Rigid Body Dynamics: A Lagrangian Approach

Introduction
Contents
Part I The Elementary Part
1 Distribution of Mass
	1.1 The Moment of Mass—The Centre of Mass
		1.1.1 Moments of a Mass Distribution
		1.1.2 Centre of Mass
	1.2 Second Moments and Inertia Matrix of a Mass Distribution
		1.2.1 Second Moments Matrix of Mass Distribution
		1.2.2 Inertia Matrix of Mass Distribution
	1.3 Properties of the Inertia Matrix
		1.3.1 The Triangle Inequalities
		1.3.2 Theorem of Parallel Axes
		1.3.3 Ellipsoid of Inertia
		1.3.4 The Gyration Ellipsoid
		1.3.5 Representation of Principal Moments of Inertia
	1.4 Relations Between the Centre of Mass and the Inertia Ellipsoid
	1.5 Solved Examples
	1.6 Exercises
2 Description of Rotation of a Rigid Body About a Fixed Point
	2.1 The Position of a Rigid Body. Euler's Angles
	2.2 The Rotation Matrix
		2.2.1 The Angle of Rotation
	2.3 Description of Finite Rotation
	2.4 Representation of Finite Rotation by Means of a Vector
		2.4.1 The Rotation Vector
	2.5 Hamilton–Rodrigues' Parameters
	2.6 The Angular Velocity Vector
	2.7 Space and Relative Time Rates of Change of a Vector
		2.7.1 Components of the Angular Velocity in the Body Axes and Space Axes
		2.7.2 The Use of the Euler–Rodrigues Parameters
	2.8 Quaternions and Representation of Finite Rotation
	2.9 Composition of Two Rotations
	2.10 Exercises
3 The Classical Problem: The Motion of a Heavy Rigid Body About a Fixed Point
	3.1 Equations of Motion
	3.2 The Heavy Rigid Body
	3.3 The Angular Momentum of a Rigid Body
	3.4 Kinetic Energy of a Moving Body
	3.5 Equations of Motion in the Moving Coordinate System
		3.5.1 The Use of the Variables ω,γ. Special Axes Related to the Inertia Matrix
	3.6 Integrals of Motion
		3.6.1 The Energy Integral
		3.6.2 The Area's Integral
		3.6.3 The Geometric Integral
		3.6.4 Exercise
	3.7 Special Axes Associated with the Inertia Matrix
	3.8 The Use of Principal Axes of Inertia of the Body
	3.9 Determination of Euler's Angles
	3.10 The Movable and Immovable Hodographs
	3.11 The Use of the Variables G,γ . Special Axes Associated with the Gyration Ellipsoid
	3.12 Equations of Motion in Generalized Coordinates
	3.13 Canonical Equations of Motion in Euler's Angles
	3.14 The Routhian Reduction
	3.15 Exercises
4 General and Conditional Integrable Cases of the Classical Problem
	4.1 Euler's Case (1758). The Torque-Free Rigid Body
		4.1.1 Explicit Time-Solution
		4.1.2 Permanent Rotations
		4.1.3 The Degeneracy of Elliptic Function
		4.1.4 The Case of Dynamically Axi-Symmetric Body
		4.1.5 Euler's Angles in Terms of Time
		4.1.6 Geometrical Interpretation of the Motion (Poinsot 1851)
	4.2 Lagrange's Case (1788). The Top with a Fixed Point
		4.2.1 The Solution
		4.2.2 The Study of the Motion
	4.3 Kowalevski's Case (1888)
		4.3.1 Integration of the Equations of Motion
	4.4 The Goryachev–Chaplygin Case: A Conditional Integrable Case
		4.4.1 The Fourth Integral
		4.4.2 Separation of Variables. Solution of the Equations of Motion
	4.5 Integrability and Nonintegrability Issues
5 The Motion of a Heavy Gyrostat
	5.1 Models of the Gyrostat
		5.1.1 The Classical Model
		5.1.2 The Free Rotor Model
		5.1.3 Joukovsky's Model
	5.2 Equations of Motion in Hamiltonian Form
	5.3 Tables of Integrable Cases
	5.4 The Case of Joukovsky and Volterra
	5.5 The Case of Axially Symmetric Gyrostat
	5.6 Yehia's Case
		5.6.1 Separation of Variables
	5.7 The Conditional Case of Sretensky
