## A Treatise on the Analytical Dynamics of Particles and Rigid Bodies |

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### Contents

CHAPTER | 1 |

Eulers theorem on rotations about a point | 2 |

The theorem of Rodrigues and Hamilton | 3 |

Chasles theorem on the most general displacement of a rigid lody | 4 |

Halphens theorem on the composition of two general displacements fl 7 Analytic representation of a displacement | 6 |

The composition of small rotations | 7 |

Eulers parametric specification of rotations round a point | 8 |

The Eulerian angles | 9 |

Expression of the curvature of a path in terms of generalised coordinates | 256 |

Appells equations | 258 |

Bertrands theorem | 261 |

CHAPTER X | 263 |

Equations arising from the Calculus of Variations | 265 |

Integralinvariants | 267 |

The variational equations | 268 |

Integralinvariants of order one | 269 |

Connexion of the Eulerian angles with the parameters ij f | 10 |

the CayleyKlein parameters | 11 |

Vectors | 13 |

Velocity and acceleration their vectorial character | 14 |

Angular velocity its vectorial character | 15 |

Determination of the components of angular velocity of a system in terms of the Eulerian angles and of the symmetrical parameters | 16 |

Timeflux of a vector whose components relative to moving axes are given | 17 |

Special resolutions of the velocity and acceleration | 18 |

Miscellaneous Examples | 22 |

CHAPTER II | 26 |

The laws which determine motion | 27 |

Force | 29 |

Work | 30 |

Forces which do no work | 31 |

The coordinates of a dynamical system | 32 |

Holonomic and nonholonomic systems | 33 |

SECTION rAat 26 Lagranges form of the equations of motion of a holonomic system | 34 |

CHAPTER V | 35 |

Conservative forces the kinetic potential | 38 |

The explicit form of Lagranges equations | 39 |

Motion of a system which is constrained to rotate uniformly round an axis | 40 |

The Lagrangian equations for quasicoordinatcs | 41 |

Forces derivable from a potentialfunction which involves the velocities | 44 |

Initial motions | 45 |

Similarity in dynamical systems | 47 |

Impulsive motion | 48 |

The Lagrangian equations of impulsive motion | 50 |

Miscellaneous Examples | 51 |

CHAPTER III | 52 |

Systems with ignorable coordinates | 54 |

Special cases of ignoration integrals of momentum and angular momentum | 58 |

The general theorem of angular momentum | 61 |

The energy equation | 62 |

Reduction of a dynamical problem to a problem with fewer degrees of freedom by means of the energy equation | 64 |

Separation of the variables dynamical systems of Liouvilles tyjc | 67 |

Miscellaneous Examples | 69 |

CHAPTER IV | 71 |

Motion in a moving tube | 74 |

Motion of two interacting free particles | 76 |

Hamiltons theorem | 77 |

The intcgrable cases of central forces problems soluble in terras of circular and elliptic functions | 80 |

Motion under the Newtonian law | 91 |

The mutual transformation of fields of central force and fields of parallel force | 93 |

Bonnets theorem | 94 |

Determination of the most general field of force under which a given curve or family of curves can be described | 95 |

The problem of two centres of gravitation | 97 |

Motion on a surface 09 | 99 |

Motion on a surface of revolution cases soluble in terms of circular and elliptic functions | 103 |

Joukovskys theorem | 109 |

Miscellaneous Examples | 111 |

THE DYNAMICAL SPECIFICATION OF BODIES SECTION PAGE 57 Definitions | 117 |

The moments of inertia of some simple bodies | 118 |

Derivation of the moment of inertia about any axis when the moment of inertia about a parallel axis through the centre of gravity is known | 121 |

Connexion between moments of inertia with respect to different sets of axes through the same origin | 122 |

The principal axes of inertia Cauchys momental ellipsoid | 124 |

Calculation of the kinetic energy of a moving rigid body | 126 |

Independence of the motion of the centre of gravity and the motion relative to it | 127 |

Miscellaneous Examples | 129 |

CHAPTER VI | 131 |

The motion of systems with two degrees of freedom | 137 |

Initial motions | 141 |

The motion of systems with three degrees of freedom | 143 |

Motion of a body about a fixed point under no forces | 144 |

Poinsots kinematical representation of the motion the polhode and herpolhode | 152 |

Motion of a top on a perfectly rough plane determination of the Eulerian angle 6 | 155 |

