| Foreword | p. xiii |
| Preface to the fourth edition | p. xix |
| Kinematical Preliminaries | |
| The displacements of rigid bodies | p. 1 |
| Euler's theorem on rotations about a point | p. 2 |
| The theorem of Rodrigues and Hamilton | p. 3 |
| The composition of equal and opposite rotations about parallel axes | p. 3 |
| Chasles' theorem on the most general displacement of a rigid body | p. 4 |
| Halphen's theorem on the composition of two general displacements | p. 5 |
| Analytic representation of a displacement | p. 6 |
| The composition of small rotations | p. 7 |
| Euler's parametric specification of rotations round a point | p. 8 |
| The Eulerian angles | p. 9 |
| Connexion of the Eulerian angles with the parameters [small xi], [small eta], [small zeta], [small chi] | p. 10 |
| The connexion of rotations with homographies: the Cayley-Klein parameters | p. 11 |
| Vectors | p. 13 |
| Velocity and acceleration; their vectorial character | p. 14 |
| Angular velocity; its vectorial character | p. 15 |
| Determination of the components of angular velocity of a system in terms of the Eulerian angles, and of the symmetrical parameters | p. 16 |
| Time-flux of a vector whose components relative to moving axes are given | p. 17 |
| Special resolutions of the velocity and acceleration | p. 18 |
| Miscellaneous Examples | p. 22 |
| The Equations of Motion | |
| The ideas of rest and motion | p. 26 |
| The laws which determine motion | p. 27 |
| Force | p. 29 |
| Work | p. 30 |
| Forces which do no work | p. 31 |
| The coordinates of a dynamical system | p. 32 |
| Holonomic and non-holonomic systems | p. 33 |
| Lagrange's form of the equations of motion of a holonomic system | p. 34 |
| Conservative forces; the kinetic potential | p. 38 |
| The explicit form of Lagrange's equations | p. 39 |
| Motion of a system which is constrained to rotate uniformly round an axis | p. 40 |
| The Lagrangian equations for quasi-coordinates | p. 41 |
| Forces derivable from a potential-function which involves the velocities | p. 44 |
| Initial motions | p. 45 |
| Similarity in dynamical systems | p. 47 |
| Motion with reversed forces | p. 47 |
| Impulsive motion | p. 48 |
| The Lagrangian equations of impulsive motion | p. 50 |
| Miscellaneous Examples | p. 51 |
| Principles Available for the Integration | |
| Problems which are soluble by quadratures | p. 52 |
| Systems with ignorable coordinates | p. 54 |
| Special cases of ignoration; integrals of momentum and angular momentum | p. 58 |
| The general theorem of angular momentum | p. 61 |
| The energy equation | p. 62 |
| Reduction of a dynamical problem to a problem with fewer degrees of freedom, by means of the energy equation | p. 64 |
| Separation of the variables; dynamical systems of Lionville's type | p. 67 |
| Miscellaneous Examples | p. 69 |
| The Soluble Problems of Particle Dynamics | |
| The particle with one degree of freedom; the pendulum | p. 71 |
| Motion in a moving tube | p. 74 |
| Motion of two interacting free particles | p. 76 |
| Central forces in general: Hamilton's theorem | p. 77 |
| The integrable cases of central forces; problems soluble in terms of circular and elliptic functions | p. 80 |
| Motion under the Newtonian law | p. 86 |
| The mutual transformation of fields of central force and fields of parallel force | p. 93 |
| Bonnet's theorem | p. 94 |
| Determination of the most general field of force under which a given curve or family of curves can be described | p. 95 |
| The problem of two centres of gravitation | p. 97 |
| Motion on a surface | p. 99 |
| Motion on a surface of revolution; cases soluble in terms of circular and elliptic functions | p. 103 |
| Joukovsky's theorem | p. 109 |
| Miscellaneous Examples | p. 111 |
| The Dynamical Specification of Bodies | |
| Definitions | p. 117 |
| The moments of inertia of some simple bodies | p. 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 | p. 121 |
