Dynamics (from Greek δυναμικός  dynamikos "powerful", from δύναμις  dynamis "power") may refer to:
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DYNAMICS (from Gr. 8vvaµcs, strength), the name of a branch of the science of Mechanics. The term was at one time restricted to the treatment of motion as affected by force, being thus opposed to Statics, which investigated equilibrium or conditions of rest. In more recent times the word has been applied comprehensively to the action of force on bodies either at rest or in motion, thus including " dynamics " (now termed kinetics) in the restricted sense and " statics." Analytical D Ynamics.  The fundamental principles of dynamics, and their application to special problems, are explained in the articles Mechanics and Motion, Laws Of, where brief indications are also given of the more general methods of investigating the properties of a dynamical system, independently of the accidents of its particular constitution, which were inaugurated by J. L. Lagrange. These methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. In determining the properties of such systems the methods of analytical geometry and of the infinitesimal calculus (or, more generally, of mathematical analysis) are necessarily employed; for this reason the subject has been named Analytical Dynamics. The following article is devoted to an outline of such portions of general dynamical theory as seem to be most important from the physical point of view.
1. General Equations of Impulsive Motion. The systems contemplated by Lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables ql, g2,...gn, each of which admits of continuous variation over a certain range, so that if x, y, z be the Cartesian coordinates of any one particle, we have for example x = f(q l, g2,... g n), y=&c., z=&c.,.. (I) where the functions f differ (of course) from particle to particle. In modern language, the variables q,, g2,.
gn are generalized co ordinates serving to specify the configuration of the system; their derivatives with respect to the time are denoted by q1, g2,...gn, and are called the generalized components of velocity. The continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called the path. For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration of impulsive motion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular coordinates of any particle m, mx mz=Z',.. (2) where X', Y', Z' are the components of the impulse on m. Now let Sx, Sy, Sz be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation 1m(xSx { j6y+zSz)= 1(X'Sx+Y'Sy +Z'Sz), . (3) where the sign E indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations Sq i, 5 Q2 ,... of the generalized coordinates, we have x = a4g 1 +a4 g2 + ..., &c., &c.. Sx=ag l Sglf ag xSq2}..., &c., &c., .
and therefore Im (xSx+pSy+zaz) = A11g1+Al2g2 +...)aq1 + (A2141 + A 2242+...) 6 q2 +..., where If we form the expression for the kinetic energy T of the system, we find 2T = 1m (x2 y2 Z2) =A 11g1 2 A 22g2 2 +...+2Al2g1g2+ ... (8) The coefficients A11, A22,...Al2,... are by an obvious analogy called the coe f ficients of inertia of the system; they are in general functions of the coordinates ql, q2,.... The equation (6) may now be written Em (x6x+/Ay +zaz) = aQ Sq 1 + a 5q21 ... (9) This maybe regarded as the cardinal formula in Lagrange's method. For the righthand side of (3) we may write (X'Sx+Y'by+Z'sz) =Q'1Sg1 +Q'2ag2+..., where Q'r= ? +Z' / g The quantities Qi, Q2, ... are called the generalized components of impulse. Comparing (9) and (Io), we have, since the variations Sq l, S Q21 ... are independent, aT a q1 = Q 1, ag = Q 2, ...
These are the general equations of impulsive motion. It is now usual to write aT p r = aq (is) r The quantities p1, p2,
represent the effects of the several component impulses on the
system, and are therefore called the generalized components of
momentum. In terms of them we have Zm(±Sx+ii
y+Sz)=p1Sg1+p2ag2+
(14) Also, since T is a homogeneous quadratic function of the
velocities g 2T =p1g1+p242+
(15) This follows independently from (14), assuming the special
variations Sx = xdt, &c., and therefore a Q1 = g l dt,
2 g 2 dt,.... Again, if the values of the velocities and
the momenta in any other motion of the system through the same
configuration be distinguished by accents, we have the identity
plg'1+p24'2+
= p'1g1 +p'2g2+
, each side being equal to the symmetrical expression A1141g'1 A
2242q ' 2
A l2(414 ' 2+4 ' 142)
The theorem (16) leads to some important reciprocal relations.
Thus, let us suppose that the momenta Pi, P2,... all vanish with
the exception of p ', and similarly that the momenta p'1, p'2,...
all vanish except p'2. We have then p1q'1 = p'2g2, or: =
q' ' 2 (18) The interpretation is simplest when the
coordinates q,, q2 are both of the same kind,
e.g. both lines or both angles. We may then conveniently
put p1= p'2, and assert that the velocity of the first type due to
an impulse of the second type is equal to the velocity of the
second type due to an equal impulse of the first type. As an
example, suppose we have a chain of straight links hinged each to
the next, extended in a straight line, and free to move. A blow at
right angles to the chain, at any point P, will produce a certain
velocity at any other point Q; the theorem asserts that an equal
velocity will be produced at P by an equal blow at Q. Again, an
impulsive couple acting on any link A will produce a certain angular velocity in
any other link B; an equal couple applied to B will produce an
equal angular velocity in A. Also if an impulse F applied at P
produce an angular velocity w in a link A, a couple Fa applied to A
will produce a linear velocity wa at P. Historically, we may note that
reciprocal relations in dynamics were first recognized by H. L. F.
