Chapter I: Part 1

</pre>

Produced by Marius Masi, Don Kretz and the Online Distributed Proofreading Team at http://www.pgdp.net

Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) dP stands for the partial-derivative symbol, or curled 'd'.

(6) [oo] stands for the infinity symbol, and [int] for the integral
symbol.

(7) Letters followed with a grave accent "`" have originally dots above.

(8) The following typographical errors have been corrected:

ARTICLE DYER, JOHN: "His poems were collected by Dodsley in 1770,
and by Mr Edward Thomas in 1903 for the Welsh Library, vol. iv."
'poems' amended from 'peoms'.

ARTICLE EAR: "The membranous semicircular canals are very much
smaller in section than the bony; in the ampulla of each is a
ridge..." 'the' amended from 'tbe'.

ARTICLE EARTH, FIGURE OF THE: "O. Callandreau, 'Memoire sur la
theorie de la figure des planetes,' Ann. obs. de Paris (1889);..."
'Callandreau' amended from 'Callendreau'.

ARTICLE EATON, THEOPHILUS: "In October 1639 a form of government
was adopted, based on the Mosaic Law, and Eaton was elected
governor..." 'Mosaic' amended from 'Mosiac'.

ARTICLE ECCLESIASTES: "A particular instance is mentioned (ix.
13-15) of a beleaguered city saved by a wise man; but the man
happened to be poor, and no one remembered him." 'beleaguered'
amended from 'beleagured'.

ARTICLE ECCLESIASTES: "Such assertions as those of ii. 26 (God
gives joy to him who pleases him, and makes the sinner toil to lay
up for the latter),..." 'and' amended from 'amd'.

ARTICLE ECCLESIASTES: "This disagreement comes largely from the
attempts made to find definitely expressed Greek philosophical
dogmas in the book; such formulas it has not, but the general air
of Greek reflection seems unmistakable. The scepticism of Koheleth
differs from that of Job in quality and scope..." 'the' originally
repeated twice.

ARTICLE ECCLESIASTICAL JURISDICTION: "In the first case, they may
be punished by the ordinary of the place, acting as delegate of the
pope without special appointment (Conc. Trid. Sess. vi. c. 3)."
'special' amended from 'speical'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME VIII, SLICE IX

Dyer to Echidna

ARTICLES IN THIS SLICE:

DYER, SIR EDWARD EAST LIVERPOOL
DYER, JOHN EAST LONDON
DYER, THOMAS HENRY EASTON
DYMOKE EAST ORANGE
DYNAMICS EASTPORT
DYNAMITE EAST PROVIDENCE
DYNAMO EAST PRUSSIA
DYNAMOMETER EASTWICK, EDWARD BACKHOUSE
DYNASTY EATON, DORMAN BRIDGMAN
DYSART EATON, MARGARET O'NEILL
DYSENTERY EATON, THEOPHILUS
DYSPEPSIA EATON, WILLIAM
DYSTELEOLOGY EATON, WYATT
DZUNGARIA EAU CLAIRE
E EAU DE COLOGNE
EA EAUX-BONNES
EABANI EAVES
EACHARD, JOHN EAVESDRIP
EADBALD EBBW VALE
EADIE, JOHN EBEL, HERMANN WILHELM
EADMER EBEL, JOHANN GOTTFRIED
EADS, JAMES BUCHANAN EBER, PAUL
EAGLE EBERBACH (town of Germany)
EAGLEHAWK EBERBACH (monastery of Germany)
EAGRE EBERHARD
EAKINS, THOMAS EBERHARD, CHRISTIAN AUGUST GOTTLOB
EALING EBERHARD, JOHANN AUGUSTUS
EAR EBERLIN, JOHANN ERNST
EARL EBERS, GEORG MORITZ
EARLE, JOHN EBERSWALDE
EARLE, RALPH EBERT, FRIEDRICH ADOLF
EARL MARSHAL EBINGEN
EARLOM, RICHARD EBIONITES
EARLSTON EBNER-ESCHENBACH, MARIE
EARLY, JUBAL ANDERSON EBOLI
EARLY ENGLISH PERIOD EBONY
EARN EBRARD, JOHANNES HEINRICH AUGUST
EARNEST EBRO
EAR-RING EBROIN
EARTH EBURACUM
EARTH, FIGURE OF THE ECA DE QUEIROZ, JOSE MARIA
EARTH CURRENTS ECARTE
EARTH-NUT ECBATANA
EARTH PILLAR ECCARD, JOHANN
EARTHQUAKE ECCELINO DA ROMANO
EARTH-STAR ECCENTRIC
EARTHWORM ECCHELLENSIS, ABRAHAM
EARWIG ECCLES
EASEMENT ECCLESFIELD
EAST, ALFRED ECCLESHALL
EAST ANGLIA ECCLESIA
EASTBOURNE ECCLESIASTES
EAST CHICAGO ECCLESIASTICAL COMMISSIONERS
EASTER ECCLESIASTICAL JURISDICTION
EASTER ISLAND ECCLESIASTICAL LAW
EASTERN BENGAL AND ASSAM ECCLESIASTICUS
EASTERN QUESTION, THE ECGBERT (king of the West Saxons)
EAST GRINSTEAD ECGBERT (archbishop of York)
EAST HAM ECGFRITH
EASTHAMPTON ECGONINE
EAST HAMPTON ECHEGARAY Y EIZAGUIRRE, JOSE
EAST INDIA COMPANY ECHELON
EAST INDIES ECHIDNA
EASTLAKE, SIR CHARLES LOCK

DYER, SIR EDWARD (d. 1607), English courtier and poet, son of Sir Thomas Dyer, Kt., was born at Sharpham Park, Somersetshire. He was educated, according to Anthony a Wood, either at Balliol College or at Broadgates Hall, Oxford. He left the university without taking a degree, and after some time spent abroad appeared at Queen Elizabeth's court. His first patron was the earl of Leicester, who seems to have thought of putting him forward as a rival to Sir Christopher Hatton in the queen's favour. He is mentioned by Gabriel Harvey with Sidney as one of the ornaments of the court. Sidney in his will desired that his books should be divided between Fulke Greville (Lord Brooke) and Dyer. He was employed by Elizabeth on a mission (1584) to the Low Countries, and in 1589 was sent to Denmark. In a commission to inquire into manors unjustly alienated from the crown in the west country he did not altogether please the queen, but he received a grant of some forfeited lands in Somerset in 1588. He was knighted and made chancellor of the order of the Garter in 1596. William Oldys says of him that he "would not stoop to fawn," and some of his verses seem to show that the exigencies of life at court oppressed him. He was buried at St Saviour's, Southwark, on the 11th of May 1607. Wood says that many esteemed him to be a Rosicrucian, and that he was a firm believer in alchemy. He had a great reputation as a poet among his contemporaries, but very little of his work has survived. Puttenham in the _Arte of English Poesie_ speaks of "Maister Edward Dyar, for Elegie most sweete, solempne, and of high conceit." One of the poems universally accepted as his is "My Mynde to me a kingdome is." Among the poems in _England's Helicon_ (1600), signed S.E.D., and included in Dr A.B. Grosart's collection of Dyer's works (_Miscellanies of the Fuller Worthies Library_, vol. iv., 1876) is the charming pastoral "My Phillis hath the morninge sunne," but this comes from the _Phillis_ of Thomas Lodge. Grosart also prints a prose tract entitled _The Prayse of Nothing_ (1585). The _Sixe Idillia_ from Theocritus, reckoned by J.P. Collier among Dyer's works, were dedicated to, not written by, him.