| Connexion between moments of inertia with respect to different sets of axes through the same origin | p. 122 |
| The principal axes of inertia; Cauchy's momental ellipsoid | p. 124 |
| Calculation of the angular momentum of a moving rigid body | p. 124 |
| Calculation of the kinetic energy of a moving rigid body | p. 126 |
| Independence of the motion of the centre of gravity and the motion relative to it | p. 127 |
| Miscellaneous Examples | p. 129 |
| The Soluble Problems of Rigid Dynamics | |
| The motion of systems with one degree of freedom; motion round a fixed axis, etc. | p. 131 |
| The motion of systems with two degrees of freedom | p. 137 |
| Initial motions | p. 141 |
| The motion of systems with three degrees of freedom | p. 143 |
| Motion of a body about a fixed point under no forces | p. 144 |
| Poinsot's kinematical representation of the motion; the polhode and herpolhode | p. 152 |
| Motion of a top on a perfectly rough plane; determination of the Eulerian angle [straight thetas] | p. 155 |
| Determination of the remaining Eulerian angles, and of the Cayley-Klein parameters; the spherical top | p. 159 |
| Motion of a top on a perfectly smooth plane | p. 163 |
| Kowalevski's top | p. 164 |
| Impulsive motion | p. 167 |
| Miscellaneous Examples | p. 169 |
| Theory of Vibrations | |
| Vibrations about equilibrium | p. 177 |
| Normal coordinates | p. 178 |
| Sylvester's theorem on the reality of the roots of the determinantal equation | p. 183 |
| Solution of the differential equations; the periods; stability | p. 185 |
| Examples of vibrations about equilibrium | p. 187 |
| Effect of a new constraint on the periods of a vibrating system | p. 191 |
| The stationary character of normal vibrations | p. 192 |
| Vibrations about steady motion | p. 193 |
| The integration of the equations | p. 195 |
| Examples of vibrations about steady motion | p. 203 |
| Vibrations of systems involving moving constraints | p. 207 |
| Miscellaneous Examples | p. 208 |
| Non-Holonomic Systems. Dissipative Systems | |
| Lagrange's equations with undetermined multipliers | p. 214 |
| Equations of motion referred to axes moving in any manner | p. 216 |
| Application to special non-holonomic problems | p. 217 |
| Vibrations of non-holonomic systems | p. 221 |
| Dissipative systems; frictional forces | p. 226 |
| Resisting forces which depend on the velocity | p. 229 |
| Rayleigh's dissipation-function | p. 230 |
| Vibrations of dissipative systems | p. 232 |
| Impact | p. 234 |
| Loss of kinetic energy in impact | p. 234 |
| Examples of impact | p. 235 |
| Miscellaneous Examples | p. 238 |
| The Principles of Least Action and Least Curvature | |
| The trajectories of a dynamical system | p. 245 |
| Hamilton's principle for conservative holonomic systems | p. 245 |
| The principle of Least Action for conservative holonomic systems | p. 247 |
| Extension of Hamilton's principle to non-conservative dynamical systems | p. 248 |
| Extension of Hamilton's principle and the principle of Least Action to non-holonomic systems | p. 249 |
| Are the stationary integrals actual minima? Kinetic foci | p. 250 |
| Representation of the motion of dynamical systems by means of geodesics | p. 253 |
| The least-curvature principle of Gauss and Hertz | p. 254 |
| Expression of the curvature of a path in terms of generalised coordinates | p. 256 |
| Appell's equations | p. 258 |
| Bertrand's theorem | p. 260 |
| Miscellaneous Examples | p. 261 |
| Hamiltonian Systems and Their Integral-Invariants | |
| Hamilton's form of the equations of motion | p. 263 |
| Equations arising from the Calculus of Variations | p. 265 |
| Integral-invariants | p. 267 |
| The variational equations | p. 268 |
| Integral-invariants of order one | p. 269 |
| Relative integral-invariants | p. 271 |
| A relative integral-invariant which is possessed by all Hamiltonian systems | p. 272 |
| On systems which possess the relative integral-invariant [integral sign][summation]p[small delta]q | p. 272 |