Helmholtz in the domain of acoustics; their use has been greatly
extended by Lord Rayleigh.
The equations (13) determine the momenta Pi, P2,... as linear functions of the velocities ql, g2,
.. Solving these,' we can express di, g2,.
as linear functions of p1, p2,
The resulting equations give us the velocities produced by any given of system of impulses. Further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta Pi, p2,
The kinetic energy, as so expressed, will be denoted by T'; thus 2T' = A' 11p1 2 +A' 22p2 2 ... +2A' 12pp2+... where A'11, A ' 22,
A'12,... are certain coefficients depending on the configuration.
They have been called by Maxwell the coefficients of mobility
of the system. When the form (19) is given, the values ax ax
ay ay az az j A rs =Em + + = AST.
ag T ag s OD. ag s ag,. aqs An= ?m (a gr? 2 +
(a gr l 2+ (agr / 2, ' 1 Y(7) (4) (5). (6).
(io). (II). (12) (19) of the velocities in terms of the momenta can
be expressed in a remarkable form due to Sir W. R. Hamilton. The formula (15)
may be written p141+p2g2+... T+T',. (20) where T is supposed
expressed as in (8), and T' as in (19). Hence if, for the moment,
we denote by S a variation affecting the velocities, and
therefore the momenta, but not the configuration, we have p15 g
1+g1Sp+p2342+425p2+... =ST +ST' OT aT 3T' aT' = aglbgl + a
sQ 2 +... + a pip1 Sp 1 +a73,3p2+
In virtue of (13) this reduces to aT' aT' 413§1+g2Sp2+
 3  13 Sp1+a sp2+....
Since Sp1, 5p ,... may be taken to be independent, we infer that aT' aT' q 1  ap 1, q 2 = a p 2, In the very remarkable exposition of the matter given by James Clerk Maxwell in his Electricity and Magnetism, the Hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument.
An important modification of the above process was introduced by E. J. Routh and Lord Kelvin and P. G. Tait. Instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it in terms of the velocities corresponding to some of the coordinates, say q1, q2,...qm, and of the momenta corresponding to the remaining coordinates, which (for the sake of distinction) we may denote by x, x', x",
.. Thus, T being expressed as a homogeneous quadratic function of Q 11 ' Q2,
Qm, X, X, x ,. ., the momenta corresponding to the coordinates x,
x', x",... may be written OT,, aT ,, aT a Z a X,, a  (24) These
equations, when written out in full, determine X, X', X",
as linear functions of 41, g2,...gm, We now consider the function
R=T  Kx  K'x'  K„Z„_...,. .. (25) supposed expressed, by means
of the above relations in terms of q1, q2, . gm, K',
K",.... Performing the operation S on both sides of
(25), we have ag 1Sglf...+aKSK}... =a Q 1Sg1+...+O
5x+...  Kai  XSK  ...,. . (26) where, for brevity, only one
term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have
aR. aR aT. ag 1 Sg1+...+ K SK+. .. =ag1Sg1+...  xSK 
..... (27) Since the variations SD, 5q ... Sq„, 5K,
SK', may be taken to be independent, we have _ p1 a _
g2 a =  An important property of the present transformation is
that, when expressed in terms of the new variables, the kinetic
energy is the sum of two homogeneous quadratic functions, thus T =?
 {  K,. ... (30) where 6 involves the velocities q1, q2,...
g alone, and K the momenta alone. For in virtue of (29) we
have, from (25), T =R  (K+ a K K' +Kn ?l t"I....),.. (31)
and it is evident that the terms in R which are bilinear in respect
of the two sets of variables 1, q2,... qm and will dis appear from
the righthand side.
It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively. Let us suppose, in the first place, that the effect that the kinetic energy is greater than if by the system is started by given impulses of certain types, but is otherwise free. J. L. F. Bertrand's theorem is to impulses of the remaining types the system were constrained to take any other course. We may suppose the coordinates to be so chosen that the constraint is expressed by the vanishing of the velocities 4 21 ... q m, whilst the given impulses are Hence the energy in the actual motion is greater than in the constrained motion by the amount 6.
Again, suppose that the system is started with prescribed velocity components 41, q2,... qm, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have = o, =o, = o,... and therefore K = o. The kinetic energy is therefore less than in any other motion consistent with the prescribed velocityconditions by the value which K assumes when represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of straight links already employed. Thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom.
Continuous Motion of a System. We may proceed to the continuous motion of a system. The equations of motion of any particle of the system are of the form mx=X, my=Y, mz=Z.. . (I) Now let 'x+Sx,' y+Sy, z+Sz be the coordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation Em(. x+iSy = E(XSx+YSy+ZSz).. (2) Lagrange's investigation consists in the transformation of (2) into an equation involving the independent variations SD, 5q ,... Sqn. It is important to notice that the symbols S and d/dt are commutative, since ax = a (x+ax)  dt = a Sx, &c.
Hence Zm(xSx}pSy+ z) =dtEm(xSx+p.y+2Sz)  Zm(xSx}1/5fj+262) d (p11g1+p2Sg2+...) by § I (14). The last member may be written POD + p 1 6 41 +p2Sg2 +§2642 OT aT aT aT Sg1  aq542  a qI Q2  ... (5) Hence, omitting the terms which cancel in virtue of § 1 (13), we find aTl (P2 lm(x+yay+z) _ ??1 Sq1+  5   6.) 6Q 2 +.... (6) For the righthand side of (2) we have Z(XSx +)(Sy+ ZSz) = QiS g 1 +Q2Sg2+..., (7) where Q,. = E (X + Y + Z;).. (8) The quantities Q1, Q2,... are called the generalized components of force acting on the system.
