Foreword / Sir William McCrea, FRS |
Preface to the fourth edition |
Kinematical Preliminaries / Chapter I: |
The displacements of rigid bodies / 1.: |
Euler's theorem on rotations about a point / 2.: |
The theorem of Rodrigues and Hamilton / 3.: |
The composition of equal and opposite rotations about parallel axes / 4.: |
Chasles' theorem on the most general displacement of a rigid body / 5.: |
Halphen's theorem on the composition of two general displacements / 6.: |
Analytic representation of a displacement / 7.: |
The composition of small rotations / 8.: |
Euler's parametric specification of rotations round a point / 9.: |
The Eulerian angles / 10.: |
Connexion of the Eulerian angles with the parameters [small xi], [small eta], [small zeta], [small chi] / 11.: |
The connexion of rotations with homographies: the Cayley-Klein parameters / 12.: |
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.: |
Time-flux of a vector whose components relative to moving axes are given / 17.: |
Special resolutions of the velocity and acceleration / 18.: |
Miscellaneous Examples |
The Equations of Motion / Chapter II: |
The ideas of rest and motion / 19.: |
The laws which determine motion / 20.: |
Force / 21.: |
Work / 22.: |
Forces which do no work / 23.: |
The coordinates of a dynamical system / 24.: |
Holonomic and non-holonomic systems / 25.: |
Lagrange's form of the equations of motion of a holonomic system / 26.: |
Conservative forces; the kinetic potential / 27.: |
The explicit form of Lagrange's equations / 28.: |
Motion of a system which is constrained to rotate uniformly round an axis / 29.: |
The Lagrangian equations for quasi-coordinates / 30.: |
Forces derivable from a potential-function which involves the velocities / 31.: |
Initial motions / 32.: |
Similarity in dynamical systems / 33.: |
Motion with reversed forces / 34.: |
Impulsive motion / 35.: |
The Lagrangian equations of impulsive motion / 36.: |
Principles Available for the Integration / Chapter III: |
Problems which are soluble by quadratures / 37.: |
Systems with ignorable coordinates / 38.: |
Special cases of ignoration; integrals of momentum and angular momentum / 39.: |
The general theorem of angular momentum / 40.: |
The energy equation / 41.: |
Reduction of a dynamical problem to a problem with fewer degrees of freedom, by means of the energy equation / 42.: |
Separation of the variables; dynamical systems of Lionville's type / 43.: |
The Soluble Problems of Particle Dynamics / Chapter IV: |
The particle with one degree of freedom; the pendulum / 44.: |
Motion in a moving tube / 45.: |
Motion of two interacting free particles / 46.: |
Central forces in general: Hamilton's theorem / 47.: |
The integrable cases of central forces; problems soluble in terms of circular and elliptic functions / 48.: |
Motion under the Newtonian law / 49.: |
The mutual transformation of fields of central force and fields of parallel force / 50.: |
Bonnet's theorem / 51.: |
Determination of the most general field of force under which a given curve or family of curves can be described / 52.: |
The problem of two centres of gravitation / 53.: |
Motion on a surface / 54.: |
Motion on a surface of revolution; cases soluble in terms of circular and elliptic functions / 55.: |
Joukovsky's theorem / 56.: |
The Dynamical Specification of Bodies / Chapter V: |
Definitions / 57.: |
The moments of inertia of some simple bodies / 58.: |
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 / 59.: |
Connexion between moments of inertia with respect to different sets of axes through the same origin / 60: |
The principal axes of inertia; Cauchy's momental ellipsoid / 61.: |
Calculation of the angular momentum of a moving rigid body / 62.: |
Calculation of the kinetic energy of a moving rigid body / 63.: |
Independence of the motion of the centre of gravity and the motion relative to it / 64.: |
The Soluble Problems of Rigid Dynamics / Chapter VI: |
The motion of systems with one degree of freedom; motion round a fixed axis, etc. / 65.: |
The motion of systems with two degrees of freedom / 66.: |
The motion of systems with three degrees of freedom / 67.: |
Motion of a body about a fixed point under no forces / 69.: |
Poinsot's kinematical representation of the motion; the polhode and herpolhode / 70.: |
Motion of a top on a perfectly rough plane; determination of the Eulerian angle [straight thetas] / 71.: |
Determination of the remaining Eulerian angles, and of the Cayley-Klein parameters; the spherical top / 72.: |
Motion of a top on a perfectly smooth plane / 73.: |
Kowalevski's top / 74.: |
Theory of Vibrations / 75.: |
Vibrations about equilibrium / 76.: |
Normal coordinates / 77.: |
Sylvester's theorem on the reality of the roots of the determinantal equation / 78.: |
