llcotp.com/textract/1-mechanics.pdf-rawText-170-176.txt

§50
General properties of motion in S dimensions
161
ordinates. For the quantities Ii are one-valued integrals of the motion in
co-ordinates which allow separation of the variables. When degeneracy occurs,
the number of one-valued integrals exceeds S, and so the choice of those
which are the desired I is no longer unique.
As an example, we may again mention Keplerian motion, which allows
separation of the variables in both spherical and parabolic co-ordinates.
In §49 it has been shown that, for finite motion in one dimension, the
action variable is an adiabatic invariant. This statement holds also for systems
with more than one degree of freedom. Here we shall give a proof valid
for the general case.
Let X(t) be again a slowly varying parameter of the system. In the canonical
transformation from the variables P, q to I, W, the generating function is, as we
know, the action So(q, I). This depends on A as a parameter and, if A is a func-
tion of time, the function So(q, I; X(t)) depends explicitly on time. In such a
case the new Hamiltonian H' is not the same as H, i.e. the energy E(I), and
by the general formulae (45.8) for the canonical transformation we have
H' E(I)+asoldt = E(I)+A, where A III (aso/ad)r. Hamilton's equations
give
ig = -
(50.15)
We average this equation over a time large compared with the fundamental
periods of the system but small compared with the time during which the
parameter A varies appreciably. Because of the latter condition we need not
average 1 on the right-hand side, and in averaging the quantities we
may regard the motion of the system as taking place at a constant value of A
and therefore as having the properties of conditionally periodic motion
described above.
The action So is not a one-valued function of the co-ordinates: when q
returns to its initial value, So increases by an integral multiple of 2I. The
derivative A = (aso/ax), is a one-valued function, since the differentiation
is effected for constant Ii, and there is therefore no increase in So. Hence A,
expressed as a function of the angle variables Wr, is periodic. The mean value
of the derivatives of such a function is zero, and therefore by (50.15)
we have also
which shows that the quantities Ii are adiabatic invariants.
Finally, we may briefly discuss the properties of finite motion of closed
systems with S degrees of freedom in the most general case, where the vari-
ables in the Hamilton-Jacobi equation are not assumed to be separable.
The fundamental property of systems with separable variables is that the
integrals of the motion Ii, whose number is equal to the number of degrees
+ To simplify the formulae we assume that there is only one such parameter, but the proof
is valid for any number.
162
The Canonical Equations
§50
of freedom, are one-valued. In the general case where the variables are not
separable, however, the one-valued integrals of the motion include only
those whose constancy is derived from the homogeneity and isotropy of space
and time, namely energy, momentum and angular momentum.
The phase path of the system traverses those regions of phase space which
are defined by the given constant values of the one-valued integrals of the
motion. For a system with separable variables and S one-valued integrals,
these conditions define an s-dimensional manifold (hypersurface) in phase
space. During a sufficient time, the path of the system passes arbitrarily close
to every point on this hypersurface.
In a system where the variables are not separable, however, the number
of one-valued integrals is less than S, and the phase path occupies, completely
or partly, a manifold of more than S dimensions in phase space.
In degenerate systems, on the other hand, which have more than S integrals
of the motion, the phase path occupies a manifold of fewer than S dimensions.
If the Hamiltonian of the system differs only by small terms from one which
allows separation of the variables, then the properties of the motion are close
to those of a conditionally periodic motion, and the difference between the
two is of a much higher order of smallness than that of the additional terms in
the Hamiltonian.
PROBLEM
Calculate the action variables for elliptic motion in a field U = -a/r.
SOLUTION. In polar co-ordinates r, in the plane of the motion we have
'max
= 1+av(m2)E)
Hence the energy, expressed in terms of the action variables, is E = It
depends only on the sum Ir+I, and the motion is therefore degenerate; the two funda-
mental frequencies (in r and in b) coincide.
The parameters P and e of the orbit (see (15.4)) are related to Ir and I by
p=
Since Ir and I are adiabatic invariants, when the coefficient a or the mass m varies slowly
the eccentricity of the orbit remains unchanged, while its dimensions vary in inverse propor-
tion to a and to m.
INDEX
Acceleration, 1
Coriolis force, 128
Action, 2, 138ff.
Couple, 109
abbreviated, 141
Cross-section, effective, for scattering,
variable, 157
49ff.
Additivity of
C system, 41
angular momentum, 19
Cyclic co-ordinates, 30
energy, 14
integrals of the motion, 13
d'Alembert's principle, 124
Lagrangians, 4
Damped oscillations, 74ff.
mass, 17
Damping
momentum, 15
aperiodic, 76
Adiabatic invariants, 155, 161
coefficient, 75
Amplitude, 59
decrement, 75
complex, 59
Degeneracy, 39, 69, 160f.
Angle variable, 157
complete, 160
Angular momentum, 19ff.
Degrees of freedom, 1
of rigid body, 105ff.
Disintegration of particles, 41ff.
Angular velocity, 97f.
Dispersion-type absorption, 79
Area integral, 31n.
