Chapter II: Part 2

If in Lagrange's equations Sec. 2 (10) we reverse the sign of the
time-element dt, the equations are unaltered. The motion is therefore
reversible; that is to say, if as the system is passing through any
configuration its velocities q`1, q`2, ..., q`_m be all
reversed, it will (if the forces be the same in the same
configuration) retrace its former path. But it is important to observe
that the statement does not in general hold of a gyrostatic system;
the terms of (16), which are linear in q`1, q`2, ..., q`_m,
change sign with dt, whilst the others do not. Hence the motion of a
gyrostatic system is not reversible, unless indeed we reverse the
cyclic motions as well as the velocities q`1, q`2, ..., q`_m.
For instance, the precessional motion of a top cannot be reversed
unless we reverse the spin.

Kinetostatics.

The _conditions of equilibrium_ of a system with latent cyclic motions
are obtained by putting q`1 = 0, q`2 = 0, ... q`_m = 0 in (16); viz.
they are

dP[Kappa] dP[Kappa]
Q1 = ---------, Q2 = ---------, ... (23)
dPq1 dPq2

These may of course be obtained independently. Thus if the system be
guided from (apparent) rest in the configuration (q1, q2, ... q_m) to
rest in the configuration q1 + [delta]q1, q2 + [delta]q2, ..., q_m +
[delta]q_m, the work done by the forces must be equal to the increment
of the kinetic energy. Hence

Q1[delta]q1 + Q2[delta]q2 + ... = [delta][Kappa], (24)

which is equivalent to (23). The conditions are the same as for the
equilibrium of a system without latent motion, but endowed with
potential energy [Kappa]. This is important from a physical point of
view, as showing how energy which is apparently potential may in its
ultimate essence be kinetic.

By means of the formulae (18), which now reduce to

dP[Kappa] dP[Kappa] dP[Kappa]
[chi]` = ---------, [chi]`' = ----------, [chi]`" = ---------- ..., (25)
dP[kappa] dP[kappa]' dP[kappa]"

[Kappa] may also be expressed as a homogeneous quadratic function of
the cyclic velocities [.[chi]], [.[chi]]', [.[chi]]", ... Denoting it
in this form by [Tau]0, we have

[delta]([Tau]0 + [Kappa] = 2[delta][Kappa] = [delta]([kappa] [chi]` + [kappa]'[chi]`' + [kappa]"[chi]`" + ...). (26)

Performing the variations, and omitting the terms which cancel by (2)
and (25), we find

dP[Tau]0 dP[Kappa] dP[Tau]0 dP[Kappa]
-------- = - ---------, -------- = - ---------, ..., (27)
dPq1 dPq1 dPq2 dPq2

so that the formulae (23) become

dP[Tau]0 dP[Tau]0
Q1 = - --------, Q2 = - --------, ... (28)
dPq1 dPq2

A simple example is furnished by the top (MECHANICS, Sec. 22). The cyclic
co-ordinates being [psi], [phi], we find

( mu - [nu] cos [theta]) squared [nu] squared
2[@] = A[theta]` squared, 2[Kappa] = ----------------------- + -----,
A sin squared [theta] C

2[Tau]0 = A sin squared[theta][psi]` squared + C([phi]` + [psi] cos [theta]) squared, (29)

whence we may verify that dP[Tau]0/dP[theta] = - dP[Kappa]/dP[theta]
in accordance with (27). And the condition of equilibrium

dP[Kappa] dPV
--------- = - --------- (30)
dP[theta] dP[theta]

gives the condition of steady precession.

6. _Stability of Steady Motion._

The small oscillations of a conservative system about a configuration
of equilibrium, and the criterion of stability, are discussed in
MECHANICS, Sec. 23. The question of the stability of given types of
motion is more difficult, owing to the want of a sufficiently general,
and at the same time precise, definition of what we mean by
"stability." A number of definitions which have been propounded by
different writers are examined by F. Klein and A. Sommerfeld in their
work _Ueber die Theorie des Kreisels_ (1897-1903). Rejecting previous
definitions, they base their criterion of stability on the character
of the changes produced in the _path_ of the system by small arbitrary
disturbing impulses. If the undisturbed path be the _limiting form_ of
the disturbed path when the impulses are indefinitely diminished, it
is said to be stable, but not otherwise. For instance, the vertical
fall of a particle under gravity is reckoned as stable, although for a
_given_ impulsive disturbance, however small, the deviation of the
particle's position at any time t from the position which it would
have occupied in the original motion increases indefinitely with t.
Even this criterion, as the writers quoted themselves recognize, is
not free from ambiguity unless the phrase "limiting form," as applied
to a path, be strictly defined. It appears, moreover, that a
definition which is analytically precise may not in all cases be easy
to reconcile with geometrical prepossessions. Thus a particle moving
in a circle about a centre of force varying inversely as the cube of
the distance will if slightly disturbed either fall into the centre,
or recede to infinity, after describing in either case a spiral with
an infinite number of convolutions. Each of these spirals has,
analytically, the circle as its limiting form, although the motion in
the circle is most naturally described as unstable.

A special form of the problem, of great interest, presents itself in
the steady motion of a gyrostatic system, when the non-eliminated
co-ordinates q1, q2, ... q_m all vanish (see Sec. 5). This has been
discussed by Routh, Lord Kelvin and Tait, and Poincare. These writers
treat the question, by an extension of Lagrange's method, as a problem
of small oscillations. Whether we adopt the notion of stability which
this implies, or take up the position of Klein and Sommerfeld, there
is no difficulty in showing that stability is ensured if V + [Kappa]
be a minimum as regards variations of q1, q2, ... q_m. The proof is
the same as that of Dirichlet for the case of statical stability.