Solution of the differential equations; the periods; stability / 79.: |
Examples of vibrations about equilibrium / 80.: |
Effect of a new constraint on the periods of a vibrating system / 81.: |
The stationary character of normal vibrations / 82.: |
Vibrations about steady motion / 83.: |
The integration of the equations / 84.: |
Examples of vibrations about steady motion / 85.: |
Vibrations of systems involving moving constraints / 86.: |
Non-Holonomic Systems. Dissipative Systems / Chapter VIII: |
Lagrange's equations with undetermined multipliers / 87.: |
Equations of motion referred to axes moving in any manner / 88.: |
Application to special non-holonomic problems / 89.: |
Vibrations of non-holonomic systems / 90.: |
Dissipative systems; frictional forces / 91.: |
Resisting forces which depend on the velocity / 92.: |
Rayleigh's dissipation-function / 93.: |
Vibrations of dissipative systems / 94.: |
Impact / 95.: |
Loss of kinetic energy in impact / 96.: |
Examples of impact / 97.: |
The Principles of Least Action and Least Curvature / Chapter IX: |
The trajectories of a dynamical system / 98.: |
Hamilton's principle for conservative holonomic systems / 99.: |
The principle of Least Action for conservative holonomic systems / 100.: |
Extension of Hamilton's principle to non-conservative dynamical systems / 101.: |
Extension of Hamilton's principle and the principle of Least Action to non-holonomic systems / 102.: |
Are the stationary integrals actual minima? Kinetic foci / 103.: |
Representation of the motion of dynamical systems by means of geodesics / 104.: |
The least-curvature principle of Gauss and Hertz / 105.: |
Expression of the curvature of a path in terms of generalised coordinates / 106.: |
Appell's equations / 107.: |
Bertrand's theorem / 108.: |
Hamiltonian Systems and Their Integral-Invariants / Chapter X: |
Hamilton's form of the equations of motion / 109.: |
Equations arising from the Calculus of Variations / 110.: |
Integral-invariants / 111.: |
The variational equations / 112.: |
Integral-invariants of order one / 113.: |
Relative integral-invariants / 114.: |
A relative integral-invariant which is possessed by all Hamiltonian systems / 115.: |
On systems which possess the relative integral-invariant [integral sign][summation]p[small delta]q / 116.: |
The expression of integral-invariants in terms of integrals / 117.: |
The theorem of Lie and Koenigs / 118.: |
The last multiplier / 119.: |
Derivation of an integral from two multipliers / 120.: |
Application of the last multiplier to Hamiltonian systems; use of a single known integral / 121.: |
Integral-invariants whose order is equal to the order of the system / 122.: |
Reduction of differential equations to the Lagrangian form / 123.: |
Case in which the kinetic energy is quadratic in the velocities / 124.: |
The Transformation-Theory of Dynamics / Chapter XI: |
Hamilton's Characteristic Function and contact-transformations / 125.: |
Contact-transformations in space of any number of dimensions / 126.: |
The bilinear covariant of a general differential form / 127.: |
The conditions for a contact-transformation expressed by means of the bilinear covariant / 128.: |
The conditions for a contact-transformation in terms of Lagrange's bracket-expressions / 129.: |
Poisson's bracket-expressions / 130.: |
The conditions for a contact-transformation expressed by means of Poisson's bracket-expressions / 131.: |
The sub-groups of Mathieu transformations and extended point-transformations / 132.: |
Infinitesimal contact-transformations / 133.: |
The resulting new view of dynamics / 134.: |
Helmholtz's reciprocal theorem / 135.: |
Jacobi's theorem on the transformation of a given dynamical system into another dynamical system / 136.: |
Representation of a dynamical problem by a differential form / 137.: |
The Hamiltonian function of the transformed equations / 138.: |
Transformations in which the independent variable is changed / 139.: |
New formulation of the integration-problem / 140.: |
Properties of the Integrals of Dynamical Systems / Chapter XII: |
Reduction of the order of a Hamiltonian system by use of the energy integral / 141.: |
Hamilton's partial differential equation / 142.: |
Hamilton's integral as a solution of Hamilton's partial differential equation / 143.: |
The connexion of integrals with infinitesimal transformations admitted by the system / 144.: |
Poisson's theorem / 145.: |
The constancy of Lagrange's bracket-expressions / 146.: |
Involution-systems / 147.: |
Solution of a dynamical problem when half the integrals are known / 148.: |
Levi-Civita's theorem / 149.: |
Systems which possess integrals linear in the momenta / 150.: |
Determination of the forces acting on a system for which an integral is known / 151.: |