DYER, JOHN (c. 1700-1758), British poet, the son of a solicitor, was born in 1699 or 1700 at Aberglasney, in Carmarthenshire. He was sent to Westminster school and was destined for the law, but on his father's death he began to study painting. He wandered about South Wales, sketching and occasionally painting portraits. In 1726 his first poem, _Grongar Hill_, appeared in a miscellany published by Richard Savage, the poet. It was an irregular ode in the so-called Pindaric style, but Dyer entirely rewrote it into a loose measure of four cadences, and printed it separately in 1727. It had an immediate and brilliant success. _Grongar Hill_, as it now stands, is a short poem of only 150 lines, describing in language of much freshness and picturesque charm the view from a hill overlooking the poet's native vale of Towy. A visit to Italy bore fruit in _The Ruins of Rome_ (1740), a descriptive piece in about 600 lines of Miltonic blank verse. He was ordained priest in 1741, and held successively the livings of Calthorp in Leicestershire, Belchford (1751), Coningsby (1752), and Kirby-on-Bane (1756), the last three being Lincolnshire parishes. He married, in 1741, a Miss Ensor, said to be descended from the brother of Shakespeare. In 1757 he published his longest work, the didactic blank-verse epic of _The Fleece_, in four books, discoursing of the tending of sheep, of the shearing and preparation of the wool, of weaving, and of trade in woollen manufactures. The town took no interest in it, and Dodsley facetiously prophesied that "Mr Dyer would be buried in woollen." He died at Coningsby of consumption, on the 15th of December 1758.

His poems were collected by Dodsley in 1770, and by Mr Edward Thomas
in 1903 for the _Welsh Library_, vol. iv.

DYER, THOMAS HENRY (1804-1888), English historical and antiquarian writer, was born in London on the 4th of May 1804. He was originally intended for a business career, and for some time acted as clerk in a West India house; but finding his services no longer required after the passing of the Negro Emancipation Act, he decided to devote himself to literature. In 1850 he published the _Life of Calvin_, a conscientious and on the whole impartial work, though the character of Calvin is somewhat harshly drawn, and his influence in the religious world generally is insufficiently appreciated. Dyer's first historical work was the _History of Modern Europe_ (1861-1864; 3rd ed. revised and continued to the end of the 19th century, by A. Hassall, 1901), a meritorious compilation and storehouse of facts, but not very readable. The _History of the City of Rome_ (1865) down to the end of the middle ages was followed by the _History of the Kings of Rome_ (1868), which, upholding against the German school the general credibility of the account of early Roman history, given in Livy and other classical authors, was violently attacked by J.R. Seeley and the _Saturday Review_, as showing ignorance of the comparative method. More favourable opinions of the work were expressed by others, but it is generally agreed that the author's scholarship is defective and that his views are far too conservative. _Roma Regalis_ (1872) and _A Plea for Livy_ (1873) were written in reply to his critics. Dyer frequently visited Greece and Italy, and his topographical works are probably his best; amongst these mention may be made of _Pompeii, its History, Buildings and Antiquities_ (1867, new ed. in Bohn's _Illustrated Library_), and _Ancient Athens, its History, Topography and Remains_ (1873). His last publication was _On Imitative Art_ (1882). He died at Bath on the 30th of January 1888.

DYMOKE, the name of an English family holding the office of king's champion. The functions of the champion were to ride into Westminster Hall at the coronation banquet, and challenge all comers to impugn the king's title (see CHAMPION). The earliest record of the ceremony at the coronation of an English king dates from the accession of Richard II. On this occasion the champion was Sir John Dymoke (d. 1381), who held the manor of Scrivelsby, Lincolnshire, in right of his wife Margaret, granddaughter of Joan Ludlow, who was the daughter and co-heiress of Philip Marmion, last Baron Marmion. The Marmions claimed descent from the lords of Fontenay, hereditary champions of the dukes of Normandy, and held the castle of Tamworth, Leicestershire, and the manor of Scrivelsby, Lincolnshire. The right to the championship was disputed with the Dymoke family by Sir Baldwin de Freville, lord of Tamworth, who was descended from an elder daughter of Philip Marmion. The court of claims eventually decided in favour of the owners of Scrivelsby on the ground that Scrivelsby was held in grand serjeanty, that is, that its tenure was dependent on rendering a special service, in this case the championship.

