as may also be seen directly from formula (49.11) and the expression (49.6)
for the period.
Conversely, if we express q and P, or any one-valued function F(p, q) of
them, in terms of the canonical variables, then they remain unchanged when
W increases by 2nd (with I constant). That is, any one-valued function F(p, q),
when expressed in terms of the canonical variables, is a periodic function of W
with period 2.
$50. General properties of motion in S dimensions
Let us consider a system with any number of degrees of freedom, executing
a motion finite in all the co-ordinates, and assume that the variables can be
completely separated in the Hamilton-Jacobi treatment. This means that,
when the co-ordinates are appropriately chosen, the abbreviated action
can be written in the form
(50.1)
as a sum of functions each depending on only one co-ordinate.
Since the generalised momenta are Pi = aso/dqi = dSi/dqi, each function
Si can be written
(50.2)
These are many-valued functions. Since the motion is finite, each co-ordinate
can take values only in a finite range. When qi varies "there and back" in this
range, the action increases by
(50.3)
where
(50.4)
the integral being taken over the variation of qi just mentioned.
Let us now effect a canonical transformation similar to that used in 49,
for the case of a single degree of freedom. The new variables are "action vari-
ables" Ii and "angle variables"
w(a(q
(50.5)
+ It should be emphasised, however, that this refers to the formal variation of the co-
ordinate qi over the whole possible range of values, not to its variation during the period of
the actual motion as in the case of motion in one dimension. An actual finite motion of a
system with several degrees of freedom not only is not in general periodic as a whole, but
does not even involve a periodic time variation of each co-ordinate separately (see below).
§50
General properties of motion in S dimensions
159
where the generating function is again the action expressed as a function of
the co-ordinates and the Ii. The equations of motion in these variables are
Ii = 0, w = de(I)/I, which give
I=constant,
(50.6)
+ constant.
(50.7)
We also find, analogously to (49.13), that a variation "there and back" of
the co-ordinate qi corresponds to a change of 2n in Wi:
Awi==2m
(50.8)
In other words, the quantities Wi(q, I) are many-valued functions of the co-
ordinates: when the latter vary and return to their original values, the Wi
may vary by any integral multiple of 2. This property may also be formulated
as a property of the function Wi(P, q), expressed in terms of the co-ordinates
and momenta, in the phase space of the system. Since the Ii, expressed in
terms of P and q, are one-valued functions, substitution of Ii(p, q) in wi(q, I)
gives a function wilp, q) which may vary by any integral multiple of 2n
(including zero) on passing round any closed path in phase space.
Hence it follows that any one-valued function F(P, q) of the state of the
system, if expressed in terms of the canonical variables, is a periodic function
of the angle variables, and its period in each variable is 2nr. It can be expanded
as a multiple Fourier series:
(50.9)
ls==
where l1, l2, ls are integers. Substituting the angle variables as functions
of time, we find that the time dependence of F is given by a sum of the form
(50.10)
lg==
Each term in this sum is a periodic function of time, with frequency
(50.11)
Since these frequencies are not in general commensurable, the sum itself is
not a periodic function, nor, in particular, are the co-ordinates q and
momenta P of the system.
Thus the motion of the system is in general not strictly periodic either as a
whole or in any co-ordinate. This means that, having passed through a given
state, the system does not return to that state in a finite time. We can say,
t Rotational co-ordinates (see the first footnote to 49) are not in one-to-one relation
with the state of the system, since the position of the latter is the same for all values of
differing by an integral multiple of 2nr. If the co-ordinates q include such angles, therefore,
these can appear in the function F(P, q) only in such expressions as cos and sin , which
are in one-to-one relation with the state of the system.
160
The Canonical Equations
§50
however, that in the course of a sufficient time the system passes arbitrarily
close to the given state. For this reason such a motion is said to be conditionally
periodic.
In certain particular cases, two or more of the fundamental frequencies
Wi = DE/DI are commensurable for arbitrary values of the Ii. This is called
degeneracy, and if all S frequencies are commensurable, the motion of the
system is said to be completely degenerate. In the latter case the motion is
evidently periodic, and the path of every particle is closed.
The existence of degeneracy leads, first of all, to a reduction in the number
of independent quantities Ii on which the energy of the system depends.
If two frequencies W1 and W2 are such that
(50.12)
where N1 and N2 are integers, then it follows that I1 and I2 appear in the energy
only as the sum n2I1+n1I2.
A very important property of degenerate motion is the increase in the
number of one-valued integrals of the motion over their number for a general
non-degenerate system with the same number of degrees of freedom. In the
latter case, of the 2s-1 integrals of the motion, only s functions of the state
of the system are one-valued; these may be, for example, the S quantities I
The remaining S - 1 integrals may be written as differences
(50.13)
The constancy of these quantities follows immediately from formula (50.7),
but they are not one-valued functions of the state of the system, because the
angle variables are not one-valued.
When there is degeneracy, the situation is different. For example, the rela-
tion (50.12) shows that, although the integral
WIN1-W2N2
(50.14)
is not one-valued, it is so except for the addition of an arbitrary integral
multiple of 2nr. Hence we need only take a trigonometrical function of this
quantity to obtain a further one-valued integral of the motion.
An example of degeneracy is motion in a field U = -a/r (see Problem).
There is consequently a further one-valued integral of the motion (15.17)
peculiar to this field, besides the two (since the motion is two-dimensional)
ordinary one-valued integrals, the angular momentum M and the energy E,
which hold for motion in any central field.
It may also be noted that the existence of further one-valued integrals
leads in turn to another property of degenerate motions: they allow a complete
separation of the variables for several (and not only one+) choices of the co-
t We ignore such trivial changes in the co-ordinates as q1' = q1'(q1), q2' = 92'(92).
§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.
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