	5.8 Some Applications of the Gyrostat Motion
	5.9 Exercises
6 Motion of a Rigid Body About a Fixed Point in the Field of a Distant Newtonian Centre and Brun's Problem
	6.1 Approximate Form of the Potential
	6.2 Brun's Problem
	6.3 Equations of Motion and Integrals of Motion
	6.4 Integrable Cases
		6.4.1 Brun's Case br (Analog of Euler's Case)
		6.4.2 The Generalization of Lagrange's Case
		6.4.3 The Place of Brun's Potential
	6.5 Exercises
7 The Motion of a Body with No Fixed Point
	7.1 General Considerations
	7.2 Poisson's Top. A Top on a Smooth Horizontal Plane
		7.2.1 Regular Precession of Poisson's Top
	7.3 Exercises
Part II The Advanced Part
8 Particular Solutions of the Classical Problem and Its Generalizations
	8.1 The Notion of a Particular Solution
	8.2 Planar Motion (Motion of the Body as a Physical Pendulum)
		8.2.1 Rotational Motion
		8.2.2 Vibrational Motion
		8.2.3 The Limiting Motion
		8.2.4 Orbital Stability
	8.3 Permanent Rotation of a Heavy Rigid Body About a Fixed Point staude (1894)
		8.3.1 Possible Axes of Permanent Rotation
		8.3.2 Description of the Motion
		8.3.3 Further Studies
		8.3.4 Excercises
	8.4 Hess' Case (1890)
		8.4.1 Equations of Motion
		8.4.2 Solution
		8.4.3 The Use of Special Axes
		8.4.4 Solution of the Case f=0
	8.5 The Case of Bobylev and Steklov (1896)
		8.5.1 Region I: The First Class of Motions
		8.5.2 Region II. The Second Class of Motions
	8.6 Steklov's Case (1899)
		8.6.1 Conditions and Solution
		8.6.2 The First Class
		8.6.3 The Second Class
		8.6.4 Some Properties of the Motion
		8.6.5 Exercises
	8.7 Goryachev's Case gor99 (1899)
		8.7.1 Conditions and Solution
		8.7.2 Properties of the Motion
		8.7.3 The History of Goryachev's Case
	8.8 Chaplygin's Case (1904)
		8.8.1 Properties of the Motion
	8.9 Kowalewski's Case kwski (1908)
	8.10 Grioli's Case (1947): The Regular Precession About a Tilted Axis
		8.10.1 Motion of the Centre of Mass
		8.10.2 Orbits of Motion on the Poisson Sphere
	8.11 Dokshevich's First Case dok66 (1966)
		8.11.1 Use of Special System of Axes
		8.11.2 Orbits on the Poisson Sphere
	8.12 The Case of Konosevich and Pozdnyakovich konspspoz,konspspoz1 (1968)
		8.12.1 The First Subcase
		8.12.2 The Second Subcase
	8.13 Dokshevich's Second Case dok70 (1970)
		8.13.1 Periodicity of the Motion
		8.13.2 Orbits on the Poisson Sphere
	8.14 Unsuccessful Cases and Incorrect Claims
		8.14.1 Shiff's Work shiff (1903)
		8.14.2 Field's Works field,field2 (1934)
		8.14.3 Corliss' Works corl,corl2 (1932–1934)
		8.14.4 Fabbri's Works fab1,fab2 (1934)
		8.14.5 Concerning Mertsalov's Work merts (1946)
		8.14.6 Gao's Work gao (2003)
		8.14.7 The Work of Yanxia and Keying yanx (2005)
		8.14.8 Ershkov's Work ersh (2014)
	8.15 Particular Solutions in the Problem of Motion of a Heavy Gyrostat
		8.15.1 Generalizations of Particular Cases Known in the Classical Problem
		8.15.2 Solvable Cases of the Gyrostat, Having No Classical Analog
		8.15.3 Known Cases of the Classical Problem, Which Are not Presently Generalized to the Gyrostat Problem
9 The Rigid Body in a Potential Field
	9.1 The Routhian in γ1,γ2,γ3 as Redundant Variables
		9.1.1 Expression of ω in Terms of γ and
		9.1.2 The Case of Complete Dynamical Symmetry of the Body
	9.2 Maximal Reduction of Order of the Differential Equations of Motion of a Rigid Body About a Fixed Point
		9.2.1 Reduction
		9.2.2 Applications
	9.3 Reduction to the Motion of a Particle on an Ellipsoid
