目錄
Introduction
The Creators of Rigid Body Dynamics
1 Rigid Body Equations of Motion and Their Integration
1.1 Poisson Brackets and Hamiltonian Formalism
1 Poisson manifolds
Poisson brackets and their properties (19). Nondegenerate brackets,
Symplectic structures (22). Symplectic foliations. Generalized Darboux theorem (22)
2 The Lie-Poisson bracket
1.2 PoincarE and PoincarE-Chetaev Equations
1 PoincarE Equations
2 PoincarE-Chetaev equations
3 Equations on Lie groups
4 Comments
1.3 Various Systems of Variables in Rigid Body Dynamics
1 Euler angles
2 Euler variables. Components of momentum and the direction cosines
3 Quaternion Rodrigues-Hamilton parameters
4 Andoyer variables
5 Comments
1.4 Different Forms of Equations of Motion
1 Equations of motion of a rigid body with a fixed point
Euler-PoincarE equations in the group SO(3) (39). Equations of motion in angular velocities and quaternions (41). Kinetic energy of a rigid body with a fixed point (41).
2 Hamiltonian form of equations of motion for different systems of variables
Equations of motion in algebraic form (42). Quaternion representation
of the equations of motion (44). Canonical equations in Euler angles
and Andoyer variables (44).
3 PoincarE section and chaotic motions
1.5 Equations of Motion of a Rigid Body in Euclidean Space
1 Lagrangian formalism and Poincare equations on the group E(3)
2 Kinetic energy of a rigid body in R3
3 Hamiltonian form of equations of motion of a rigid body in R3
1.6 Examples and Similar Problems
1 Motion of a rigid body with a fixed point in a superposition of constant uniform force fields
2 A free rigid body in a quadratic potential
3 Motion of a rigid body with a fixed point in a rotating coordinate system
Gyroscope and Foucault pendulum (56). A satellite on a circular earth orbit (56).
4 Relative motion of a rigid body with a fixed point
5 Motion of a rigid body sliding on a smooth plane
6 Gyroscope in the Cardan suspension
7 Motion of a rigid body in an ideal incompressible fluid and the Kirchhoff equations
8 Falling rigid body in an ideal fluid
9 The Levitron
1.7 Theorems on Integrability and Methods of Integration
1 Hamiltonian systems. The Liouville-Arnold theorem
2 Theory of the last multiplier. Euler-Jacobi theorem
3 Separation of variables. Hamilton-Jacobi method
Geodesic flow on the ellipsoid (Jacobi problem) (73). System with
4 Linear Integrals and Reduction
5 Generalizations of Integrability Cases. Explicit Integration
6 Periodic Solutions, Nonintegrability, and Transition to Chaos
A Derivation of the KirchhoffPoincare一Zhukovskii, and Four·Dimensional Top Equations
B The Lie Algebra e(4)and Its Orbits
C Quaternion Equations and L-A Pair for the Generalized Goryachev——Chaplygin Top
D The Hess Case and Quantization of the Rotation Number
E Ferromagnetic Dynamics in a Magnetic Field
F The Landau—Lifshitz Equation, Discrete Systems, and the Neumann Problem
G Dynamics of Tops and Material Points on Spheres and Ellipsoids
H On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation
I The Hamiltonian Dynamics of Self-gravitating Fluid and Gas Ellipsoids
Bibliography
Glossary
Index
[