Sir Thomas Dymoke (1428?-1471) joined a Lancastrian rising in 1469, and, with his brother-in-law Richard, Lord Willoughby and Welles, was beheaded in 1471 by order of Edward IV. after he had been induced to leave sanctuary on a promise of personal safety. The estates were restored to his son Sir Robert Dymoke (d. 1546), champion at the coronations of Richard III., Henry VII. and Henry VIII., who distinguished himself at the siege of Tournai and became treasurer of the kingdom. His descendants acted as champions at successive coronations. Lewis Dymoke (d. 1820) put in an unsuccessful claim before the House of Lords for the barony of Marmion. His nephew Henry (1801-1865) was champion at the coronation of George IV. He was accompanied on that occasion by the duke of Wellington and Lord Howard of Effingham. Henry Dymoke was created a baronet; he was succeeded by his brother John, rector of Scrivelsby (1804-1873), whose son Henry Lionel died without issue in 1875, when the baronetcy became extinct, the estate passing to a collateral branch of the family. After the coronation of George IV. the ceremony was allowed to lapse, but at the coronation of King Edward VII. H.S. Dymoke bore the standard of England in Westminster Abbey.

DYNAMICS (from Gr. [Greek: dynamis], strength), the name of a branch of the science of Mechanics (q.v.). 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 DYNAMICS.--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 q1, q2, ... q_n,
each of which admits of continuous variation over a certain range, so
that if x, y, z be the Cartesian co-ordinates of any one particle, we
have for example

x = [f](q1, q2, ... q_n), y = &c., z = &c., (1)

where the functions [f] differ (of course) from particle to particle.
In modern language, the variables q1, q2, ... q_n are _generalized
co-ordinates_ serving to specify the _configuration_ of the system;
their derivatives with respect to the time are denoted by q`1, q`2,
... q`_n, 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_.

Impulsive motion.

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 co-ordinates of any particle m,

mx` = X', my` = Y', mz` = Z', (2)

where X', Y', Z' are the components of the impulse on m. Now let
[delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y,
z which are consistent with the connexions of the system, and let us
form the equation

[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z), (3)

where the sign [Sigma] indicates (as throughout this article) a
summation extending over all the particles of the system. To transform
(3) into an equation involving the variations [delta]q1, [delta]q2,
... of the generalized co-ordinates, we have

dPx dPx
x` = ---- q`1 + ---- q`2 + ..., &c., &c. (4)
dPq1 dPq2

dPx dPx
[delta]x = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c. (5)
dPq1 dPq2

and therefore

[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
(A11q`1 + A12q`2 + ...)[delta]q1 +
(A21q`1 + A22q`2 + ...)[delta]q2 + ..., (6)

where
_ _
| / dPx \ squared / dPy \ squared / dPz \ squared | \
A_rr = [Sigma]m | ( ----- ) + ( ----- ) + ( ----- ) |, |
|_ \dPq_r/ \dPq_r/ \dPq_r/ _| |
_ _ > (7)
| dPx dPx dPy dPy dPz dPz | |
A_rs = [Sigma]m | ----- ----- + ----- ----- + ----- ----- | = A_sr. |
|_dPq_r dPq_s dPq_r dPq_s dPq_r dPq_s _| /

If we form the expression for the kinetic energy [Tau] of the system,
we find

2[Tau] = [Sigma]m(x` squared + y` squared + z` squared) = A11q`1 squared + A22q`2 squared + ...
+ 2A12q`1q`2 + ... (8)

The coefficients A11, A22, ... A12, ... are by an obvious analogy
called the _coefficients of inertia_ of the system; they are in
general functions of the co-ordinates q1, q2, ... . The equation (6)
may now be written

dP[Tau] dP[Tau]
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ------- [delta]q1 + ------- [delta]q2 + ... (9)
dPq`1 dPq`2