Application to the case of a particle whose equations of motion possess an integral quadratic in the velocities / 152.: |
General dynamical systems possessing integrals quadratic in the velocities / 153.: |
The Reduction of the Problem of Three Bodies / Chapter XIII: |
Introduction / 154.: |
The differential equations of the problem / 155.: |
Jacobi's equation / 156.: |
Reduction to the 12th order, by use of the integrals of motion of the centre of gravity / 157.: |
Reduction to the 8th order, by use of the integrals of angular momentum and elimination of the nodes / 158.: |
Reduction to the 6th order / 159.: |
Alternative reduction of the problem from the 18th to the 6th order / 160.: |
The problem of three bodies in a plane / 161.: |
The restricted problem of three bodies / 162.: |
Extension to the problem of n bodies / 163.: |
The Theorems of Bruns and Poincare / Chapter XIV: |
Bruns' theorem / 164.: |
Statement of the theorem / (i): |
Expression of an integral in terms of the essential coordinates of the problem / (ii): |
An integral must involve the momenta / (iii): |
Only one irrationality can occur in the integral / (iv): |
Expression of the integral as a quotient of two real polynomials / (v): |
Derivation of integrals from the numerator and denominator of the quotient / (vi): |
Proof that [phi][subscript 0] does not involve the irrationality / (vii): |
Proof that [phi][subscript 0] is a function only of the momenta and the integrals of angular momentum / (viii): |
Proof that [phi][subscript 0] is a function of T, L, M, N / (ix): |
Deduction of Bruns' theorem, for integrals which do not involve t / (x): |
Extension of Bruns' result to integrals which involve the time / (xi): |
Poincare's theorem / 165.: |
The equations of motion of the restricted problem of three bodies |
Statement of Poincare's theorem |
Proof that [phi][subscript 0] is not a function of H[subscript 0] |
Proof that [phi][subscript 0] cannot involve the variables q[subscript 1], q[subscript 2] |
Proof that the existence of a one-valued integral is inconsistent with the result of (iii) in the general case |
Removal of the restrictions on the coefficients B[subscript m1, m2] |
Deduction of Poincare's theorem |
The General Theory of Orbits / Chapter XV: |
Periodic solutions / 166.: |
A criterion for the discovery of periodic orbits / 168.: |
Asymptotic solutions / 169.: |
The orbits of planets in the relativity-theory / 170.: |
The motion of a particle on an ellipsoid under no external forces / 171.: |
Ordinary and singular periodic solutions / 172.: |
Characteristic exponents / 173.: |
Characteristic exponents when t does not occur explicitly / 174.: |
The characteristic exponents of a system which possesses a one-valued integral / 175.: |
The theory of matrices / 176.: |
The characteristic exponents of a Hamiltonian system / 177.: |
The asymptotic solutions of [section] 170 deduced from the theory of characteristic exponents / 178.: |
The characteristic exponents of "ordinary" and "singular" periodic solutions / 179.: |
Lagrange's three particles / 180.: |
Stability of Lagrange's particles: periodic orbits in the vicinity / 181.: |
The stability of orbits as affected by terms of higher order in the displacement / 182.: |
Attractive and repellent regions of a field of force / 183.: |
Application of the energy integral to the problem of stability / 184.: |
Application of integral-invariants to investigations of stability / 185.: |
Synge's "Geometry of Dynamics" / 186.: |
Connexion with the theory of surface transformations / 187.: |
Integration by Series / Chapter XVI: |
The need for series which converge for all values of the time; Poincare's series / 188.: |
The regularisation of the problem of three bodies / 189.: |
Trigonometric series / 190.: |
Removal of terms of the first degree from the energy function / 191.: |
Determination of the normal coordinates by a contact-transformation / 192.: |
Transformation to the trigonometric form of H / 193.: |
Other types of motion which lead to equations of the same form / 194.: |
The problem of integration / 195.: |
Determination of the adelphic integral in Case I / 196.: |
An example of the adelphic integral in Case I / 197.: |
The question of convergence / 198.: |
Use of the adelphic integral in order to complete the integration / 199.: |
The fundamental property of the adelphic integral / 200.: |
Determination of the adelphic integral in Case II / 201.: |
An example of the adelphic integral in Case II / 202.: |
Determination of the adelphic integral in Case III / 203.: |
An example of the adelphic integral in Case III / 204.: |
Completion of the integration of the dynamical system in Cases II and III / 205.: |
Index of Authors Quoted |
Index of Terms Employed |