| The expression of integral-invariants in terms of integrals | p. 274 |
| The theorem of Lie and Koenigs | p. 275 |
| The last multiplier | p. 276 |
| Derivation of an integral from two multipliers | p. 279 |
| Application of the last multiplier to Hamiltonian systems; use of a single known integral | p. 280 |
| Integral-invariants whose order is equal to the order of the system | p. 283 |
| Reduction of differential equations to the Lagrangian form | p. 284 |
| Case in which the kinetic energy is quadratic in the velocities | p. 285 |
| Miscellaneous Examples | p. 286 |
| The Transformation-Theory of Dynamics | |
| Hamilton's Characteristic Function and contact-transformations | p. 288 |
| Contact-transformations in space of any number of dimensions | p. 292 |
| The bilinear covariant of a general differential form | p. 296 |
| The conditions for a contact-transformation expressed by means of the bilinear covariant | p. 297 |
| The conditions for a contact-transformation in terms of Lagrange's bracket-expressions | p. 298 |
| Poisson's bracket-expressions | p. 299 |
| The conditions for a contact-transformation expressed by means of Poisson's bracket-expressions | p. 300 |
| The sub-groups of Mathieu transformations and extended point-transformations | p. 301 |
| Infinitesimal contact-transformations | p. 302 |
| The resulting new view of dynamics | p. 304 |
| Helmholtz's reciprocal theorem | p. 304 |
| Jacobi's theorem on the transformation of a given dynamical system into another dynamical system | p. 305 |
| Representation of a dynamical problem by a differential form | p. 307 |
| The Hamiltonian function of the transformed equations | p. 309 |
| Transformations in which the independent variable is changed | p. 310 |
| New formulation of the integration-problem | p. 310 |
| Miscellaneous Examples | p. 311 |
| Properties of the Integrals of Dynamical Systems | |
| Reduction of the order of a Hamiltonian system by use of the energy integral | p. 313 |
| Hamilton's partial differential equation | p. 314 |
| Hamilton's integral as a solution of Hamilton's partial differential equation | p. 316 |
| The connexion of integrals with infinitesimal transformations admitted by the system | p. 318 |
| Poisson's theorem | p. 320 |
| The constancy of Lagrange's bracket-expressions | p. 321 |
| Involution-systems | p. 322 |
| Solution of a dynamical problem when half the integrals are known | p. 323 |
| Levi-Civita's theorem | p. 325 |
| Systems which possess integrals linear in the momenta | p. 328 |
| Determination of the forces acting on a system for which an integral is known | p. 331 |
| Application to the case of a particle whose equations of motion possess an integral quadratic in the velocities | p. 332 |
| General dynamical systems possessing integrals quadratic in the velocities | p. 335 |
| Miscellaneous Examples | p. 336 |
| The Reduction of the Problem of Three Bodies | |
| Introduction | p. 339 |
| The differential equations of the problem | p. 340 |
| Jacobi's equation | p. 342 |
| Reduction to the 12th order, by use of the integrals of motion of the centre of gravity | p. 343 |
| Reduction to the 8th order, by use of the integrals of angular momentum and elimination of the nodes | p. 344 |
| Reduction to the 6th order | p. 347 |
| Alternative reduction of the problem from the 18th to the 6th order | p. 348 |
| The problem of three bodies in a plane | p. 351 |
| The restricted problem of three bodies | p. 353 |
| Extension to the problem of n bodies | p. 356 |
| Miscellaneous Examples | p. 356 |
| The Theorems of Bruns and Poincare | |
| Bruns' theorem | |
| Statement of the theorem | p. 358 |
| Expression of an integral in terms of the essential coordinates of the problem | p. 358 |
| An integral must involve the momenta | p. 359 |
| Only one irrationality can occur in the integral | p. 360 |