We can illustrate this condition from the case of the top, where, in
our previous notation,

( mu - [nu]cos [theta]) squared [nu] squared
V + [Kappa] = Mgh cos[theta] + ---------------------- + -----. (1)
2A sin squared [theta] 2C

To examine whether the steady motion with the centre of gravity
vertically above the pivot is stable, we must put mu = [nu]. We then
find without difficulty that V + [Kappa] is a minimum provided [nu] squared
[>=] 4AMgh. The method of small oscillations gave us the condition
[nu] squared > 4AMgh, and indicated instability in the cases [nu] squared [=<]
4AMgh. The present criterion can also be applied to show that the
steady precessional motions in which the axis has a constant
inclination to the vertical are stable.

The question remains, as before, whether it is _essential_ for
stability that V + [Kappa] should be a minimum. It appears that from
the point of view of the theory of small oscillations it is not
essential, and that there may even be stability when V + [Kappa] is a
maximum. The precise conditions, which are of a somewhat elaborate
character, have been formulated by Routh. An important distinction
has, however, been established by Thomson and Tait, and by Poincare,
between what we may call _ordinary_ or _temporary_ stability (which is
stability in the above sense) and _permanent_ or _secular_ stability,
which means stability when regard is had to possible dissipative
forces called into play whenever the co-ordinates q1, q2, ... q_m
vary. Since the total energy of the system at any instant is given (in
the notation of Sec. 5) by an expression of the form [@] + V +
[Kappa], where [@] cannot be negative, the argument of Thomson
and Tait, given under MECHANICS, Sec. 23, for the statical question,
shows that it is a necessary as well as a sufficient condition for
secular stability that V + [Kappa] should be a minimum. When a system
is "ordinarily" stable, but "secularly" unstable, the operation of the
frictional forces is to induce a gradual increase in the amplitude of
the free vibrations which are called into play by accidental
disturbances.

There is a similar theory in relation to the constrained systems
considered in Sec. 3 above. The equation (21) there given leads to the
conclusion that for secular stability of any type of motion in which
the velocities q`1, q`2, ... q`_n are zero it is necessary and
sufficient that the function V - [Tau]0 should be a minimum.

The simplest possible example of this is the case of a particle at the
lowest point of a smooth spherical bowl which rotates with constant
angular velocity ([omega]) about the vertical diameter. This position
obviously possesses "ordinary" stability. If a be the radius of the
bowl, and [theta] denote angular distance from the lowest point, we
have

V - [Tau]0 = mga(1 - cos [theta]) - 1/2m[omega] squareda squared sin squared [theta]; (2)

this is a minimum for [theta] = 0 only so long as [omega] squared < g/a. For
greater values of [omega] the only position of "permanent" stability
is that in which the particle rotates with the bowl at an angular
distance cos^(-1) (g/[omega] squareda) from the lowest point. To examine the
motion in the neighbourhood of the lowest point, when frictional
forces are taken into account, we may take fixed ones, in a horizontal
plane, through the lowest point. Assuming that the friction varies as
the relative velocity, we have

x" = -p squaredx - k(x` + [omega]y), \ (3)
y" = -p squaredy - k(y` - [omega]x), /

where p squared = g/a. These combine into

z" + kz` + (p squared - ik[omega])z = 0, (4)

where z = x + iy, i = [root]-1. Assuming z = Ce^([lambda]t), we find

[lambda] = -1/2k(1 [-+] [omega]/p) +- ip, (5)

if the square of k be neglected. The complete solution is then

x + iy = C1e^(-ss1t)e^(ipt) + C2e^(-ss2t)e^(-ipt), (6)

where ss1 = 1/2k(1 - [omega]/p), ss2 = 1/2k(1 + [omega]/p). (7)

This represents two superposed circular vibrations, in opposite
directions, of period 2[pi]/p. If [omega] < p, the amplitude of each
of these diminishes asymptotically to zero, and the position x = 0, y
= 0 is permanently stable. But if [omega] > p the amplitude of that
circular vibration which agrees in sense with the rotation [omega]
will continually increase, and the particle will work its way in an
ever-widening spiral path towards the eccentric position of secular
stability. If the bowl be not spherical but ellipsoidal, the vertical
diameter being a principal axis, it may easily be shown that the
lowest position is permanently stable only so long as the period of
the rotation is longer than that of the slower of the two normal
modes in the absence of rotation (see MECHANICS, Sec. 13).

7. _Principle of Least Action._

Stationary Action.

The preceding theories give us statements applicable to the system at
any one instant of its motion. We now come to a series of theorems
relating to the whole motion of the system between any two
configurations through which it passes, viz. we consider the actual
motion and compare it with other imaginable motions, differing
infinitely little from it, between the same two configurations. We use
the symbol [delta] to denote the transition from the actual to any one
of the hypothetical motions.

The best-known theorem of this class is that of _Least Action_,
originated by P.L.M. de Maupertuis, but first put in a definite form
by Lagrange. The "action" of a single particle in passing from one
position to another is the space-integral of the momentum, or the
time-integral of the _vis viva_. The action of a dynamical system is
the sum of the actions of its constituent particles, and is
accordingly given by the formula
_ _ _
/ / /
A = [Sigma] | mvds = [Sigma] | mv squareddt = 2 | [Tau]dt. (1)
_/ _/ _/

The theorem referred to asserts that the free motion of a conservative
system between any two given configurations is characterized by the
property

[delta]A = 0, (2)

provided the total energy have the same constant value in the varied
motion as in the actual motion.