Dissipative function, 76f.
Dummy suffix, 99n.
Beats, 63
Brackets, Poisson, 135ff.
Eccentricity, 36
Eigenfrequencies, 67
Canonical equations (VII), 131ff.
Elastic collision, 44
Canonical transformation, 143ff.
Elliptic functions, 118f.
Canonical variables, 157
Elliptic integrals, 26, 118
Canonically conjugate quantities, 145
Energy, 14, 25f.
Central field, 21, 30
centrifugal, 32, 128
motion in, 30ff.
internal, 17
Centrally symmetric field, 21
kinetic, see Kinetic energy
Centre of field, 21
potential, see Potential energy
Centre of mass, 17
Equations of motion (I), 1ff.
system, 41
canonical (VII), 131ff.
Centrifugal force, 128
integration of (III), 25ff.
Centrifugal potential, 32, 128
of rigid body, 107ff.
Characteristic equation, 67
Eulerian angles, 110ff.
Characteristic frequencies, 67
Euler's equations, 115, 119
Closed system, 8
Collisions between particles (IV), 41ff.
Finite motion, 25
elastic, 44ff.
Force, 9
Combination frequencies, 85
generalised, 16
Complete integral, 148
Foucault's pendulum, 129f.
Conditionally periodic motion, 160
Frame of reference, 4
Conservation laws (II), 13ff.
inertial, 5f.
Conservative systems, 14
non-inertial, 126ff.
Conserved quantities, 13
Freedom, degrees of, 1
Constraints, 10
Frequency, 59
equations of, 123
circular, 59
holonomic, 123
combination, 85
Co-ordinates, 1
Friction, 75, 122
cyclic, 30
generalised, 1ff.
Galilean transformation, 6
normal, 68f.
Galileo's relativity principle, 6
163
164
Index
General integral, 148
Mechanical similarity, 22ff.
Generalised
Molecules, vibrations of, 70ff.
co-ordinates, 1ff.
Moment
forces, 16
of force, 108
momenta, 16
of inertia, 99ff.
velocities, 1ff.
principal, 100ff.
Generating function, 144
Momentum, 15f.
angular, see Angular momentum
Half-width, 79
generalised, 16
Hamiltonian, 131f.
moment of, see Angular momentum
Hamilton-Jacobi equation, 147ff.
Multi-dimensional motion, 158ff.
Hamilton's equations, 132
Hamilton's function, 131
Hamilton's principle, 2ff.
Newton's equations, 9
Holonomic constraint, 123
Newton's third law, 16
Nodes, line of, 110
Impact parameter, 48
Non-holonomic constraint, 123
Inertia
Normal co-ordinates, 68f.
law of, 5
Normal oscillations, 68
moments of, 99ff.
Nutation, 113
principal, 100ff.
principal axes of, 100
One-dimensional motion, 25ff., 58ff.
tensor, 99
Oscillations, see Small oscillations
Inertial frames, 5f.
Oscillator
Infinite motion, 25
one-dimensional, 58n.
Instantaneous axis, 98
space, 32, 70
Integrals of the motion, 13, 135
Jacobi's identity, 136
Particle, 1
Pendulums, 11f., 26, 33ff., 61, 70, 95,
Kepler's problem, 35ff.
102f., 129f.
Kepler's second law, 31
compound, 102f.
Kepler's third law, 23
conical, 34
Kinetic energy, 8, 15
Foucault's, 129f.
of rigid body, 98f.
spherical, 33f.
Perihelion, 36
Laboratory system, 41
movement of, 40
Lagrange's equations, 3f.
Phase, 59
Lagrangian, 2ff.
path, 146
for free motion, 5
space, 146
of free particle, 6ff.
Point transformation, 143
in non-inertial frame, 127
Poisson brackets, 135ff.
for one-dimensional motion, 25, 58
Poisson's theorem, 137
of rigid body, 99
Polhodes, 117n.
for small oscillations, 58, 61, 66, 69, 84
Potential energy, 8, 15
of system of particles, 8ff.
centrifugal, 32, 128
of two bodies, 29
effective, 32, 94
Latus rectum, 36
from period of oscillation, 27ff.
Least action, principle of, 2ff.
Potential well, 26, 54f.
Legendre's transformation, 131
Precession, regular, 107
Liouville's theorem, 147
L system, 41
Rapidly oscillating field, motion in, 93ff.
Reactions, 122
Mass, 7
Reduced mass, 29
additivity of, 17
Resonance, 62, 79
centre of, 17
in non-linear oscillations, 87ff.
reduced, 29
parametric, 80ff.
Mathieu's equation, 82n.
Rest, system at, 17
Maupertuis' principle, 141
Reversibility of motion, 9
Index
165
Rigid bodies, 96
Space
angular momentum of, 105ff.
homogeneity of, 5, 15
in contact, 122ff.
isotropy of, 5, 18
equations of motion of, 107ff.
Space oscillator, 32, 70
motion of (VI), 96ff.
Rolling, 122
Time
Rotator, 101, 106
homogeneity of, 5, 13ff.