Determination of the remaining Eulerian angles and of the CayleyKlein parameters the spherical top | 159 |

Motion of a top on a perfectly smooth plane | 163 |

Kowalevskis top | 164 |

SECTION PAOK 165 Poincares theorem | 165 |

Impulsive motion | 167 |

Miscellaneous Examples | 169 |

CHAPTER VII | 177 |

Normal coordinates | 178 |

Sylvesters theorem on the reality of the roots of the determinantal equation | 183 |

Solution of the differential equations the periods stability | 185 |

Examples of vibrations about equilibrium | 187 |

Effect of a new constraint on the periods of a vibrating system | 191 |

The stationary character of normal vibrations | 192 |

Vibrations about steady motion | 193 |

The integration of the equations | 195 |

Examples of vibrations about steady motion | 203 |

Vibrations of systems involving moving constraints | 207 |

Miscellaneous Examples | 208 |

CHAPTER VIII | 214 |

Equations of motion referred to axes moving in any manner | 216 |

Application to special nonholonomic problems | 217 |

Vibrations of nonholonomic systems | 221 |

Dissipative systems frictional forces | 226 |

Resisting forces which depend on the velocity | 229 |

Eayleighs dissipationfunction | 230 |

Vibrations of dissipative systems | 232 |

Impact | 234 |

Examples of impact | 235 |

MISCELLANEOUS EXAMPLES | 238 |

CHAPTER IX | 245 |

The principle of Least Action for conservative holonomic systems | 247 |

Extension of Hamiltons principle to nonconservative dynamical systems | 248 |

Extension of Hamiltons principle and the principle of Least Action to nonholonomic systems | 249 |

Are the stationary integrals actual minima? Kinetic foci | 250 |

Representation of the motion of dynamical systems by means of geodesies | 253 |

The leastcurvature principle of Gauss and Hertz | 254 |

Relative integralinvariants | 271 |

A relative integralinvariant which is possessed by all Hamiltonian systems | 272 |

The expression of integralinvariants in terms of integrals | 274 |

The theorem of Lie and Koenigs | 275 |

The last multiplier | 276 |

SECTION PAGE 120 Derivation of an integral from two multipliers | 279 |

Application of the last multiplier to Hamiltonian systems use of a single known integral | 280 |

Integralinvariants whose order is equal to the order of the system | 283 |

Reduction of differential equations to the Lagrangian form | 284 |

Case in which the kinetic energy is quadratic in the velocities | 285 |

Miscellaneous Examples | 286 |

CHAPTER XI | 288 |

Contacttransformations in space of any number of dimensions | 292 |

The bilinear covariant of a general differential form | 296 |

The conditions for a contacttransformation expressed by means of the bilinear covariant | 297 |

The conditions for a contacttransformation in terms of Lagranges bracket expressions | 298 |

Poissons bracketexpressions | 299 |

The conditions for a contacttransformation expressed by moans of Poissons bracketexpressions | 300 |

The subgroups of Mathieu transformations and extended pointtransforma tions | 301 |

Infinitesimal contacttransformations | 302 |

The resulting new view of dynamics | 304 |

Jacobis theorem on the transformation of a given dynamical system intu another dynamical system | 305 |

Representation of a dynamical problem by a differential forru | 307 |

The Hamiltonian function of the transformed equations | 309 |

Transformations in which the independent variable is changed | 310 |

Miscellaneous Examples | 311 |

CHAPTER XII | 313 |

Hamiltons partial differential equation | 314 |

Hamiltons integral as a solution of Hamiltons partial differential equation | 318 |

Poissons theorem | 320 |

The constancy of Lagranges bracketexpressions | 321 |

Involutionsystems | 322 |

SECTION PAOE 148 Solution of a dynamical problem when half the integrals arc known | 323 |

LeviCi vitas theorem | 328 |

Determination of the forces acting on a system for which an integral is known | 331 |

Application to the case of a particle whose equations of motion possess an integral quadratic in the velocities | 332 |

General dynamical systems possessing integrals quadratic in the velocities | 335 |

Miscellaneous Examples | 336 |

CHAPTER XIII | 339 |

The differential equations of the problem | 340 |

Jacobis equation | 342 |

Reduction to the 12th order by use of the integrals of motion of the centre of gravity | 343 |

Reduction to the 8th order by use of the integrals of angular momentum and elimination of the nodes | 344 |