If t, t' be the times of passing through the initial and final
configurations respectively, we have
_
/ t'
[delta]A = [delta] | [Sigma]m(x` squared + y` squared + z` squared)dt
_/t
_
/ t'
= 2 | [delta][Tau]dt + 2[Tau]'[delta]t' + 2[Tau][delta]t, (3)
_/t

since the upper and lower limits of the integral must both be regarded
as variable. This may be written
_ _
/ t' / t'
[delta]A = | [delta][Tau]dt + | [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t
_/t _/t
_ _ _
/ t' | | t'
= | [delta][Tau]dt + | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z |
_/t |_ _| t
_
/ t'
- | [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t. (4)
_/t

Now, by d'Alembert's principle,

[Sigma]m( x"[delta]x + y"[delta]y + z"[delta]z ) = -[delta]V, (5)

and by hypothesis we have

[delta]([Tau] + V) = 0. (6)

The formula therefore reduces to
_ _
| |t'
[delta]A = | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z) | + 2[Tau]'[delta]t' - 2[Tau][delta]t. (7)
|_ _|t

Since the terminal configurations are unaltered, we must have at the
lower limit

[delta]x + x`[delta]t = 0, [delta]y + y`[delta]t = 0, [delta]z + z`[delta]t = 0, (8)

with similar relations at the upper limit. These reduce (7) to the
form (2).

The equation (2), it is to be noticed, merely expresses that the
variation of A vanishes _to the first order_; the phrase _stationary
action_ has therefore been suggested as indicating more accurately
what has been proved. The action in the free path between two given
configurations is in fact not invariably a minimum, and even when a
minimum it need not be the _least possible_ subject to the given
conditions. Simple illustrations are furnished by the case of a single
particle. A particle moving on a smooth surface, and free from
extraneous force, will have its velocity constant; hence the theorem
in this case resolves itself into
_
/
[delta] | ds = 0, (9)
_/

i.e. the path must be a geodesic line. Now a geodesic is not
necessarily the _shortest_ path between two given points on it; for
example, on the sphere a great-circle arc ceases to be the shortest
path between its extremities when it exceeds 180 deg.. More generally,
taking any surface, let a point P, starting from O, move along a
geodesic; this geodesic will be a minimum path from O to P until P
passes through a point O' (if such exist), which is the intersection
with a consecutive geodesic through O. After this point the minimum
property ceases. On an anticlastic surface two geodesics cannot
intersect more than once, and each geodesic is therefore a minimum
path between any two of its points. These illustrations are due to
K.G.J. Jacobi, who has also formulated the general criterion,
applicable to all dynamical systems, as follows:--Let O and P denote
any two configurations on a natural path of the system. If this be the
sole free path from O to P with the prescribed amount of energy, the
action from O to P is a minimum. But if there be several distinct
paths, let P vary from coincidence with O along the first-named path;
the action will then cease to be a minimum when a configuration O' is
reached such that two of the possible paths from O to O' coincide. For
instance, if O and P be positions on the parabolic path of a
projectile under gravity, there will be a second path (with the same
energy and therefore the same velocity of projection from O), these
two paths coinciding when P is at the other extremity (O', say) of the
focal chord through O. The action from O to P will therefore be a
minimum for all positions of P short of O'. Two configurations such as
O and O' in the general statement are called conjugate _kinetic foci_.
Cf. VARIATIONS, CALCULUS OF.

Before leaving this topic the connexion of the principle of stationary
action with a well-known theorem of optics may be noticed. For the
motion of a particle in a conservative field of force the principle
takes the form
_
/
[delta] | vds = 0. (10)
_/

On the corpuscular theory of light v is proportional to the refractive
index mu of the medium, whence
_
/
[delta] | muds = 0. (11)
_/

Hamiltonian principle.

In the formula (2) the energy in the hypothetical motion is
prescribed, whilst the time of transit from the initial to the final
configuration is variable. In another and generally more convenient
theorem, due to Hamilton, the time of transit is prescribed to be the
same as in the actual motion, whilst the energy may be different and
need not (indeed) be constant. Under these conditions we have
_
/t'
[delta] | ([Tau] - V)dt = 0, (12)
_/t

where t, t' are the prescribed times of passing through the given
initial and final configurations. The proof of (12) is simple; we have
_ _ _
/t' /t' /t'
[delta] | ([Tau] - V)dt = | ([delta][Tau] - [delta]V)dt = | {[Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) - [delta]V}dt
_/t _/t _/t
_ _
| |t'
= | [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) |
|_ _|t
_
/t'
- | {[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) + [delta]V}dt (13)
_/t

The integrated terms vanish at both limits, since by hypothesis the
configurations at these instants are fixed; and the terms under the
integral sign vanish by d'Alembert's principle.

The fact that in (12) the variation does not affect the time of
transit renders the formula easy of application in any system of
co-ordinates. Thus, to deduce Lagrange's equations, we have

_ _
/t' /t'/dP[Tau] dP[Tau] dPV \
| ([delta][Tau]-[delta]V)dt = | ( -------[delta]q`1 + -------[delta]q1 + ... - ----[delta]q1 - ...)dt
_/t _/t \ dPq`1 dPq1 dPq1 /
_ _
| |t'
= | p1[delta]q1 + p2[delta]q2 + ... |
|_ _|t

_ _ _
/t'| / dP[Tau] dPV \ / dP[Tau] dPV\ |
- | | (p`1 - ------- + ---- )[delta]q1 + (p`2 - ------- + -----)[delta]q2 + ...|dt. (14)
_/t |_ \ dPq1 dPq1/ \ dPq2 dPq2/ _|

The integrated terms vanish at both limits; and in order that the
remainder of the right-hand member may vanish it is necessary that the
coefficients of [delta]q1, [delta]q2, ... under the integral sign
should vanish for all values of t, since the variations in question
are independent, and subject only to the condition of vanishing at the
limits of integration. We are thus led to Lagrange's equation of
motion for a conservative system. It appears that the formula (12) is
a convenient as well as a compact embodiment of the whole of ordinary
dynamics.

Extension to cyclic systems.