Rough surface, 122
isotropy of, 8f.
Routhian, 134f.
Top
Rutherford's formula, 53f.
asymmetrical, 100, 116ff.
"fast", 113f.
spherical, 100, 106
Scattering, 48ff.
symmetrical, 100, 106f., 111f.
cross-section, effective, 49ff.
Torque, 108
Rutherford's formula for, 53f.
Turning points, 25, 32
small-angle, 55ff.
Two-body problem, 29
Sectorial velocity, 31
Separation of variables, 149ff.
Uniform field, 10
Similarity, mechanical, 22ff.
Sliding, 122
Variation, 2, 3
Small oscillations, 22, (V) 58ff.
first, 3
anharmonic, 84ff.
Velocity, 1
damped, 74ff.
angular, 97f.
forced, 61ff., 77ff.
sectorial, 31
free, 58ff., 65ff.
translational, 97
linear, 84
Virial, 23n.
non-linear, 84ff.
theorem, 23f.
normal, 68
Smooth surface, 122
Well, potential, 26, 54f.
PHYSICS
The enormous increase in the number
and size of scientific journals has led to a
qualitative change in the problem of
scientific communication. The policies
of most journals are based on the old
need to ensure that no valid science
was lost to the scientific public by being
rejected ; the problem now seems to be
whether almost all good science will
be buried among mountains of valid
but mediocre work, or secreted in
specialized publications. The scientist
reads only a tiny fraction of physics,
either sharply specialized or selected at
random, by rumour or by the author's
reputation.
PHYSICS will help its readers to find
at least some of the first-rate new work,
particularly outside their speciality, it
will help to maintain the unity of
physics against an increasing tendency
toward specialization and to keep high
standards of presentation and possibly
of creative scientific work.
Write for an Information and Index
Leaflet giving full details including
subscription rates.
Pergamon Press
Headington Hill Hall, Oxford OX3 OBW
Maxwell House, Fairview Park,
Elmsford, New York 10523
4 & 5 Fitzroy Square, London W1
2 & 3 Teviot Place, Edinburgh 1
207 Queen's Quay West, Toronto 1
19A Boundary Street, Rushcutters Bay,
N.S.W. 2011, Australia
24 rue des Ecoles, Paris 5e
Vieweg & Sohn GmbH, Burgplatz 1,
Braunschweig
Printed in Great Britain/Bradley
COURSE OF THEORETICAL PHYSICS
by L.D. LANDAU and E.M. LIFSHITZ
Institute of Physical Problems, USSR Academy of Sciences
The complete Course of Theoretical Physics by Landau and Lifshitz, recognized as two
of the world's outstanding physicists, is being published in full by Pergamon Press. It
comprises nine volumes, covering all branches of the subject translations from the Russian
are by leading scientists.
Typical of many statements made by experts reviewing the series, are the following
"The titles of the volumes in this series cover a vast range of topics, and there seems to be
little in physics on which the authors are not very well informed."
Nature
"The remarkable nine-volume Course of Theoretical Physics
the
clearness
and
accuracy
of the authors' treatment of theoretical physics is well maintained."
Proceedings of the Physical Society
Of individual volumes, reviewers have written
MECHANICS
"The entire book is a masterpiece of scientific writing. There is not a superfluous sentence
and the authors know exactly where they are going
It is certain that this volume will
be able to hold its own amongst more conventional texts in classical mechanisms, as a
scholarly and economic exposition of the subject."
Science Progress
QUANTUM MECHANICS (Non-relativistic Theory)
throughout the five hundred large pages, the authors' discussion proceeds with the
clarity and succinctness typical of the very best works on theoretical physics."
Technology
FLUID MECHANICS
"The ground covered includes ideal fluids, viscous fluids, turbulence, boundary layers,
conduction and diffusion, surface phenomena and sound. Compressible fluids are treated
under the headings of shock waves, one-dimensional gas flow and flow past finite bodies.
There is a chapter on the fluid dynamics of combustion while unusual topics discussed are
relativistic fluid dynamics, dynamics of superfluids and fluctuations of fluid dynamics
a
valuable addition to any library covering the mechanics of fluids."
Science Progress
THE CLASSICAL THEORY OF FIELDS (Second Edition)
"This is an excellent and readable volume. It is a valuable and unique addition to the
literature of theoretical physics."
Science
STATISTICAL PHYSICS
stimulating reading, partly because of the clarity and compactness of some of the
treatments put forward, and partly by reason of contrasts with texts on statistical mechanics
and statistical thermodynamics better known to English sciences
Other features
attract attention since they do not always receive comparable mention in other textbooks."
New Scientist
THEORY OF ELASTICITY
"I shall be surprised if this book does not come to be regarded as a masterpiece."
Journal of the Royal Institute of Physics
ELECTRODYNAMICS OF CONTINUOUS MEDIA
"Within the volume one finds everything expected of a textbook on classical electricity
and magnetism, and a great deal more. It is quite certain that this book will remain unique
and indispensable for many years to come."
Science Progress
08 006466 3