This maybe regarded as the cardinal formula in Lagrange's method. For
the right-hand side of (3) we may write

[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z) = Q'1[delta]q1 + Q'2[delta]q2 + ... , (10)

where

/ dPx dPy dPz \
Q'_r = [Sigma]( X'----- + Y'----- + Z'----- ). (11)
\ dPq_r dPq_r dPq_r/

The quantities Q1, Q2, ... are called the _generalized components of
impulse_. Comparing (9) and (10), we have, since the variations
[delta]q1, [delta]q2,... are independent,

dP[Tau] dP[Tau]
------- = Q'1, ------- = Q'2, ... (12)
dPq`1 dPq`2

These are the general equations of impulsive motion. It is now usual
to write

dP[Tau]
p_r = ------- (13)
dPq`_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

[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = p1[delta]q1 + p2[delta]q2 + ... (14)

Also, since [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2 ...,

2[Tau] = p1q`1 + p2q`2 + ... (15)

This follows independently from (14), assuming the special variations
[delta]x = x`dt, &c., and therefore [delta]q1 = q`1dt, [delta]q2 =
q`2dt, ...

Reciprocal theorems.

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

p1q`'1 + p2q`'2 + ... = p'1q`1 + p'2q`2 + ..., (16)

each side being equal to the symmetrical expression

A11q`1q''1 + A22q`2q`'2 + ... + A12(q`1q`'2 + q`'1q`2) + ... (17)

The theorem (16) leads to some important reciprocal relations. Thus,
let us suppose that the momenta p1, p2, ... all vanish with the
exception of p1, and similarly that the momenta p'1, p'2, ... all
vanish except p'2. We have then p1q`'1 = p'2q`2, or

q`2 : p1 = q`'1 : p'2 (18)

The interpretation is simplest when the co-ordinates q1, 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 [omega] in a link A, a couple Fa applied to A will
produce a linear velocity [omega]a 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.

Velocities in terms of momenta.

The equations (13) determine the momenta p1, p2,... as linear
functions of the velocities q`1, q`2,... Solving these, we can
express q`1, q`2 ... as linear functions of p1, p2,... The
resulting equations give us the velocities produced by any given
system of impulses. Further, by substitution in (8), we can express
the kinetic energy as a homogeneous quadratic function of the momenta
p1, p2,... The kinetic energy, _as so expressed_, will be denoted by
[Tau]'; thus

2[Tau]' = A'11p1 squared + A'22p2 squared + ... + 2A'12p - p2 + ... (19)

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
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

p1q`1 + p2q`2 + ... = [Tau] + [Tau]', ... (20)

where [Tau] is supposed expressed as in (8), and [Tau]' as in (19).
Hence if, for the moment, we denote by [delta] a variation affecting
the velocities, and therefore the momenta, but not the configuration,
we have

p1[delta]q`1 + q`1[delta]p + p2[delta]q`2 + q`2[delta]p2 + ... = [delta][Tau] + [delta][Tau]'

dP[Tau] dP[Tau] dP[Tau]' dP[Tau]'
= ------- [delta]q`1 + ------- [delta]q`2 + ... + -------- [delta]p1 + -------- [delta]p2 + ... (21)
dPq`1 dPq`2 dPp1 dPp2

In virtue of (13) this reduces to

dP[Tau]' dP[Tau]'
q`1[delta]p1 + q`2[delta]p2 + ... = ------- [delta]p1 + ------- [delta]p2 + ... (22)
dPp1 dPp2

Since [delta]p1, [delta]p2, ... may be taken to be independent, we
infer that

dP[Tau]' dP[Tau]'
q`1 = -------, q`2 = -------, ... (23)
dPp1 dPp2

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.

Routh's modification.