The modification of the Hamiltonian principle appropriate to the case
of cyclic systems has been given by J. Larmor. If we write, as in Sec. 1
(25),

R = T - [kappa][chi]` - [kappa]'[chi]`' - [kappa]''[chi]`" - ..., (15)

we shall have
_
/t'
[delta] | (R - V)dt = 0, (16)
_/t

provided that the variation does not affect the cyclic momenta
[kappa], [kappa]', [kappa]", ..., and that the configurations at times
t and t' are unaltered, so far as they depend on the palpable
co-ordinates q1, q2, ... q_m. The initial and final values of the
ignored co-ordinates will in general be affected.

To prove (16) we have, on the above understandings,
_ _
/t' /t'
[delta] | (R - V)dt = | ([delta][Tau] - [kappa][delta][chi]` - ... -[delta]V)dt
_/t _/t
_
/t' /dP[Tau] dP[Tau] \
= | ( -------[delta]q`1 + ... + -------[delta]q1 + ... - [delta]V )dt, (17)
_/t \ dPq`1 dPq1 /

where terms have been cancelled in virtue of Sec. 5 (2). The last member
of (17) represents a variation of the integral
_
/t'
| ([Tau] - V)dt
_/t

on the supposition that [delta]X = 0, [delta]X' = 0, [delta]X" = 0,
... throughout, whilst [delta]q1, [delta]q2, [delta]q_m vanish at
times t and t'; i.e. it is a variation in which the initial and final
configurations are absolutely unaltered. It therefore vanishes as a
consequence of the Hamiltonian principle in its original form.

Larmor has also given the corresponding form of the principle of least
action. He shows that if we write
_
/
A = |(2[Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ...)dt, (18)
_/

then

[delta]A = 0, (19)

provided the varied motion takes place with the same constant value of
the energy, and with the same constant cyclic momenta, between the
same two configurations, these being regarded as defined by the
palpable co-ordinates alone.

Sec. 8. _Hamilton's Principal and Characteristic Functions._

Principal function.

In the investigations next to be described a more extended meaning is
given to the symbol [delta]. We will, in the first instance, denote by
it an infinitesimal variation of the most general kind, affecting not
merely the values of the co-ordinates at any instant, but also the
initial and final configurations and the times of passing through
them. If we put
_
/t'
S = | (T - V)dt, (1)
_/t

we have, then,
_
/t'
[delta]S = (T' - V')[delta]t' - (T - V)[delta]t + | ([delta]T - [delta]V)dt
_/t
_ _
| |t'
= (T' - V')[delta]t' - (T - V)[delta]t + |[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)| (2)
|_ _|t

Let us now denote by x' + [delta]x', y' + [delta]y', z' + [delta]z',
the final co-ordinates (i.e. at time t' + [delta]t') of a particle m.
In the terms in (2) which relate to the upper limit we must therefore
write [delta]x' - x`'[delta]t', [delta]y' - y`'[delta]t',
[delta]z' - z`'[delta]t' for [delta]x, [delta]y, [delta]z. With a
similar modification at the lower limit, we obtain

[delta]S = - H[delta][tau] + [Sigma]m(x`'[delta]x' + y`'[delta]y' + z`'[delta]z')
- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z), (3)

where H(= T + V) is the constant value of the energy in the free
motion of the system, and [tau](= t' - t) is the time of transit. In
generalized co-ordinates this takes the form

[delta]S = - H[delta][tau] + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (4)

Now if we select any two arbitrary configurations as initial and
final, it is evident that we can in general (by suitable initial
velocities or impulses) start the system so that it will of itself
pass from the first to the second in any prescribed time [tau]. On
this view of the matter, S will be a function of the initial and final
co-ordinates (q1, q2, ... and q'1, q'2, ...) and the time [tau], as
independent variables. And we obtain at once from (4)

dPS dPS \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (5)
dPS dPS |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /

dPS
and H = - -------. (6)
dP[tau]

S is called by Hamilton the _principal function_; if its general form
for any system can be found, the preceding equations suffice to
determine the motion resulting from any given conditions. If we
substitute the values of p1, p2, ... and H from (5) and (6) in the
expression for the kinetic energy in the form [Tau]' (see Sec. 1), the
equation

Tš + V = H (7)

becomes a partial differential equation to be satisfied by S. It has
been shown by Jacobi that the dynamical problem resolves itself into
obtaining a "complete" solution of this equation, involving n + 1
arbitrary constants. This aspect of the subject, as a problem in
partial differential equations, has received great attention at the
hands of mathematicians, but must be passed over here.

Characteristic function.

There is a similar theory for the function
_
/
A = 2 | Tdt = S + H[tau] (8)
_/

It follows from (4) that

[delta]A = [tau][delta]H + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (9)

This formula (it may be remarked) contains the principle of "least
action" as a particular case. Selecting, as before, any two arbitrary
configurations, it is in general possible to start the system from one
of these, with a prescribed value of the total energy H, so that it
shall pass through the other. Hence, regarding A as a function of the
initial and final co-ordinates and the energy, we find

dPA dPA \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (10)
dPA dPA |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /

dPA
and [tau] = --- (11)
dPH

A is called by Hamilton the _characteristic function_; it represents,
of course, the "action" of the system in the free motion (with
prescribed energy) between the two configurations. Like S, it
satisfies a partial differential equation, obtained by substitution
from (10) in (7).