Comparing (6) and (7) we find i l  Q P ?Q (9) or, restoring the values of pi, p2,..., d aT a T d T dt aT  5F2= Q2, .... These are Lagrange's general equations of motion. Their number is of course equal to that of the coordinates q1, q2,... to be determined.
Analytically, the above proof is that given by Lagrange, but the terminology employed is of much more recent date, having been first introduced by Lord Kelvin and P. G. Tait; it has greatly promoted the physical application of the subject. Another proof of the equations (to), by direct transformation of coordinates, has been given by Hamilton and independently by other writers (see but the variational method of Lagrange is that which stands in closest relation to the subsequent developments of the subject. The chapter of Maxwell, already referred to, is a most instructive commentary on the subject from the physical point of view, although the proof there attempted of the equations (to) is fallacious.
In a " conservative system " the work which would have to be done by extraneous forces to bring the system from rest in some standard configuration to rest in the configuration (Q 11 q2,... is independent of the path, and may therefore be regarded as a definite function of q1, q2,... Denoting this function (the potential energy) by V, we have, if there be no extraneous force on the system, Z(XSx+)(Sy+ZIz) =  SV,. .. (I I) and therefore = Q2 and. (28) (3) (4) (21) (23) (Io) Hence the typical Lagrange's equation may be now written in the form d aTl aT av dt alt./ aq r aqr' or, again, fir=   T). (14) It, has been proposed by Helmholtz to give the name kinetic potential to the combination V  T.
As shown under Mechanics, § 22, we derive from (Io) d, ? = Qlgl+Q242 +
, and therefore in the case of a conservative system free from
extraneous force, dt (T +V) = 0 or T+V =const.,.. (16)
which is the equation of energy. For examples of the application of
the formula (13) see Mechanics, § 22.
3. Constrained Systems. It has so far been assumed that the geometrical relations, if varying any, which exist between the various parts of the system are of the type § I (I), and so do not contain t explicitly. The extension of Lagrange's equations to the case of " varying relations " of the type x = f(t, 41, 4 2 ,...g n), y =&c., z=&c.,. (I) was made by J. M. L. Vieille. We now have ax ax. ax x  at + Q1 41+ Q2 42+... &c., &c., ax ax ax =5qiF662+..., &c., &c., .
so that the expression § I (8) for the kinetic energy is to be replaced by 2T= ao+2a141+2a2g2+ +A1141 2 +A2242 2 +
+Al24142+..., (4) where a =Fim ('it  ) (Oy)
C4) ar = " m at 'aq r + at
+ a oqr ax ax ay ay az az and the forms of A rr, A ra are
as given by § I (7). It is to be remembered that the coefficients
ao, al, a2, ...A11, A22,... Al2... will in general involve
t explicitly as well as implicitly through the coordinates
ql, q2,.... Again, we find Em(±Sx+ySy+ZSz) = (al +Au41+Al242+
)Sq1 + (a2+A2141+A2242+
)0q2+
. aT aT =a QSg1+a 4 6q2+... = PlSgl+P2642+
, . (6) where p r is defined as in § I (13). The
derivation of Lagrange's equations then follows exactly as before.
It is to be noted that the equation § 2 (15) does not as a rule now
hold. The proof involved the assumption that T is a homogeneous
quadratic function of the velocities ql, q2.
It has been pointed out by R. B. Hayward that Vieille's case can be
brought under Lagrange's by introducing a new coordinate (x) in place of
t, so far as it appears explicitly in the relations (I).
We have then 2T = aoX 2 +2(a141+a242+...)X+A1141 2 +A2242 2 +...
+2Al24142+.... (7) The equations of motion will be as in § 2 (to),
with the additional equation d aT aT _ X (8) dt
aX ax, .
where X is the force corresponding to the coordinate x. We may suppose X to be adjusted so as to make X = o, and in the remaining equations nothing is altered if we write t for x before, instead of after, the differentiations. The reason why the equation § 2 (15) no longer holds is that we should require to add a term XX on the righthand side; this represents the rate at which work is being done by the constraining forces required to keep X constant.
As an example, let x, y, z be the coordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. If 4 be the angular coordinate of the solid, we find without difficulty 2T =m(x2+.0+z2)+2(bm(xy  y±)+{I +m (x2 + y2)1m 2, (9) where I is the moment of inertia of the solid. The equations of motion, viz.
d aT aT _ X become m(z2 y  x¢ 2  y) = 'm(' y +2 x  yq2 +x(b) =Y, naz= Z,(12) and d }{I+m(x2+y2)}?+m(xy  yx)] =4)... (13) suppose the moment of inertia I to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz.