	9.4 The Use of Elliptic Coordinates on the Inertia Ellipsoid kol
	9.5 The Use of Isometric Coordinates on the Inertia Ellipsoid
	9.6 Reduction to the Simplest Form of Orbital Differential Equation
		9.6.1 Sphero-Conic Coordinates on the Poisson Sphere yv
		9.6.2 Reduction to a Single Differential Equation yv
		9.6.3 Special Cases
	9.7 Separable Potentials in Rigid Body Dynamics (Conditional Integrable Problems)
		9.7.1 Potentials Separable for Axi-Symmetric Body
		9.7.2 Potentials Separable for an Asymmetric Body
		9.7.3 Potentials Separable for a Body of Spherical Dynamical Symmetry
	9.8 Exercises
10 The Problem of Motion of a Body  in a Liquid
	10.1 Equations of Motion
		10.1.1 Kirchhoff's Equations
		10.1.2 Example: Permanent Translational Motions
		10.1.3 Clebsch's Form of Kirchhoff's Equations
	10.2 Thomson-Lamb's Equations
	10.3 On Different Forms of the Equations of Motion
	10.4 A New Form of the Equations of Motion
	10.5 Steklov and Kharlamov Analogies and Their Generalization
		10.5.1 The Equivalent Problem of Motion About a Fixed Point
		10.5.2 Steklov's Analogy
		10.5.3 Kharlamov's Analogy
	10.6 Completing the Solution
		10.6.1 Solution of the Equivalent Problem
		10.6.2 Solution of the Original Problem
	10.7 Uniform Translational-Rotational Motion of a Body …
	10.8 Stationary Motions About an Axis Inclined  to the Vertical
	10.9 A Several-Parameter Particular Solution
	10.10 Alternative Hamiltonian Formulation
	10.11 The Uniform Precession Transformation yjm2
		10.11.1 Direct Derivation
		10.11.2 Lagrangian Derivation
		10.11.3 Physical and Mechanical Significance  of the Transformation
		10.11.4 Uniform Precession Transformation in Hamiltonian Formalism
	10.12 Generalization of General Integrable Cases
		10.12.1 Generalization of the Integrable Case Found  by Sokolov
		10.12.2 Steklov's Case and Its Generalizations
	10.13 Generalization of Conditional Integrable Cases
		10.13.1 Generalization of Goryachev–Chaplygin's, Sretensky's and Sokolov–Tsiganov Cases
	10.14 Generalizations of Particular Solvable Cases
		10.14.1 Example 1. Equilibria and Permanent Rotations About a Vertical Axis
		10.14.2 Example 2. Permanent Rotations About a Tilted Axis and Precessional Motions About the Vertical
		10.14.3 Example 3. generalization of grioli's precession  yjpall,yzamp
		10.14.4 Example 4. Regularly Precessing Pendulum
	10.15 Tables of Integrable Cases of Motion of a Rigid Body in a Liquid
		10.15.1 General Integrable Cases
		10.15.2 Conditional Integrable Cases on the Level f=0
	10.16 Further Studies on Integrable Cases
		10.16.1 Separation of Variables, Explicit Solutions and
		10.16.2 Topological Classification of Integrable Cases
	10.17 Chaplygin's Case of Integrability
		10.17.1 Separation of Variables
		10.17.2 Forms of Motion on the Poisson Sphere
		10.17.3 Explicit Solution
	10.18 Integrability Issues
		10.18.1 Results Concerning Kirchhoff's Equations
	10.19 Remark Concerning Particular Solutions  of the Problem
	10.20 The Donetsk School of Mechanics and Its Attitude  to Competing Works
		10.20.1 The Attitude to the Uniform Precession Transformation
		10.20.2 The Attitude to the Equations of Motion in the Form (10.45)
	10.21 Exercises
11 The General Problem of Motion of a Rigid Body Acted upon by a Coaxial Combination of Potential and Gyroscopic Forces
	11.1 Introduction
	11.2 Equations of Motion
	11.3 Relation to Grioli's and Kharlamov's Equations