The preceding theorems are easily adapted to the case of cyclic
systems. We have only to write
_ _
/t' /t'
S = | (R - V)dt= | (T - [kappa][chi]` - [kappa]'[chi]`' - ... - V)dt (12)
_/t _/t

in place of (1), and
_
/
A = | (2T - [kappa][chi]` - [kappa]'[chi]`' - ...)dt, (3)
_/

in place of (8); cf. Sec. 7 ad fin. It is understood, of course, that in
(12) S is regarded as a function of the initial and final values of
the palpable co-ordinates q1, q2, ... q_m, and of the time of transit
[tau], the cyclic momenta being invariable. Similarly in (13), A is
regarded as a function of the initial and final values of q1, q2, ...
q_m, and of the total energy H, with the cyclic momenta invariable. It
will be found that the forms of (4) and (9) will be conserved,
provided the variations [delta]q1, [delta]q2, ... be understood to
refer to the palpable co-ordinates alone. It follows that the
equations (5), (6) and (10), (11) will still hold under the new
meanings of the symbols.

9. _Reciprocal Properties of Direct and Reversed Motions._

Lagrange's formula.

We may employ Hamilton's principal function to prove a very remarkable
formula connecting any _two_ slightly disturbed natural motions of the
system. If we use the symbols [delta] and [Delta] to denote the
corresponding variations, the theorem is

d
--[Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r) = 0; (1)
dt

or integrating from t to t',

[Sigma]([delta]p'_r.[Delta]q'_r - [Delta]q'_r.[delta]q'_r) = [Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r). (2)

If for shortness we write

dP squaredS dP squaredS
(r,s) = ----------, (r,s') = -----------, (3)
dPq_rdPq_s dPq_rdPq'_s

we have

dPp_r = - [Sigma]_s(r,s)[delta]q_s - [Sigma]_s(r,s')[delta]q'_s (4)

with a similar expression for [Delta]p_r. Hence the right-hand side of
(2) becomes

- [Sigma]_r{[Sigma]_s(r,s)[delta]q_s + [Sigma]_s(r,s')[delta]q'_s}[Delta]q_r
+ [Sigma]_r{[Sigma]_s(r,s)[Delta]q_s + [Sigma]_s(r,s')[Delta]q'_s}[delta]q_r
= [Sigma]_r[Sigma]_s(r,s'){[delta]q_r.[Delta]q'_s - [Delta]q_r.[delta]q'_s}. (5)

The same value is obtained in like manner for the expression on the
left hand of (2); hence the theorem, which, in the form (1), is due to
Lagrange, and was employed by him as the basis of his method of
treating the dynamical theory of _Variation of Arbitrary Constants_.

Helmholtz's reciprocal theorems.

The formula (2) leads at once to some remarkable reciprocal relations
which were first expressed, in their complete form, by Helmholtz.
Consider any natural motion of a conservative system between two
configurations O and O' through which it passes at times t and t'
respectively, and let t' - t = [tau]. As the system is passing through
O let a small impulse [delta]p_r be given to it, and let the
consequent alteration in the co-ordinate q_s after the time [tau] be
[delta]q'_s. Next consider the _reversed_ motion of the system, in
which it would, if undisturbed, pass from O' to O in the same time
[tau]. Let a small impulse [delta]p'_s be applied as the system is
passing through O', and let the consequent change in the co-ordinate
q_r after a time [tau] be [delta]q_r. Helmholtz's first theorem is to
the effect that

[delta]q_r : [delta]p'_s = [delta]q'_s : [delta]p_r. (6)

To prove this, suppose, in (2), that all the [delta]q vanish, and
likewise all the [delta]p with the exception of [delta]p_r. Further,
suppose all the [Delta]q' to vanish, and likewise all the [Delta]p'
except [Delta]p'_s, the formula then gives

[delta]p_r.[Delta]q_r = - [Delta]p'_s.[delta]q'_s, (7)

which is equivalent to Helmholtz's result, since we may suppose the
symbol [Delta] to refer to the reversed motion, provided we change
the signs of the [Delta]p. In the most general motion of a top
(MECHANICS, Sec. 22), suppose that a small impulsive couple about the
vertical produces after a time [tau] a change [delta][theta] in the
inclination of the axis, the theorem asserts that in the reversed
motion an equal impulsive couple in the plane of [theta] will produce
after a time [tau] a change [delta][psi], in the azimuth of the axis,
which is equal to [delta][theta]. It is understood, of course, that
the couples have no components (in the generalized sense) except of
the types indicated; for instance, they may consist in each case of a
force applied to the top at a point of the axis, and of the
accompanying reaction at the pivot. Again, in the corpuscular theory
of light let O, O' be any two points on the axis of a symmetrical
optical combination, and let V, V' be the corresponding velocities of
light. At O let a small impulse be applied perpendicular to the axis
so as to produce an angular deflection [delta][theta], and let ss'
be the corresponding lateral deviation at O'. In like manner in the
reversed motion, let a small deflection [delta][theta]' at O' produce
a lateral deviation ss at O. The theorem (6) asserts that

ss ss'
----------------- = ---------------, (8)
V'[delta][theta]' V[delta][theta]

or, in optical language, the "apparent distance" of O from O' is to
that of O' from O in the ratio of the refractive indices at O' and O
respectively.

Helmholtz's second reciprocal theorem.

In the second reciprocal theorem of Helmholtz the configuration O is
slightly varied by a change [delta]q_r in one of the co-ordinates, the
momenta being all unaltered, and [delta]q'_s is the consequent
variation in one of the momenta after time [tau]. Similarly in the
reversed motion a change [delta]p'_s produces after time [tau] a
change of momentum [delta]p_r. The theorem asserts that

[delta]p'_s : [delta]q_r = [delta]p_r : [delta]q'_s (9)

This follows at once from (2) if we imagine all the [delta]p to
vanish, and likewise all the [delta]q save [delta]q_r, and if
(further) we imagine all the [Delta]p' to vanish, and all the
[Delta]q' save [Delta]q'_s. Reverting to the optical illustration, if
F, F', be principal foci, we can infer that the convergence at F' of a
parallel beam from F is to the convergence at F of a parallel beam
from F' in the inverse ratio of the refractive indices at F' and F.
This is equivalent to Gauss's relation between the two principal focal
lengths of an optical instrument. It may be obtained otherwise as a
particular case of (8).

