The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point

In the theory of motion of several coupled rigid bodies about a fixed point one can distinguish three basic ramifications. 1. The first, the so-called classical direction of investigations, is concerned with particular cases of integrability ot the equations of motion of a single rigid body about a fixed point,1 and with their geo­ metrical interpretation. This path of thought was predominant until the beginning of the 20th century and its most illustrious represen­ tatives are L. EULER (1707-1783), J L. LAGRANGE (1736-1813), L. POINSOT (1777-1859), S. V. KOVALEVSKAYA (1850-1891), and others. Chapter I of the present monograph intends to reflect this branch of investigations. For collateral reading on the general questions dealt with in this chapter the reader is referred to the following textbooks and reports: A. DOMOGAROV [1J, F. KLEIN and A. SOMMERFELD [11, 1 , 1 J, A. G. 2 3 GREENHILL [10J, A. GRAY [1J, R. GRAMMEL [4 J, E. J. ROUTH [21' 2 , 1 2 31' 32J, J. B. SCARBOROUGH [1J, and V. V. GOLUBEV [1, 2J.

I: Single rigid body.- I: Heavy rigid body.- A. General solution of the Euler and Poisson equations.- § 1. The Euler angles.- 1.1 Definition.- 1.2 The direction cosines of 0x, 0y, 0z as functions of the Euler angles.- 1.3 The components of the angular velocity ?? as functions of the Euler angles.- § 2. The Euler and Poisson equations of motion.- 2.1 The dynamical equations of Euler.- 2.2 The Poisson kinematical equations.- 2.3 Finding of the first integrals.- 2.4 On the number of independent integrals.- § 3. Case of Euler and Poinsot.- 3.1 The first integrals.- 3.2 Symmetric notations for the constants I and h.- 3.3 Calculation of the instantaneous rotation.- 3.4 Calculation of the Euler angles.- § 4. Calculation of the Poinsot motion.- 4.1 Introductory remarks.- 4.2 The angular velocity components and the Euler angles.- 4.3 Proper rotation of the Poinsot motion.- 4.4 Precessional rotation of the Poinsot motion.- 4.5 Nutation of the Poinsot motion.- 4.6 Estimation of the validity of the above results.- 4.7 On some finite relations among the Euler angles.- § 5. Case of Lagrange and Poisson.- 5.1 The first integrals.- 5.2 Reduction of the Euler equations of motion.- 5.3 The sign of the precession.- 5.4 Upper and lower bounds for the apsidal angle.- 5.41 The spherical pendulum.- 5.5 Stability of a particular solution.- 5.6 Cases differing slightly from those of Euler and Poinsot, and Lagrange and Poisson.- § 6. Case of Kovalevskaya.- 6.1 The first integrals.- 6.2 Introduction of the new variables s1 and s2.- 6.3 Transformation of the elliptic differential % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaWcaaWdaeaapeGaamizaiaadIhaa8aabaWdbmaakaaapaqaa8qa % caWGsbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaSqabaaaaa % aa!3BFE!$$\frac{{dx}}{{\sqrt {R\left( x \right)} }}$$.- 6.4 Differential relations between s and x, and s and t.- 6.5 Expressions for p and q in terms of s1 and s2.- 6.6 Expressions for r, ?, ? and ? in terms of s1 and s2.- 6.7 Stability of a particular solution.- 6.8 Concluding remarks concerning the Euler and Poisson equations.- § 7. Existence of single-valued solutions.- 7.1 Introduction.- 7.2 Existence of algebraic integrals.- 7.3 Lyapunov’s theorem.- 7.31 Arbitrary initial values.- 7.32 Real initial values.- 7.33 Real initial values with % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaHXoqypaWaa0baaSqaa8qacaaIWaaapaqaa8qacaaIYaaaaOGa % ey4kaSIaeqOSdi2damaaDaaaleaapeGaaGimaaWdaeaapeGaaGOmaa % aakiabgUcaRiabeo7aN9aadaqhaaWcbaWdbiaaicdaa8aabaWdbiaa % ikdaaaGccqGH9aqpcaaIXaaaaa!4441!$$\alpha _0^2 + \beta _0^2 + \gamma _0^2 = 1$$.- B. Particular solutions of the Euler and Poisson equations.- § 8. Particular cases of integrability.- 8.1 Introduction.- 8.2 Case of a loxodromic pendulum.- 8.3 Permanent rotations.- 8.31 The mass center cone and the mass center curve.- 8.32 Special cases B = C and A = B.- 8.33 Stability of permanent rotations.- 8.34 Applications to particular cases of motion.- 8.4 Case of Steklov and Bobylev.- 8.5 Case of Gorya?ev and ?aplygin.- 8.6 Other cases of Gorya?ev, Steklov and ?aplygin.- 8.61 Second case of Gorya?ev.- 8.62 Second case of Steklov.- 8.63 Second case of Caplygin.- 8.7 Case of Mercalov.- 8.8 Center of mass lies in the characteristic plane.- 8.9 Case of N. Kowalewski.- 8.10 Cases of Corliss and Field.- 8.11 Center of mass lies on one of the principal planes of inertia.- 8.12 Regular precessions about nonvertical axes.- 8.13 Case of Mrs. Harlamova.- 8.14 Linear integrals.- 8.15 The principle of gyroscopic effect.- 8.16 Intrinsic equations of motion.- 8.17 Other cases of integrability.- C. Application of Lie series to the Euler and Poisson equations.- § 9. Lie series and their application to the study of motion of a heavy rigid body about a fixed point.- 9.1 Definition of generalized Lie series.- 9.2 Convergence of generalized Lie series.- 9.3 Operations with generalized Lie series.- 9.4 First integrals of a system of ordinary differential equations.- 9.5 First integrals of canonical equations.- 9.6 First integrals in the problem of motion of a heavy rigid body about a fixed point.