If we suppose 1 adjusted so as to maintain =o, or (again) if we m(2wy  w 2 x) =X, m(y+2wx  w 2 y) =Y, mz=Z,. (14) where w has been written for. These are the equations which we should have obtained by applying Lagrange's rule at once to the formula 2T =m(2 +? 2 + 22 ) + 2 m w (xy  yx)+m w2 (x 2 +y 2), (is) which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity w. (See Mechanics, § 13.) More generally, let us suppose that we have a certain group of coordinates x, x', x",
 whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocitycomponents X, X', X",... are maintained constant. The remaining coordinates being denoted by qn, we may write 2T = 6+To+ 2 (a141 +a242+...)+2(a'lgl+a'242+...)X' +..., (16) where 6 is a homogeneous quadratic function of the velocities l , 42,...4n of the type §I (8), whilst To is a homogeneous quadratic function of the velocities X, X', X",... alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (io) of § 2 give n equations of the type (Lt(Or)  % 6 +(r, 1)41+ (r, 2)g2 +...  ?4?=Qr where aa r aa 3 aa 'r aa' a, (r, s) = ( x+( X+.... a o r ,. These quantities (r, s) are subject to the relations (r, s) =  (s, r), (r, r) = o. . (29) The remaining dynamical equations, equal in number to the co ordinates x, x', x yield expressions for the forces which must be applied in order to maintain the velocities X, X', X",
constant; they need not be written down. If we follow the method by
which the equation of energy was established in § 2, the equations
(17) lead, on taking account of
the relations (19), to am  To =Q141+Q242+...+Qn g n,. .
(20) or, in case the forces Q r depend only on the coordinates
qi, g2,...gn and are conservative, 6+V  To =const.. ..
(21) The conditions that the equations (27) should be
satisfied by zero values of the
velocities 41, 42, . 4, are Qr=  aq, aTo. (22) or in the
case of conservative forces a g r (V  To) =0, (23) i.e.
the value of V  To must be stationary. We may apply this
to the case of a system whose configuration relative to axes
rotating with constant angular velocity (w) is defined by means of
the n coordinates ql, g2,
.gn This is important on account of its bearing on the kinetic theory of the tides. Since the Cartesian coordinates x, y, z of any particle m of the system relative to the moving axes are functions of qi, g2,...gn, of the form § I (I), we have, by (25) 26=Em(x 2 + y2 +t 2), 2To ar = Em (x ag yag), whence (r, s) = 2w.E ma(x y ') (26) a (4a, qr) The conditions of relative equilibrium are given by (23).
It will be noticed that this expression V  To, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious " centrifugal forces." The question of stability of relative equilibrium will be noticed later (§ 6).
It should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. From (15) and (14) we find 27  D(6 +To) ± /ni (xp  = d t (6  To) +w.E(xY  yX)... (27) (13) (15) (5) (25) d aT aT d aT aT ( dt a?  ay = Y' (11_ az  az Z , 10) d aT aT at a  4;  a? = ' and =w2Em (x2 2), which follow at once from § I (23), since V does not involve pi, we obtain a complete system of differential equations of the first order for the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since these make dH __ aH. aH. aH a H .
ata pi ?'1+appzP2+... + gi q ' +agq2 + ... = 0. The Hamiltonian transformation is extended to the varying relations as follows. Instead of (4) we write This must be equal to the rate at which the forces acting on the system do work, viz. to col (xY yX)+Qig1+Q2g2+
+Qngn, where the first term represents the work done in virtue of
the rotation.
have still to notice the modifications which Lagrange's equations undergo when the coordinates q i, are not all independently variable. In the first place, we may suppose them connected by a number m (<n) of relations of the type A(t, qi, q2, ...q n) =0, B (t, qi, q 2, ... qn) =o, &c. (28) These may be interpreted as introducing partial constraints into a previously free system. The variations Sq i, SQ 21 ...Sg n in the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations aA aA aB aB a gi S g1 + a Q2 s g2+ ... =0, agiS g i+ aQ2 Sg2 + ... = 0, &c. (29) Introducing indeterminate multipliers X, /2 ,..., one for each of these equations, we obtain in the usual manner n equations of the type d aT aT dt aq  aq = A + a q +...,
(30) in place of § 2 (to). These equations, together with (28), serve to determine the n coordinates q i, q 2,
q„ and the m multipliers When t does not occur explicitly
in the relations (28) the system is said to be holonomic.
The term connotes the existence of integral (as opposed to
differential) relations between the coordinates, independent of
the time.
Again, it may happen that although there are no prescribed relations between the coordinates qi, g2,...gn, yet from the circumstances of the problem certain geometrical conditions are imposed on their variations, thus Aibgi+A25g2+... =o, B i ag i +B 2 8g 2 + ... = o, &c., (31) where the coefficients are functions of qi, 42,
q, and (possibly) of t. It is assumed that these
equations are not integrable as regards the variables qi, q2, ...
qn; otherwise, we fall back on the previous conditions. Cases of
the present type arise, for instance, in ordinary dynamics when we
have a solid rolling on a (fixed or moving) surface. The six
coordinates which serve to specify the position of the solid at
any instant are not subject to any necessary relation, but the
conditions to be satisfied at the point of contact impose three
conditions of the form (31). The general equations of motion are
obtained, as before, by the method of indeterminate multipliers,
thus aT aT =Q r+ aA,+uBr+.... . (32) dt aq T
 aqT The coordinates qi, q2, ... qn, and the indeterminate
multipliers X, µ,..., are determined by these equations and by the
velocityconditions corresponding to (31). When t does not
appear explicitly in the coefficients, these velocityconditions
take the forms A141+A242+...=0, Bigi+B2g2+...=0, &c. (33)
Systems of this kind, where the relations (31) are not integrable,
are called nonholonomic. 4. Hamiltonian Equations of
Motion. In the Hamiltonian form of the equations of motion of a
conservative system with unvarying relations, the kinetic energy is
supposed expressed in terms of the momenta p i, p 2, ...