We have by no means exhausted the inferences to be drawn from
Lagrange's formula. It may be noted that (6) includes as particular
cases various important reciprocal relations in optics and acoustics
formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and
Tait, and Lord Rayleigh. In applying the theorem care must be taken
that in the reversed motion the reversal is complete, and extends to
every velocity in the system; in particular, in a cyclic system the
cyclic motions must be imagined to be reversed with the rest.
Conspicuous instances of the failure of the theorem through incomplete
reversal are afforded by the propagation of sound in a wind and the
propagation of light in a magnetic medium.

It may be worth while to point out, however, that there is no such
limitation to the use of Lagrange's formula (1). In applying it to
cyclic systems, it is convenient to introduce conditions already laid
down, viz. that the co-ordinates q_r are the palpable co-ordinates and
that the cyclic momenta are invariable. Special inference can then be
drawn as before, but the interpretation cannot be expressed so neatly
owing to the non-reversibility of the motion.

AUTHORITIES.--The most important and most accessible early authorities
are J.L. Lagrange, _Mecanique analytique_ (1st ed. Paris, 1788, 2nd
ed. Paris, 1811; reprinted in _Oeuvres_, vols. xi., xii., Paris,
1888-89); Hamilton, "On a General Method in Dynamics," _Phil. Trans._
1834 and 1835; C.G.J. Jacobi, _Vorlesungen ueber Dynamik_ (Berlin,
1866, reprinted in _Werke_, Supp.-Bd., Berlin, 1884). An account of
the extensive literature on the differential equations of dynamics and
on the theory of variation of parameters is given by A. Cayley,
"Report on Theoretical Dynamics," _Brit. Assn. Rep._ (1857),
_Mathematical Papers_, vol. iii. (Cambridge, 1890). For the modern
developments reference may be made to Thomson and Tait, _Natural
Philosophy_ (1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord
Rayleigh, _Theory of Sound_, vol. i. (1st ed. London, 1877; 2nd ed.
London, 1894); E.J. Routh, _Stability of Motion_ (London, 1877), and
_Rigid Dynamics_ (4th ed. London, 1884); H. Helmholtz, "Ueber die
physikalische Bedeutung des Prinzips der kleinsten Action," _Crelle_,
vol. c., 1886, reprinted (with other cognate papers) in _Wiss. Abh._
vol. iii. (Leipzig, 1895); J. Larmor, "On Least Action," _Proc. Lond.
Math. Soc._ vol. xv. (1884); E.T. Whittaker, _Analytical Dynamics_
(Cambridge, 1904). As to the question of stability, reference may be
made to H. Poincare, "Sur l'equilibre d'une masse fluide animee d'un
mouvement de rotation" _Acta math._ vol. vii. (1885); F. Klein and A.
Sommerfeld, _Theorie des Kreisels_, pts. 1, 2 (Leipzig, 1897-1898); A.
Lioupanoff and J. Hadamard, _Liouville_, 5me serie, vol. iii. (1897);
T.J.I. Bromwich, Proc. Lond. Math. Soc. vol. xxxiii. (1901). A
remarkable interpretation of various dynamical principles is given by
H. Hertz in his posthumous work _Die Prinzipien der Mechanik_
(Leipzig, 1894), of which an English translation appeared in 1900.
(H. Lb.)

DYNAMITE (Gr. [Greek: dynamis], power), the name given to several explosive preparations containing nitroglycerin (q.v.) which are almost exclusively used for blasting purposes. The first practical application of nitroglycerin in this way was made by A. Nobel in 1863. He soaked gunpowder with the liquid and fired the gunpowder by an ordinary fuse. Later he found that nitroglycerin could be detonated by the explosion of several materials such as fulminate of mercury, the use of which as a detonator he patented in 1867. In 1866-1867 he experimented with charcoal and other substances, and found the infusorial earth known as kieselguhr, which consists mainly of silica (nearly 95%), eminently adapted to the purpose, as it was inert, non-combustible, and after a little heating and preparation very porous, retaining a large amount of nitroglycerin as water is held in a sponge, without very serious exudation on standing. This kieselguhr dynamite is generally made by incorporating three parts of nitroglycerin with one part of the dry earth, the paste being then formed into cylindrical cartridges. This work is done by hand. Generally a small percentage of the kieselguhr is replaced by a mixture containing sodium and ammonium carbonates, talc and ochre. This product is known as dynamite No. 1. Disabilities attaching to kieselguhr dynamite are that when placed in water the nitroglycerin is liable to be exuded or displaced, also that, like nitroglycerin itself, it freezes fairly easily and thawing the frozen cartridges is a dangerous operation. Other substances, e.g. kaolin, tripoli, magnesia alba (magnesium carbonate), alumina, sugar, charcoal, some powdered salts and mixtures of sawdust and salts, have been shown to be absorbents more or less adapted to the purpose of making a dynamite. Charcoal from cork is said to absorb about 90% of its weight of nitroglycerin. With the idea of obtaining greater safety, mixtures have been made of nitroglycerin with wood fibre, charcoal and metallic nitrates. Lithofracteur, for instance, consists of 50% nitroglycerin and a mixture of prepared sawdust, kieselguhr and barium nitrate. Carbonite contains 25% of nitroglycerin, the remainder being a mixture of wood-meal and alkali nitrates, with about 1% of sulphur. Dualin, atlas dynamite and potentite are other modifications.