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 co-ordinates, say q1, q2, ... q_m, and of the momenta
corresponding to the remaining co-ordinates, which (for the sake of
distinction) we may denote by [chi], [chi]', [chi]", .... Thus, [Tau]
being expressed as a homogeneous quadratic function of q`1, q`2, ...
q`_m, [chi]`, [chi]`', [chi]`", ..., the momenta corresponding to the
co-ordinates [chi], [chi]', [chi]", ... may be written

dP[Tau] dP[Tau] dP[Tau]
[kappa] = --------, [kappa]' = ---------, [kappa]" = ------------, ... (24)
dP[chi]` dP[chi]`' dP[.[chi]`"

These equations, when written out in full, determine [chi]`, [chi]`',
[chi]`", ... as linear functions of q`1, q`2, ... q`_m, [kappa],
[kappa]', [kappa]",... We now consider the function

R = [Tau] - [kappa][chi]' - [kappa]'[chi]]`' - [kappa]"[chi]]`" - ..., (25)

supposed expressed, by means of the above relations in terms of q`1,
q`2, ... q`_m, [kappa], [kappa]', [kappa]",... Performing the
operation [delta] on both sides of (25), we have

dPR dPR dP[Tau] dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... + -------- [delta][chi]` + ...
dPq`1 dP[kappa] dPq`1 dP[chi]`

- [kappa]dP[chi]` - [chi]`[delta][kappa] - ... , (26)

where, for brevity, only one term of each type has been exhibited.
Omitting the terms which cancel in virtue of (24), we have

dPR dPR dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... - [chi]`[delta][kappa] - ... (27)
dPq`1 dP[kappa] dPq`1

Since the variations [delta]q1, [delta]q2, ... [delta]q_m,
[delta][kappa], [delta][kappa]', [delta][kappa]", ... may be taken to
be independent, we have

dP[Tau] dPR dP[Tau] dPR
p1 = ------- = -----, p2 = ------- = -----, ... (28)
dPq`1 dPq`1 dPq`2 dPq`2

and

dPR dPR dPR
[chi]` = - ---------, [chi]`' = - ----------, [chi]]`" = - ---------, ... (29)
dP[kappa] dP[kappa]' dP[kappa]"

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

[Tau] = [@] + K, (30)

where [@] involves the velocities q`1, q`2, ... q`_m alone, and K the
momenta [kappa], [kappa]', [kappa]", ... alone. For in virtue of (29)
we have, from (25),

/ dPR dPR dPR \
[Tau] = R - ( [kappa] --------- + [kappa]' ---------- + [kappa]" ----------- + ... ), (31)
\ dP[kappa] dP[kappa]' dP[kappa]" /

and it is evident that the terms in R which are bilinear in respect of
the two sets of variables q`1, q`2, ... q`_m and [kappa], [kappa]',
[kappa]", ... will disappear from the right-hand side.

Maximum and minimum energy.

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 system is started by
given impulses of certain types, but is otherwise free. J.L.F.
Bertrand's theorem is to the effect that the kinetic energy is
_greater_ than if by impulses of the remaining types the system were
constrained to take any other course. We may suppose the co-ordinates
to be so chosen that the constraint is expressed by the vanishing of
the velocities q`1, q`2, ... q`_m, whilst the given impulses are
[kappa], [kappa]', [kappa]",... Hence the energy in the actual motion
is greater than in the constrained motion by the amount [@].

Again, suppose that the system is started with prescribed velocity
components q`1, q`2, ... q`_m, by means of proper impulses of
the corresponding types, but is otherwise free, so that in the motion
actually generated we have [kappa] = 0, [kappa]' = 0, [kappa]" = 0,
... and therefore K = 0. The kinetic energy is therefore _less_ than
in any other motion consistent with the prescribed velocity-conditions
by the value which K assumes when [kappa], [kappa]', [kappa]", ...
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.

2. _Continuous Motion of a System._

Lagrange's equations.