		11.3.1 Grioli's Equations
		11.3.2 M. Kharlamov's Equations
	11.4 Potential of, and Torques on, a Heavy, Magnetized  and Electrically Charged Body
	11.5 On General and Conditional Integrable Cases in Rigid Body Dynamics
	11.6 Transformation of the Equations of Motion
	11.7 Maximal Reduction of the Order of the Equations  of Motion
		11.7.1 The Case of Complete Dynamical Symmetry
	11.8 Extensions of Integrable Problems
	11.9 Transformations of Cyclic Variables
12 The Most General Integrable Cases  in Rigid Body Dynamics
	12.1 General Integrable Cases
		12.1.1 Table of General Integrable Extensions of General Integrable Cases
		12.1.2 About the Hamiltonian Formulation
	12.2 Conditional Integrable Deformations of General Integrable Cases
		12.2.1 Table of Cases
		12.2.2 Example of Physical Application
13 Miscellaneous Cases Integrable on a Single Level of the Areas Integral
	13.1 Cases with a Quadratic Integral
		13.1.1 Separable Integrable Potentials
		13.1.2 Non-separable Cases with a Quadratic Integral
	13.2 Cases with a Cubic Integral
	13.3 Cases with a Quartic Integral
		13.3.1 Cases Stemming from Kowalevski's Case
		13.3.2 Cases Stemming from Chaplygin's Case
		13.3.3 Cases Combining Kowalevski's and Chaplygin's Cases
		13.3.4 A Case with a Quartic Integral Outside the Above Classification
		13.3.5 Two Conditional Cases Valid on a Single, Not Necessary Zero, Level of the Linear Integral
	13.4 Integrable Extensions of Conditional Integrable Cases
14 The Rigid Body Acted upon by a Skew Combination of Fields
	14.1 Equations of Motion
		14.1.1 Interpretation of Forces
		14.1.2 The Motion of a Magnetizable Rigid Body in an Ideal Fluid and In a Uniform Magnetic Field
		14.1.3 Example: The Motion of a Satellite in a Circular Orbit
	14.2 The Rigid Body (Gyrostat) Acted upon by More than One Uniform Field
		14.2.1 The Motion of a Body Acted upon by Two Uniform Fields
		14.2.2 The Motion of a Body Acted upon by Three Irreducible Uniform Fields
	14.3 Integrable Cases of a Body with a Homogeneous Quadratic Potential
		14.3.1 Brun's Case of the Asymmetric Body in an Asymmetric Gravitational Field
		14.3.2 Case of Dynamically Spherical Body
	14.4 The Motion of an Axi-Symmetric Body Under the Action of Asymmetric Forces
		14.4.1 Description of the Problem
		14.4.2 General Integrable Cases
		14.4.3 Conditional Integrable Cases
	14.5 Motion of a Body with Combined (Quaternion) Symmetry
		14.5.1 Introduction
		14.5.2 Routhian Reduction
		14.5.3 Equivalence of Two Problems
		14.5.4 Basic Equivalent Integrable Problems
		14.5.5 Generalization Through Transformation
Appendix A Some Useful Identities
Appendix B Kowalevski's Case: Appelrot's Four Classes  of Simple Motions
B.1 The First Class of Simplest Motions (Known as Delone's Case)
B.1.1 A Special Case
B.1.2 A Case of Rational Solution
B.2 The Second Class of Simplest Motions
B.3 The Third Class of Simplest Motions
B.4 The Fourth Class of Simplest Motions
B.5 Intersection of the Four Classes
Appendix C Particularly Simple Classes of Motions in Goryachev–Chaplygin's Case
C.1 Solution on the Boundary I (The Case of Goryachev)
C.1.0.1 The Subcase of Goryachev
C.1.1 The Solution
C.1.1.1 When Ein[-1,1)
C.1.1.2 When E>1
C.2 Solution on Boundary II
C.3 Solution on Boundary III
Appendix D Gyrostatic Generalization of the Appelrot Classes
Appendix E The Conditional Case of Sretensky
Appendix  Bibliography