| Expression of the integral as a quotient of two real polynomials | p. 361 |
| Derivation of integrals from the numerator and denominator of the quotient | p. 362 |
| Proof that [phi][subscript 0] does not involve the irrationality | p. 366 |
| Proof that [phi][subscript 0] is a function only of the momenta and the integrals of angular momentum | p. 371 |
| Proof that [phi][subscript 0] is a function of T, L, M, N | p. 374 |
| Deduction of Bruns' theorem, for integrals which do not involve t | p. 376 |
| Extension of Bruns' result to integrals which involve the time | p. 378 |
| Poincare's theorem | |
| The equations of motion of the restricted problem of three bodies | p. 380 |
| Statement of Poincare's theorem | p. 381 |
| Proof that [phi][subscript 0] is not a function of H[subscript 0] | p. 381 |
| Proof that [phi][subscript 0] cannot involve the variables q[subscript 1], q[subscript 2] | p. 382 |
| Proof that the existence of a one-valued integral is inconsistent with the result of (iii) in the general case | p. 383 |
| Removal of the restrictions on the coefficients B[subscript m1, m2] | p. 384 |
| Deduction of Poincare's theorem | p. 385 |
| The General Theory of Orbits | |
| Introduction | p. 386 |
| Periodic solutions | p. 386 |
| A criterion for the discovery of periodic orbits | p. 386 |
| Asymptotic solutions | p. 389 |
| The orbits of planets in the relativity-theory | p. 389 |
| The motion of a particle on an ellipsoid under no external forces | p. 393 |
| Ordinary and singular periodic solutions | p. 395 |
| Characteristic exponents | p. 397 |
| Characteristic exponents when t does not occur explicitly | p. 398 |
| The characteristic exponents of a system which possesses a one-valued integral | p. 399 |
| The theory of matrices | p. 400 |
| The characteristic exponents of a Hamiltonian system | p. 402 |
| The asymptotic solutions of [section] 170 deduced from the theory of characteristic exponents | p. 405 |
| The characteristic exponents of "ordinary" and "singular" periodic solutions | p. 406 |
| Lagrange's three particles | p. 406 |
| Stability of Lagrange's particles: periodic orbits in the vicinity | p. 409 |
| The stability of orbits as affected by terms of higher order in the displacement | p. 412 |
| Attractive and repellent regions of a field of force | p. 413 |
| Application of the energy integral to the problem of stability | p. 416 |
| Application of integral-invariants to investigations of stability | p. 417 |
| Synge's "Geometry of Dynamics" | p. 417 |
| Connexion with the theory of surface transformations | p. 420 |
| Miscellaneous Examples | p. 420 |
| Integration by Series | |
| The need for series which converge for all values of the time; Poincare's series | p. 423 |
| The regularisation of the problem of three bodies | p. 424 |
| Trigonometric series | p. 425 |
| Removal of terms of the first degree from the energy function | p. 426 |
| Determination of the normal coordinates by a contact-transformation | p. 427 |
| Transformation to the trigonometric form of H | p. 430 |
| Other types of motion which lead to equations of the same form | p. 431 |
| The problem of integration | p. 432 |
| Determination of the adelphic integral in Case I | p. 433 |
| An example of the adelphic integral in Case I | p. 436 |
| The question of convergence | p. 437 |
| Use of the adelphic integral in order to complete the integration | p. 438 |
| The fundamental property of the adelphic integral | p. 442 |
| Determination of the adelphic integral in Case II | p. 443 |
| An example of the adelphic integral in Case II | p. 444 |
| Determination of the adelphic integral in Case III | p. 446 |
| An example of the adelphic integral in Case III | p. 447 |
| Completion of the integration of the dynamical system in Cases II and III | p. 449 |
| Miscellaneous Examples | p. 449 |
| Index of Authors Quoted | p. 451 |
| Index of Terms Employed | p. 453 |
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