- II: Self ― excited rigid body.- § 10. Self-excited symmetric rigid body.- 10.1 Introduction.- 10.2 The angular velocity of a rigid body subject to a time-independent self-excitement with a fixed direction in the body.- 10.3 Formulas describing rotations.- 10.4 The angles of rotation of a rigid body subject to a time-independent self-excitement with a fixed direction in the body.- 10.5 Self-excited symmetric rigid body in a viscous medium.- 10.51 Equations of motion.- 10.52 The angular velocity of a rigid body.- 10.53 Time-independent torque vector fixed in direction within the body.- 10.54 The asymptotic motion of the spin vector.- § 11. Self-excited asymmetric rigid body.- 11.1 Torque vector fixed along the axis of either the largest or the smallest principal moment of inertia.- 11.11 Torque vector fixed along the largest principal axis.- 11.111 A qualitative discussion of the motion.- 11.112 The motion of the spin vector in the unperturbed case.- 11.12 Torque vector fixed along the smallest principal axis.- 11.2 Torque vector fixed along the middle principal axis.- 11.21 Equations of motion and their integration.- 11.22 A qualitative discussion of the motion of the spin vector ? with respect to the moving trihedral in the unperturbed case.- § 12. Approximate solutions.- 12.1 Periodic solutions.- 12.11 Periodic solutions in the case of a time-dependent torque vector fixed along the largest principal axis.- 12.12 Periodic solutions in the case of a time-dependent torque vector fixed along the middle principal axis.- 12.2 Iterative solutions.- § 13. Regulation of rotations about fixed axes by self-excitements with fixed axes.- 13.1 Time-independent rotations caused by time-independent self-excitement.- 13.11 Stability of time-independent rotations.- 13.12 Stabilization of unstable time-independent rotations.- 15.21 Time-dependent rotations about fixed axes caused by self-excitements with fixed axes.- 13.22 Time-dependent rotations about fixed axes caused by self-excitements with variable axes.- III: Externally excited rigid body.- § 14. Symmetric rigid body subject to a periodic torque.- 14.1 Statement of the problem.- 14.11 Alternating field parallel to the constant field.- 14.12 Alternating field orthogonal to the constant field.- 14.2 Symmetric rigid body subject to gravitation and an orthogonal sinusoidal periodic force.- 14.3 Other types of torques.- II: Several coupled rigid bodies.- IV: Gyrostats.- § 15. Permanent axes of rotation of a heavy gyrostat about a fixed point.- 15.1 Equations of motion of a gyrostat.- 15.2 Permanent rotations of a gyrostat.- 15.3 Other systems for which permanent rotations are possible.- 15.4 The existence of first integrals.- 15.5 Regular precession of a gyrostat.- 15.6 Equations of motion in terms of non-Eulerian angles.- §16. Asymmetric body subject to a self-excitement in the equatorial plane.- 16.1 Equations of motion.- 16.2 Symmetric body.- 16.3 Asymmetric body.- 16.4 Gyroscopic function W.- 16.5 Nonstationary solutions for ??.- 16.6 Determination of the position relative to a fixed coordinate trihedral.- 16.7 Permanent rotations.- 16.8 Motions corresponding to the stationary solutions of ??.- V: Gyroscope in a Cardan suspension.- §17. Aspects of the Cardan suspension of gyroscopes.- 17.1 Introduction.- 17.2 Statement of the problem.- 17.3 Equations of motion and their first integrals.- 17.4 Solution in the case where the axis of the outer gimbal ring is vertical.- 17.5 Regular precessions.- 17.6 Stability in the case where the position of the axis of the outer gimbal ring is vertical.- 17.7 Stability in the case where the position of the axis of the outer gimbal ring is inclined.- 17.8 Gyroscopes subject to various perturbing moments.- 17.9 Gyroscopes on elastic foundations and moving bases.- 17.10 Application to inertial guidance systems.- III: Gyroscopes and artificial Earth satellites.- VI: Rigid body in a central Newtonian field of forces.- § 18. Motion of a rigid body with a fixed point in a central NEWTONian field of forces.- 18.1 Calculation of the force function.- 18.2 Calculation of the resultant force and moment.- 18.3 Equations of motion and their first integrals.- 18.4 Stability of rotation of a body fixed at its mass center.- 18.5 Stability of rotation in the case A = B and U = U(?).- 18.6 Stability of permanent rotations.- 18.7 Motion and stability of a gyroscope in a Cardan suspension.- 18.8 Gyrostat in a central NEWTONian force field.- VII: Motion of an artificial Earth satellite about its mass center.- § 19. The problem of separating the general motion of mutually attracting rigid bodies into translations of their mass centers and rotations about the latter.- 19.1 Resultant force and moment.- 19.2 Equations of motion.- 19.3 The first integrals.- 19.4 Separation of the system of differential equations of motion.- 19.5 Relative differential equations of motion.- 19.6 The two-body problem.- 19.7 Particular solutions.- § 20. Motion of an artificial Earth satellite.- 20.1 Introduction.- 20.2 Equations of motion.- 20.3 Integration of the equations of motion.- Author Index.