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 (1)

Now let x + [delta]x, y + [delta]y, z + [delta]z be the co-ordinates
of m in any arbitrary motion of the system differing infinitely little
from the actual motion, and let us form the equation

[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) =
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) (2)

Lagrange's investigation consists in the transformation of (2) into an
equation involving the independent variations [delta]q1, [delta]q2,
... [delta]q_n.

It is important to notice that the symbols [delta] and d/dt are
commutative, since

d dx d
[delta]x` = --(x + [delta]x) - -- = --[delta]x, &c. (3)
dt dt dt

Hence

d
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = -- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)
dt
- [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)

d
= --(p1[delta]q1 + p2[delta]q2 + ...) - [delta][Tau], (4)
dt

by Sec. 1 (14). The last member may be written

p`1[delta]q1 + p1[delta]q`1 + p`2[delta]q2 + p2[delta]q`2 + ...

dP[Tau] dP[Tau] dP[Tau] dP[Tau]
- ------- [delta]q`1 - ------- [delta]q1 - ------- [delta]q`2 - ------- [delta]q2 - ... (5)
dPq`1 dPq1 dPq`2 dPq2

Hence, omitting the terms which cancel in virtue of Sec. 1 (13), we find

/ dP[Tau]\ / dP[Tau]\
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = (p`1 - ------- ) [delta]q1 + (p`2 - ------- ) [delta]q2 + ... (6)
\ dPq1 / \ dPq2 /

For the right-hand side of (2) we have

[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ..., (7)

/ dPx dPy dPz \
where Q_r = [Sigma]( X ----- + Y ----- + Z ----- ) (8)
\ dPq_r dPq_r dPq_r/

The quantities Q1, Q2, ... are called the _generalized components of
force_ acting on the system.

Comparing (6) and (7) we find

dP[Tau] dP[Tau]
p`1 - ------- = Q1, p`2 - ------- = Q2, ..., (9)
dPq`1 dPq`2

or, restoring the values of p1, p2, ...,

d /dP[Tau]\ dP[Tau] d /dP[Tau]\ dP[Tau]
-- ( ------- ) - ------- = Q1, -- ( ------- ) - ------- = Q2, ... (10)
dt \ dPq`1 / dPq1 dt \ dPq`2 / dPq2

These are Lagrange's general equations of motion. Their number is of
course equal to that of the co-ordinates 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
(10), by direct transformation of co-ordinates, has been given by
Hamilton and independently by other writers (see MECHANICS), 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 (10) 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 (q1, q2, ... q_n) is
independent of the path, and may therefore be regarded as a definite
function of q1, q2, ... q_n. Denoting this function (the _potential
energy_) by V, we have, if there be no extraneous force on the system,

[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = - [delta]V, (11)

and therefore

dPV dPV
Q1 = - ----, Q2 = - ----, .... (12)
dPq1 dPq2

Hence the typical Lagrange's equation may be now written in the form

d /dP[Tau]\ dP[Tau] dPV
-- ( ------- ) - ------- = - -----, (13)
dt \dPq`_r / dPq_r dPq_r

or, again,

dP
p`_r = - ----- (V - [Tau]) (14)
dPq_r

It has been proposed by Helmholtz to give the name _kinetic potential_
to the combination V - [Tau].

As shown under MECHANICS, Sec. 22, we derive from (10)

d[Tau]
------ = Q1q`1 + Q2q`2 + ..., (15)
dt

and therefore in the case of a conservative system free from
extraneous force,

d
--([Tau] + V) = 0 or [Tau] + V = const., (16)
dt

which is the equation of energy. For examples of the application of
the formula (13) see MECHANICS, Sec. 22.

3. _Constrained Systems._

Case of varying relations.