and the coordinates q i, Q 2, ..., as in § I (19). Since the symbol S now denotes a
variation extending to the coordinates as well as to the momenta,
we must add to the last member of § I (21) terms of the types
aT Sqi Sq i +
. (1) Since the variations Sp i, Sp 2, ... Sq i, Q2, ... may be taken to be independent, we infer the equations § I (23) as before, together with aT aT' aT aqi =  aqi' 8q2 = a2, ..., Hence the Lagrangian equations § 2 (14) transform into p i = (T' +V), p2= (T' +V), ... If we write H=T'+V,. .
so that H denotes the total energy of the system, supposed in terms of the new variables, we get aH aH (5) (I P2 =  aq?, If to these we join the equations aH aH A l = a p , g2 = ap 2, ..., (7) case of H = pigs+p242+...  T+V,. .. (8) and imagine H to be expressed in terms of the momenta pi, p 2, ..., the coordinates q i, q 21 .., and the time. The internal forces of the system are assumed to be conservative, with the potential energy V. Performing the variation S on both sides, we find aT av SH =giapi+
 agi Sgi+ agi Sq+
, (9) terms which cancel in virtue of the definition of p i,
p2,... being omitted. Since Sp i, 3p 21
, Sq i, SQ 21
may be taken to be independent, we infer OH OH = 2 = and
(T  V)=  a4, (2(T  V)=  aq,..... (II) It follows
from (II) that aH aH pi   4., The equations (to) and (12) have
the same form as above, but H is no longer equal to the energy of
the system.
5. Cyclic Systems. A cyclic or gyrostatic system is characterized by the following properties. In the first place, the kinetic energy is not affected if we alter the absolute values of certain of the coordinates, which we will denote by x, x', x",
, provided the remaining coordinates q i, q 2, ... q„,
and the velocities, including of course the velocities X, x', x",
..., are unaltered. Secondly, there are no forces acting on the
system of the types x, x',x",
. This case arises, for example, when the system includes gyrostats
which are free to rotate about their axes, the coordinates x,
x', x ",
then being the angular coordinates of the gyrostats relatively to
their frames. Again, in theoretical hydrodynamics we have the problem of
moving solids in a frictionless liquid; the ignored coordinates x,
x', x",
then refer to the fluid, and are infinite in number. The same question presents
itself in various physical speculations where certain phenomena are
ascribed to the existence of latent motions in the
ultimate constituents of matter. The general theory of such systems
has been treated by E. J. Routh, Lord Kelvin, and H. L. F.
Helmholtz.
If we suppose the kinetic energy T to be expressed, as in Lagrange's method, in terms of the coordinates and the velocities, the equations of motion corresponding to x, x', x",
reduce, in virtue of the above hypotheses, to the forms d
=O, d aT = d a? = 0, .,. at dt ax °' at ax whence
aT_ , aT _ „ aX  K, K, where K,
K', K", ... are the constant momenta corresponding to the
cyclic coordinates x, x', x",
These equations are linear in x, x', x", ...; solving them with respect to these quantities and substituting in the remaining. Lagrangian equations, we obtain m differential equations to determine the remaining coordinates q i, g 2, ... qm. The object of the present investigation is to ascertain the general form of the resulting equations. The retained coordinates qi, q2, ... qn, may be called (for distinction) the palpable coordinates of the system; in many practical questions they are the only coordinates directly in evidence.
If, as in § 1 (25), we write R=T  Kx  K'x'  K»x„  ..., (3) and imagine R to be expressed by means of (2) as a quadratic function of q i , Q 21 ... q m, K, K', K", ... with coefficients which are in general functions of the coordinates q i, q 2,
q,,,, then, performing the operation S on both sides, we
find aRS ...}  aRaK+... + aRSgi+...=a
r Sgi+...+a1Sgi+... aq i at, aqi agi a qi + az &X+...+ aX
Sqi+...KSxx3K...
(4) We. (2) (3) (4) expressed. (6). (12) aT aX (I)  (2) Omitting the terms which cancel by (2), we find aT aR aT aR a41 =aq1' 342  342'
aT aR aT aR ' '= OD'' aq2 =aq2, x=  a K, X =  aK
', aR aR „ aR Substituting in § 2
(io), we have d 'aR R ' d '_ _ = dt
a q l a q l QI' crt. a42  a Q = These are
Routh's forms of the modified Equivalent forms were obtained
independently by Helmholtz at a later date.