A convenient form in which nitroglycerin can be made up for blasting purposes, especially in wet ground, is the gelatinous material obtained by the action of nitroglycerin, either alone or with the help of solvents, on low-grade or soluble gun-cottons. It is known as blasting gelatin, and was first made by Nobel by incorporating 6 or 7% of low nitrated cellulose (collodion cotton or soluble gun-cotton) with slightly warmed nitroglycerin. The result is a transparent plastic material, of specific gravity 1.5 to 1.6, which may be kept under water for a long time without appreciable change. It is less sensitive to detonation than ordinary dynamite, and although its explosion is slightly slower it is more powerful than dynamite and much superior to the liquid nitroglycerin. Blasting gelatin also freezes and is sensitive to percussion in this state. Camphor and other substances have been added to blasting gelatin to render it more solid and less sensitive. Some modifications of blasting gelatin, e.g. gelignite, contain wood-meal and such oxygen-containing salts as potassium nitrate. Experience has conclusively shown that dynamites are more satisfactory, quicker, and more intense in action than liquid nitroglycerin.

To prevent nitroglycerin and some of the forms of dynamite from freezing it has been proposed to add to them small quantities of either monochlor-dinitroglycerin or of a nitrated poly-glycerin. The former is obtained by first acting upon glycerin with hydrogen chloride to produce _u-_chlorhydrin or chlor-propylene glycol, C3H7O2Cl, which is then nitrated as in the case of glycerin. The latter is obtained by heating glycerin for six or seven hours to about 300 deg. C., whereby water is split off in such manner that a diglycerin C6H14O5, for the most part, results. This on nitration in the usual manner gives a product C6H{10}N4O{13}, which burns and explodes in a similar manner to ordinary nitroglycerin, but is less sensitive and does not so easily freeze. The mono- and di-nitrates of glycerin have also been proposed as additions to ordinary nitroglycerin (q.v.) for the same purpose. (W. R. E. H.)

DYNAMO (a shortened form of "dynamo-electric machine," from Gr. [Greek: dynamis], power), a machine for converting mechanical into electrical energy.

The dynamo ranks with the telegraph and telephone as one of the three striking applications of electrical and magnetic science to which the material progress that marked the second half of the 19th century was in no small measure due. Since the discovery of the principle of the dynamo by Faraday in 1831 the simple model which he first constructed has been gradually developed into the machines of 5000 horse-power or more which are now built to meet the needs of large cities for electric lighting and power, while at the same time the numbers of dynamos in use have increased almost beyond estimate. Yet such was the insight of Faraday into the fundamental nature of the dynamo that the theory of its action which he laid down has remained essentially unchanged. His experiments on the current which was set up in a coil of wire during its movement across the poles of a magnet led naturally to the explanation of induced electromotive force as caused by the linking or unlinking of magnetic lines of flux with an electric circuit. For the more definite case of the dynamo, however, we may, with Faraday, make the transition from line-linkage to the equivalent conception of "line-cutting" as the source of E.M.F.--in other words, to the idea of electric conductors "cutting" or intersecting[1] the lines of flux in virtue of relative motion of the magnetic field and electric circuit. On the 28th of October 1831 Faraday mounted a copper disk so that it could be rotated edgewise between the poles of a permanent horse-shoe magnet. When so rotated, it cut the lines of flux which passed transversely through its lower half, and by means of two rubbing contacts, one on its periphery and the other on its spindle, the circuit was closed through a galvanometer, which indicated the passage of a continuous current so long as the disk was rotated (fig. 1). Thus by the invention of the first dynamo Faraday proved his idea that the E.M.F. induced through the interaction of a magnetic field and an electric circuit was due to the passage of a portion of the electric circuit _across_ the lines of flux, or vice versa, and so could be maintained if the cutting of the lines were made continuous.[2] In comparison with Faraday's results, the subsequent advance is to be regarded as a progressive perfecting of the mechanical and electro-magnetic design, partly from the theoretical and partly from the practical side, rather than as modifying or adding to the idea which was originally present in his mind, and of which he already saw the possibilities.

A dynamo, then, is a machine in which, by means of continuous relative motion, an electrical conductor or system of conductors forming part of a circuit is caused to cut the lines of a magnetic field or fields; the cutting of the magnetic flux induces an electromotive force in the conductors, and when the circuit is closed a current flows, whereby mechanical energy is converted into electrical energy.

Little practical use could be made of electrical energy so long as its
only known sources were frictional machines and voltaic batteries. The
cost of the materials for producing electrical currents on a large
scale by chemical action was prohibitive, while the frictional machine
only yielded very small currents at extremely high potentials. In the
dynamo, on the other hand, electrical energy in a convenient form
could be cheaply and easily obtained by mechanical means, and with its
invention the application of electricity to a wide range of commercial
purposes became economically possible. As a converter of energy from
one form to another it is only surpassed in efficiency by another
electrical appliance, namely, the transformer (see TRANSFORMERS). In
this there is merely conversion of electrical energy at a high
potential into electrical energy at a low potential, or vice versa,
but in the dynamo the mechanical energy which must be applied to
maintain the relative movement of magnetic field and conductor is
absorbed, and reappears in an electrical form. A true transformation
takes place, and the proportion which the rate of delivery of
electrical energy bears to the power absorbed, or in other words the
_efficiency_, is the more remarkable. The useful return or "output" at
the terminals of a large machine may amount to as much as 95% of the
mechanical energy which forms the "input." Since it needs some prime
mover to drive it, the dynamo has not made any direct addition to our
sources of energy, and does not therefore rank with the primary
battery or oil-engine, or even the steam-engine, all of which draw
their energy more immediately from nature. Yet by the aid of the
dynamo the power to be derived from waterfalls can be economically and
conveniently converted into an electrical form and brought to the
neighbouring factory or distant town, to be there reconverted by
motors into mechanical power. Over any but very short distances energy
is most easily transmitted when it is in an electrical form, and
turbine-driven dynamos are very largely and successfully employed for
such transmission. Thus by conducing to the utilization of water-power
which may previously have had but little value owing to its
disadvantageous situation, the dynamo may almost be said to have added
another to our available natural resources.