Reduction to the 6th order | 347 |

Alternative reduction of the problem from the 18th to the 6th order | 348 |

The problem of three bodies in a plane | 351 |

The restricted problem of three bodies | 353 |

Extension to the problem of n bodies | 356 |

CHAPTER XIV | 358 |

iii An integral must involve the momenta | 359 |

iv Only one irrationality can occur in the integral | 360 |

v Expression of the integral as a quotient of two real polynomials | 361 |

vi Derivation of integrals from the numerator and denominator of the quotient | 362 |

vii Proof that o does not involve the irrationality | 366 |

viii Proof that f0 is a function only of the momenta and the integrals of angular momentum | 371 |

ix Proof that if0 is a function of T L M N | 374 |

x Deduction of Bruns theorem for integrals which do not involve I | 376 |

xi Extension of Bruns result to integrals which involve the time | 378 |

i The equations of motion of the restricted problem of three bodies | 380 |

ii Statement of Poincares theorem | 381 |

iii Proof that is not a function of U0 | 382 |

v Proof that the existence of a onevalued integral is inconsistent with the result of iii in the general case | 383 |

vi Removal of the restrictions on the coefficients Dm m | 384 |

vii Deduction of Poincares theorem | 385 |

CHAPTER XV | 386 |

Asymptotic solutions | 389 |

The motion of a particle on an ellipsoid under no external forces | 393 |

Ordinary and singular periodic solutions | 395 |

Characteristic exponents | 397 |

Characteristic exponents when t does not occur explicitly | 398 |

The characteristic exponents of a system which possesses a onovalued integral | 399 |

The theory of matrices | 400 |

The characteristic exponents of a Hamiltonian system | 402 |

The asymptotic solutions of 170 deduced from the theory of characteristic exponents | 405 |

The characteristic exponents of ordinary and singular periodic solutions | 406 |

periodic orbits in the vicinity | 409 |

The stability of orbits as affected by terms of higher order in the displacement | 412 |

Attractive and repellent regions of a field of force | 413 |

Application of the energy integral to the problem of stability | 416 |

Application of integralinvariants to investigations of stability | 417 |

Connexion with the theory of surface transformations | 420 |

CHAPTER XVI | 423 |

The regularination of the problem of three bodies | 424 |

Trigonometric series | 425 |

Removal of terms of the first degree from the energy function | 426 |

SECTION PAGE 192 Determination of the normal coordinates by a contacttransformation | 427 |

Transformation to the trigonometric form of H | 430 |

Other types of motion which lead to equations of the same form | 431 |

The problem of integration | 432 |

Determination of the adelphic integral in Case I | 433 |

An example of the adelphic integral in Case I | 436 |

The question of convergence | 437 |

Use of the adelphic integral in order to complete the integration | 438 |

The fundamental property of the adelphic integral | 442 |

Determination of the adelphic integral in Case II | 443 |

An example of the adelphic integral in Case II | 444 |

Determination of the adelphic integral in Case III | 446 |

An example of the adelphic integral in Case III | 447 |

Completion of the integration of the dynamical system in Cases II and III | 449 |

451 | |

453 | |

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies E. T. Whittaker Limited preview - 1988 |

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies E. T. Whittaker Limited preview - 1988 |

### Common terms and phrases

angle angular momentum angular velocity arbitrary constants axis centre of force centre of gravity coefficients Coll components constants of integration constraints contact-transformation corresponding curvature curve degrees of freedom denote determine differential form disc displacement distance dt dpr dynamical system elliptic elliptic functions equations of motion Exam Example expressed in terms external forces fixed point given Hamilton's Hamiltonian system holonomic holonomic systems horizontal ignorable coordinate impulse inertia initial integral of energy integral-invariant kinetic energy kinetic potential Lagrangian equations last multiplier linear mass Math moment of inertia obtained orbit partial differential equation perfectly rough perpendicular plane position potential energy problem of three qlt qt quadratic quantities radius respectively rigid body roots rotation satisfied shew solution sphere steady motion suppose surface theorem three bodies trajectory transformation values variables vector vertical vibrations zero

### Popular passages

Page ix - The moment of inertia of a body or system of bodies about any axis is equal to the moment of inertia about a parallel axis, through the centre of gravity, plus the moment of inertia of the whole mass collected at the centre of gravity about the original axis.