The function R is made up of three parts, thus R = R2,0+R1,i+Rc,2,
(9) where R 1, 0 is a homogeneous quadratic function of 4'1 i g21...4,,,, i is a homogeneous quadratic function of whilst R 111 consists of products of the velocities g l, g2,
.gm into the momenta / Hence from (3) and (7) we have T=R  (+ 4+K'+...) =R2,0  R0,2. . (io) If, as in § i (30), we write this in the form T=6+K,. .
then (3) may be written R = U K (12) where Ni, 12,
are linear functions of say R r= aric  FerK'  Fa"ric"+
..,
(13) the coefficients a,, a' r, a being in general
functions of the co .ordinates ql, q2,... qm. Evidently
(3 r denotes that part of the momentumcomponent
aR/a4 which is due to the cyclic motions. Now d
aR d (06+ Rr d 'a6 + aar 4 1+ a ' g2 + ..., (14)
dt a4r  a4 r  dt a 4 r aq l aq2 aR 36 aK,
as2. a 4r = a 4r  a4r  a gr gl+ a gr
g2+.... Hence, substituting in (8), we obtain the
typical of a gyrostatic system in the form d 'a6 06
1)4'14(' r , 2)42+
+(r, s)',+.. dt a4r  a4r where (r, s) _ q  a/3r
308 a 8 a4r' This form is due to Lord Kelvin. When ql,
q2,... qm have been determined, as functions of the time, the
velocities corresponding to the cyclic coordinates can be found,
if required, from the relations (7), which may be written 3K X = 
a141  a2g2  ...,, aK &c., &c.
It is to be particularly noticed that (r, r) = o, (r, s)=  (s, r). .. . (19) Hence, if in (16) we put r = I, 2, 3,... m, and multiply by q i, 42,... qm respectively, and add, we find dt(v+K) = Q1 g 1+Q2g2+
, . (20) or, in the case of a conservative system 6+V+K=const.,..
(21) which is the equation of energy.
The equation (16) includes § 3 (17) as a particular case, the eliminated coordinate being the angular coordinate of a rotating solid having an infinite moment of inertia.
In the particular case where the cyclic momenta are all zero, (16) reduces to dt d '36 36' (22) a. 4r a4 r = r. The form is the same as in § 2, and the system now behaves, as regards the coordinates qi, q2,... qm, exactly like the acyclic type there contemplated. These coordinates do not, however, now fix the position of every particle of the system. For example, if by suitable forces the system be brought back to its initial con figuration (so far as this is defined by q l, q 2 ,... q,,), after performing any evolutions, the ignored coordinates x, x', x",
will not in general return to their original values.
If in Lagrange's equations § 2 (io) we reverse the sign of the timeelement 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 41, q'2,.. qm 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 g i, q2,... qm, 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 41, q2,... qm. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.
The conditions of equilibrium of a system with latent cyclic motions, are obtained by putting a l = o, 4 2 = o,... q m = o in (16); viz. they are Q = 3K, Q2 = a ..... .. (23) These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration (qi, q2,... qm) to rest in the configuration (q1+5q,, q215g2, ..gm+64m), the work done by the forces must be equal to the increment of the kinetic energy. Hence Ql3ql+Q23q2 +
. =SK, which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic.
with potential energy K. This is important from a physical point By means of the formulae (18), which now reduce to 3K ., 3K „ OK
(25) x _ aK, x x
K may also be expressed as a homogeneous quadratic function of
the cyclic velocities X, X', X",.... Denoting it in this form by
To, we have S(To+K) =2SK=5(4+K'x'+KY +...). . (26)
Performing the variations, and omitting the terms which cancel by
(2) and (25), we find aT o aK aTo _ aK (27)
a q i =  3 q 1' aq2   3Q2' ..., so that the formulae
(23) become Qi =  aTo, Q 2   aTo ..... .. (28)  Tv", aq2 A
simple example is furnished by the top (Mechanics, § 22). The cyclic
coordinates being 1 ', 4), we find 26=A9 2, 2K = 
v cos 0)2 +v2 A sin 2 0 C' 2T 0 =A sin 2 01, 2 +C(.3+1G cos 0) 2,.
(29) whence we may verify that aTo/ae=  0K/ae in
accordance with (27). And the condition of equilibrium OK
aV (30) ae =  ae 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, § 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 Ober die Theorie des Kreisels (18971903). 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 (24) (5). (6) (7) . (8) (15) equation of motion aK +a qr = Qr (16) (17) 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 noneliminated coordinates q l, q2,
qm all vanish (see § 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+K be a
minimum as regards variations of qi, q2, ... qm. 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, V+K =Mgh coso+ (P  v o)2 + v2 (I) 2A sin 0 2C To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put m= V. We then find without difficulty that V+K is a minimum provided 4AMgh. The method of small oscillations gave us the condition v2>4AMgh, and indicated instability in the cases v c4AMgh. 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+K 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+K 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 coordinates ql, q2,... qm vary. Since the total energy of the system at any instant is given (in the notation of § 5) by an expression of the form 6+V+K, where 6 cannot be negative, the argument of Thomson and Tait, given under Mechanics, § 23, for the statical question, shows that it is a necessary as well as a sufficient condition for secular stability that V+K 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 § 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 l, Q 21 .. qn are zero it is necessary and sufficient that the function V  To 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 (w) about the vertical diameter. This position obviously possesses " ordinary " stability. If a be the radius of the bowl, and 0 denote angular distance from the lowest point, we have V  To = mga (I  cos o)  1mw2a2 sin 0; (2) this is a minimum for o =o only so long as w 2 <g/a. For greater values of w the only position of " permanent " stability is that in which the particle rotates with the bowl at an angular distance cos (g/w a) 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=  p2x  k(x+wy), y=  ply  k(y  wx), where p1 = g/a. These combine into z+kz +(p ikw)z =o, where z = x+iy, i =1/  i. Assuming z = Ce we find A=  2k(I=w/p) yip (5) if the square of k be neglected. The complete solution is then x + iy = C i e P i i e ipt +C 2 e  P 2 t tint,. . (6) where 131 2k(I w/p), 1 2 z k + w /p). (7) This represents two superposed circular vibrations, in opposite directions, of period 27r/p. If w
p the amplitude of that circular vibration which agrees in sense with the rotation w will continually increase, and the particle will work its way in an everwidening 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, § 13).
