A Treatise on the Analytical Dynamics of Particles and Rigid Bodies

Front Cover
CUP Archive, Jan 2, 1937 - Mathematics - 456 pages
1 Review
 

What people are saying - Write a review

User Review - Flag as inappropriate

Nice, and Informal book, really helps if you read!

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
Index of authors quoted
451
Index of terms employed
453

Other editions - View all

Common terms and phrases

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.

References to this book

All Book Search results »

About the author (1937)

Whittaker, Professor of Mathematics in the University of Edinburgh.

Bibliographic information