The two essential parts of the dynamo, as required by its definition, may be illustrated by the original disk machine of Faraday. They are (1) the _iron magnet_, between the poles of which a magnetic field exists, and (2) the _electrical conductors_, represented by the rotating copper disk. The sector of the disk cutting the lines of the field forms part of a closed electric circuit, and has an E.M.F. induced in it, by reason of which it is no longer simply a conductor, but has become "active." In its more highly developed form the simple copper disk is elaborated into a system of many active wires or bars which form the "winding," and which are so interconnected as to add up their several E.M.F.'s. Since these active wires are usually mounted on an iron structure, which may be likened to the keeper or "armature" of a magnet rotating between its poles, the term "armature" has been extended to cover not only the iron core, but also the wires on it, and when there is no iron core it is even applied to the copper conductors themselves. In the dynamo of Faraday the "armature" was the rotating portion, and such is the case with modern continuous-current dynamos; in alternators, however, the magnet, or a portion of it, is more commonly rotated while the armature is stationary. It is in fact immaterial to the action whether the one or the other is moved, or both, so long as their relative motion causes the armature conductors to cut the magnetic flux. As to the ultimate reason why an E.M.F. should be thereby induced, physical science cannot as yet yield any surer knowledge than in the days of Faraday.[3] For the engineer, it suffices to know that the E.M.F. of the dynamo is due to the cutting of the magnetic flux by the active wires, and, further, is proportional to the rate at which the lines are cut.[4]

The equation of the _electromotive force_ which is required in order to render this statement quantitative must contain three factors, namely, the density of the flux in the air-gap through which the armature conductors move, the active length of these wires, and the speed of their movement. For given values of the first and third factors and a single straight wire moved parallel to itself through a uniform field, the maximum rate of cutting is evidently obtained when the three directions of the lines of the conductor's length and of the relative motion are respectively at right angles to each other, as shown by the three co-ordinate axes of fig. 2. The E.M.F. of the single wire is then

E = B_gLV x 10^(-8) volts (1)

where B_g is the density of the flux within the air-gap expressed in C.G.S. lines per square centimetre, L is the active length of the conductor within the field in centimetres, and V is the velocity of movement in centimetres per second. Further, the direction in which the E.M.F. has the above maximum value is along the length of the conductor, its "sense" being determined by the direction of the movement[5] in relation to the direction of the field.

The second fundamental equation of the dynamo brings to light its mechanical side, and rests on H.C. Oersted's discovery of the interaction of a magnetic field and an electric current. If a straight electric conductor through which a current is passing be so placed in a magnetic field that its length is not parallel to the direction of the lines of flux, it is acted on by a force which will move it, if free, in a definite direction relatively to the magnet; or if the conductor is fixed and the magnet is free, the latter will itself move in the opposite direction. Now in the dynamo the active wires are placed so that their length is at right angles to the field; hence when they are rotated and an electric current begins to flow under the E.M.F. which they induce, a mutual force at once arises between the copper conductors and the magnet, and the direction of this force must by Lenz's law be opposed to the direction of the movement. Thus as soon as the disk of fig. 1 is rotated and its circuit is closed, it experiences a mechanical pull or drag which must be overcome by the force applied to turn the disk. While the magnet must be firmly held so as to remain stationary, the armature must be of such mechanical construction that its wires can be forcibly driven through the magnetic field against the mutual pull. This law of electrodynamic action may be quantitatively stated in an _equation of mechanical force_, analogous to the equation (I.) of electromotive force, which states the law of electromagnetic induction. If a conductor of length L cm., carrying a current C amperes, is immersed in a field of uniform density B_g, and the length of the conductor is at right angles to the direction of the lines, it is acted on by a force

F = B_gLC x 10^(-1) dynes, (2)

and the direction of this force is at right angles to the conductor and to the field. The rate at which electrical energy is developed, when this force is overcome by moving the conductor as a dynamo through the field, is EC = B_gLVC x 10^(-8) watts, whence the equality of the mechanical power absorbed and the electrical power developed (as required by the law of the conservation of energy) is easily established. The whole of this power is not, however, available at the terminals of the machine; if R_a be the resistance of the armature in ohms, the passage of the current C_a through the armature conductors causes a drop of pressure of C_aR_a volts, and a corresponding loss of energy in the armature at the rate of C_a squaredR_a watts. As the resistance of the external circuit R_e is lowered, the current C = E_a/(R_e + R_a) is increased. The increase of the current is, however, accompanied by a progressive increase in the loss of energy over the armature, and as this is expended in heating the armature conductors, their temperature may rise so much as to destroy the insulating materials with which they are covered. Hence the temperature which the machine may be permitted to attain in its working is of great importance in determining its output, the current which forms one factor therein being primarily limited by the heating which it produces in the armature winding. The lower the resistance of the armature, the less the rise of its temperature for a given current flowing through it; and the reason for the almost universal adoption of copper as the material for the armature conductors is now seen to lie in its high conductivity.[6]

Since the voltage of the dynamo is the second factor to which its output is proportional, the conditions which render the induced E.M.F. a maximum must evidently be reproduced as far as possible in practice, if the best use is to be made of a given mass of iron and copper. The first problem, therefore, in the construction of the dynamo is the disposition of the wires and field in such a manner that the three directions of field, length of active conductors, and movement are at right angles to one another, and so that the relative motion is continuous. Reciprocating motion, such as would be obtained by direct attachment of the conductors to the piston of a steam-engine, has been successfully employed only in the special case of an "oscillator,"[7] producing a small current very rapidly changing in direction. Rotary motion is therefore universally adopted, and with this two distinct cases arise. Either (A) the active length of the wire is parallel to the axis of rotation, or (B) it is at right angles to it.