7. Principle of Least 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 Stationary between any two configurations through which it passes, action. 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 S to denote the transition from the actual to any one of the hypothetical motions.
The bestknown 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 spaceintegral of the momentum, or the timeintegral 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=E f muds=Efmv 2 dt=2 (Tdt.. .. (1) The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property SA=o,. (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 IA=S = 2 f 2T3t,. .. (3) since the upper and lower limits of the integral must both be regarded as variable. This may be written IA f t' STdt+ f t'Em(xSx+ySy+ zSi)dt+2T  2TSt t = f 1' oTdt+ [Em (xSx+yby+zSz)] V.
<p>  f Em(x x
+y5y+Iz)dt+2T'St'2TSt. 

Now, by d'Alembert's principle, Zm(Ix+ysy+Sz) =  and by hypothesis we have 

S(T+V) =o. 
. 
(6) 
The formula therefore reduces to 

SA [m(x+y+z)] +2T'St'2T6t. 
Since the terminal configurations are unaltered, we must have at the lower limit Sx+xSt= o, Sy+ySt=o, Sz+iIt=o,. . (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 Si 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 greatcircle arc ceases to be the shortest path between its extremities when it exceeds 180°. More generally, taking any surface, let a point P, starting from 0, move along a geodesic; this geodesic will be a minimum path from 0 to P until P passes through a point 0' (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 0 to P with the prescribed amount of energy, the action from 0 to P is a minimum. But if (3) (4) where terms have been cancelled in virtue of § 5 (2). The last member of (17) represents a variation of the integral there be several distinct paths, let P vary from coincidence with 0 along the firstnamed path; the action will then cease to be a minimum when a configuration 0' is reached such that two of the possible paths from 0 to 0' coincide. For instance, if 0 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 0), these two paths coinciding when P is at the other extremity (0', say) of the focal chord through O. The action from 0 to P will therefore be a minimum for all positions P short of 0'. Two configurations such as 0 and 0' 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 wellknown theorem of optics may be noticed. For the motion of a particle in a conservative field of force the principle takes the form S f vds=0. . (io) On the corpuscular theory of light v is proportional to the refractive index /.t. of the medium, whence a f µds =O... . (II) 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 Under these conditions we have Si t (T  V)dt =0,.. . (12) where t, t' are the prescribed times of passing through the given initial and final configurations. The proof of (12) is simple; we have 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 coordinates. Thus, to deduce Lagrange's equations, we have t?
i (ST  SV)dt= a 541+ag 6q +...  a4 Sg  ... (dt i+  + r aq2 +... 4 The integrated terms vanish at both limits; and in order that the remainder of the righthand member may vanish it is necessary that the coefficients of Sq l, Sq 2 ,... 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.
The modification of the Hamiltonian principle appropriate to the case of cyclic systems has been given by J. Larmor. If we write, as in § I (25), R=T  KX  KT  K"X"  ..... (15) we shall have Sf (R  V)dt =0, . . (16) provided that the variation does not affect the cyclic momenta K, K', K",..., and that the configurations at times t and t' are unaltered, so far as they depend on the palpable coordinates qi, g2,...gm. The initial and final values of the ignored coordinates will in general be affected.
To prove (16) we have, on the above understandings, bi t ?(R  V)dt = f (ST  KSx  ...aV)dt ' aT a t (  5V) dt, . (17) (T  V)dt on the supposition that SX = o, SX' = o, SX" = o,... throughout, whilst Sq l, 6Q 21 Sq 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= f(2T  KX  K'X'  K"X"  ...)dt, . (18) sA=O,.. (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 coordinates alone.
§ 8. Hamilton's Principal and Characteristic Functions. In the investigations next to be described a more extended meaning is given to the symbol S. We will, in the first instance, denote by it an infinitesimal variation of the most general kind, affecting not merely the values of the coordinates at any instant, but also the initial and final configurations and the times of passing through them. If we put S =i (T  V)dt,. . (I) we have, then, SS=(T'  V')St'  (T  V)St+ f t'(ST  SV)dt = (T'  V')at'  (T  V)at+ [m(x+iÖ +âz)] t . (2) Let us now denote by z'+Sz', the final coordinates (i.e. at time t'+St) of a particle m. In the terms in (2) which relate to the upper limit we must therefore write Sx'  c;'St', Sy' Sz'  5'St for Sx,' Sy, Sz. With a similar modification at the lower limit, we obtain SS =HST+Em (x'Sx'+p'Sy'+z'Sz')  Em(±Sx+ySy+zSz),
(3) where H(=T+V) is the constant value of the energy in the
free generalized coordinates this takes the form motion of the
system, and T(=t'  t) is the time of transit. In aS
=  HaT+p'iaq'i+p'23q'2+...  p,agl  p2ag2  ....
