
\[ \operatorname{tg}\theta_0^* = \frac{|z_0(C-A)|R_1^2} {\sqrt{x_0^2L_2^2+y_0^2L_3^2}}, \qquad \text{if } z_0(x_0^2+y_0^2)(3B-2C)\ne0. \]
Condition (3) may hold for all values of \(\theta_0\), or on the whole interval
\(0<\theta_0<\pi/2\), or on one of its parts
\(0<\theta_0<\theta_0^*\), \(\theta_0^*<\theta_0<\pi/2\),
depending on the sign of the quantity \(z_0(C-A)\) and on the sign before the radical in the expression for \(s^2\).
Formulas (6), (7.4) \((^2)\), which depend on four arbitrary constants \(\theta_0\) (in the indicated domains), \(\varphi_0,\psi_0,r_0\) (\(r_0\) large), make it possible to study the motion of a heavy rigid body in the case under consideration.
- From the results obtained and the results of paper \((^2)\), corresponding to the case \(\omega={}^{1}/_{2}\), and of paper \((^3)\), the following theorems follow.
Theorem 1. The equations (1), under the conditions (2) and the condition \(\omega=\frac12\), possess periodic solutions satisfying the relations (5) or (8) when the conditions (3), (4), or (7), respectively, are fulfilled.
To these periodic solutions there correspond expressions for the Euler angles (6), (7.4) \(^2\), depending on four arbitrary constants \(\theta_0\) (in the indicated domains), \(\varphi_0\), \(\psi_0\), \(r_0\) (\(r_0\) large).
For any periodic solution of the equations (1) satisfying the conditions (2) and the condition \(\omega=\frac12\) and the corresponding relations (3), (4), or (7), the expressions for the Euler angles are given by formulas (6), (7.4).
Theorem 2. For the system (1), under the conditions (2) and the condition \(\omega=\frac12\), to have new periodic solutions to which: 1) there correspond expressions for the Euler angles depending on five arbitrary constants, or 2) the relation (5) is satisfied, it is necessary that
\[ z_0(C-A)>0,\qquad (x_0^2+y_0^2)(3B-2C)=0. \]
In the Kovalevskaya case \((A=B=2C,\ y_0=z_0=0)\), the relations between the moments of inertia satisfy the condition \(\omega=\frac12\). Therefore, on the basis of the results obtained, we have
Theorem 3. In the Kovalevskaya case, any periodic solutions of the equations (1) under the conditions (2) satisfy the relation \(p(0,0)q(0,0)=0\), and the corresponding expressions for the Euler angles can depend only on four arbitrary constants.
Moscow State University
named after M. V. Lomonosov
Received
8 V 1964
CITED LITERATURE
\(^1\) Yu. A. Arkhangel’skii, DAN, 158, No. 2 (1964).
\(^2\) Yu. A. Arkhangel’skii, Prikl. matem. i mekh., 27, no. 5 (1963).
\(^3\) Yu. A. Arkhangel’skii, Prikl. matem. i mekh., 28, no. 5 (1964).