(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 T. On this view of the matter, S will be a
function of the initial and final coordinates (q i, Q21 ...
and q'i, q'2,
.) and the time T, as independent variables. And we obtain at once from (4) aS, i P2  ag,, as H=  ar and 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 p i, p 2 ,... and H from (5) and (6) in the expression for the kinetic energy in the form T' (see § I), the equation T '+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+I 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.
There is a similar theory for the function (8) It follows from (4) that SA =TSH+p'1Sq'i+p'2Sg'2+...
 p 1 Sq,  p2Sg2  ...
(9) This formula (it may be remarked) contains the principle of " least Of ? (T  V)dt= f (S T  SV)dt= f m(+ i'  l ol)  V1dt = E[m(±öx+iy,y+.&)] t a?  a (Em(xsx+,ijSy+zSz) +SV} dt.. (is) then A =2 f Tdt =S+Hr.. constant.
} (6) 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 coordinates and the energy, we find 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 (to) in (7).
The preceding theorems are easily adapted to the case of cyclic systems. We have only to write S=f: (R  V)dt= f (TKXK'x'... V)dt . (12) in place of (I), and (3) in place of (8); cf. § 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 coordinates q l, g 2 ,...g m , and of the time of transit T, the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of q i, q2,...q,,,,, 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 Sq l, Sq 2 ,... be understood to refer to the palpable coordinates alone. It follows that the equations (5), (6) and (10), (I I) will still hold under the new meanings of the symbols.
d Fi(SPr.z qr OPr.Sgr)=0; or, integrating from t to t', Z(Sp' r .Aq' r Oq' r Sq' r) =E(3Pr.L'gr  'Pr Sgr) If for shortness we write(r, s) a gr a gs, ( r, s')= agrag, s, 
9. Reciprocal Properties of Direct and Reversed Motions. 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 S and A to denote the corresponding variations, the theorem is we have Sp r =  E s (r, s)SgsE 5 (r, s')Sq' s. . (4) with a similar expression for Ap r. Hence the righthand side of (2) becomes s)Sgs+Zs(r, s')bq'slAgr+Zir{Zs(r, s)Ogs+Zs(r, s')Oq's}Sqr =Er2s(r, s'){0g r .oq' s  Og r .aq' 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 (I), 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. 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 con servative system between two configurations 0 and 0' holtz's through which it passes at times t and t' respectively, reciprocal . and let t  t= T. As the system is passing through 0 let a small impulse Sp r be given to it, and let the consequent alteration in the coordinate q s after the time T be Sq' s . Next consider the reversed motion of the system, in which it would, if undisturbed, pass from O' to 0 in the same time T. Let a small impulse op' s be applied as the system is passing through 0', and let the consequent change in the coordinate q r after a time T be Sqr. Helmholtz's first theorem is to the effect that Sqr: Sp' s =Sq' 8 : Spr.
(6) To prove this, suppose, in (2), that all the Sq vanish, and likewise all the Op with the exception of op,. Further, suppose all the Oq' to vanish, and likewise all the op' except Op' s, the formula then gives Spr.oqr =  op' s .Sg' s, which is equivalent to Helmholtz's result, since we may suppose the symbol A to refer to the reversed motion, provided we change the signs of the op. In the most general motion of a top (Mechanics, § 22), suppose that a small impulsive couple about the vertical produces after a time T a change 00 in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of 0 will produce after a time r a change SIP, in the azimuth of the axis, which is equal to So. 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 0, 0' be any two points on the axis of a symmetrical optical combination, and let V, V' be the corresponding velocities of light. At 0 let a small impulse be applied perpendicular to the axis so as to produce an angular deflection 00, and let /3 be the corresponding lateral deviation at 0'. In like manner in the reversed motion, let a small deflection 00 at 0' produce a lateral deviation 13 at O. The theorem (6) asserts that or, in optical language, the " apparent distance " of 0 from 0' is to that of 0' from 0 in the ratio of the refractive indices at 0' and 0 respectively.
In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change Sq, in one of the coordinates, the momenta being all unaltered, and Sq is the consequent variation in one of the momenta after time T. Similarly in the reversed motion a change Op' s produces after time T a change of momentum Sp . The theorem asserts that Sp' s :Sg r =Sp r :Sq's. (9) This follows at once from (2) if we imagine all the Sp to vanish, and likewise all the Sq save Sq r , and if (further) we imagine all the Op' to vanish, and all the zq' save Oq'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 coordinates q r are the palpable coordinates 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 nonreversibility of the motion.
The most important and most accessible early authorities are J. L. Lagrange, Mecanique analytique (1st ed. Paris, 1788, 2nd ed. Paris, 1811; reprinted in Ouvres, vols. xi., xii., Paris, 188889); Hamilton, "On a General Method in Dynamics,"Phil. Trans. 1834 and 1835; C. G. J. Jacobi, Vorlesungen uber 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 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, " ilber 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 " Ada math. vol. vii. (1885); F. Klein and A. Sommerfeld, Theorie des Kreisels, pts. I, 2 (Leipzig, 18971898); 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.) P, aA, _ aA  q l, P2  '.,' ... aA OA p1 = , P2 =  ... a a A T and A= f (2T KXK'X'...)dt, (7) (8)
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