It has so far been assumed that the geometrical relations, if any,
which exist between the various parts of the system are of the type Sec.
1 (1), 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, q1, q2,...q_n), y = &c., z = &c., (1)

was made by J.M.L. Vieille. We now have

dPx dPx dPx
x` = --- + ---- q`1 + ---- q`2 + ..., &c., &c., (2)
dPt dPq1 dPq2

dPx dPx
dPx = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c., (3)
dPq1 dPq2

so that the expression Sec. 1 (8) for the kinetic energy is to be
replaced by

2[Tau] = [alpha]0 + 2[alpha]1q`1 + 2[alpha]2q`2 + ... + A11q`1 squared + A22q`2 squared + ... + A12q`1q`2 + ..., (4)

where
_ _ \
| /dPx\ squared /dPy\ squared /dPz\ squared | |
a0 = [Sigma]m |( --- ) + ( --- ) + ( --- ) |, |
|_\dPt/ \dPt/ \dPt/ _| |
_ _ > (5)
| dPx dPx dPy dPy dPz dPz | |
a_r = [Sigma]m | --- ----- + --- ----- + --- ----- |, |
|_dPt dPq_r dPt dPq_r dPt dPq_r_| |
/

and the forms of A_rr, A_rs are as given by Sec. 1 (7). It is to be
remembered that the coefficients [alpha]0, [alpha]1, [alpha]2, ...
A11, A22, ... A12 ... will in general involve t explicitly as well as
implicitly through the co-ordinates q1, q2,... Again, we find

[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =

([alpha]1 + A11q`1 + A12q`2 + ...)[delta]q1
+ ([alpha]2 + A21q`1 + A22q`2 + ...)dPq2 + ...

dP[Tau] dP[Tau]
= ------- [delta]q1 + ------- [delta]q2 + ...
dPq`1 dPq`2

= p1[delta]q1 + p2[delta]q2 + ..., (6)

where p_r is defined as in Sec. 1 (13). The derivation of Lagrange's
equations then follows exactly as before. It is to be noted that the
equation Sec. 2 (15) does not as a rule now hold. The proof involved the
assumption that [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2....

It has been pointed out by R.B. Hayward that Vieille's case can be
brought under Lagrange's by introducing a new co-ordinate ([chi]) in
place of t, so far as it appears explicitly in the relations (1). We
have then

2[Tau] = [alpha]0[chi]` squared + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ A11q`1 squared + A22q`2 squared + ... + 2A12q`1q`2 + .... (7)

The equations of motion will be as in Sec. 2 (10), with the additional
equation

d dP[Tau] dP[Tau]
-- -------- - ------- = X, (8)
dt dP[chi]` dP[chi]

where X is the force corresponding to the co-ordinate [chi]. We may
suppose X to be adjusted so as to make [chi]" = 0, and in the
remaining equations nothing is altered if we write t for [chi] before,
instead of after, the differentiations. The reason why the equation Sec.
2 (15) no longer holds is that we should require to add a term X[chi]`
on the right-hand side; this represents the rate at which work is
being done by the constraining forces required to keep [chi]`
constant.

As an example, let x, y, z be the co-ordinates of a particle relative
to axes fixed in a solid which is free to rotate about the axis of z.
If [phi] be the angular co-ordinate of the solid, we find without
difficulty

2[Tau] = m(x` squared + y` squared +z` squared) + 2[phi]`m(xy` - yx`) + {I + m(x squared + y squared)}[phi]` squared, (9)

where I is the moment of inertia of the solid. The equations of
motion, viz.

d dP[Tau] dP[Tau] d dP[Tau] dP[Tau] d dP[Tau] dP[Tau]
-- ------ - ------ = X, -- ------- - ------- = Y, -- ------- - ------- = Z, (10)
dt dPx` dPx dt dPy` dPy dt dPz` dPz

d dP[Tau] dP[Tau]
and -- -------- - ------- = [Phi], (11)
dt dP[phi]` dP[phi]

become

m(x" - 2[phi]`y` - x[phi]` squared - y[phi]") = X, m(y" + 2[phi]`x` - y[phi]` squared + x[phi]`) = Y, mz" = Z, (12)
_ _
d | / \ |
and -- |(I + m(x squared + y squared)) [phi]` + m(xy` - yx`)| = [Phi]. (13)
dt |_\ / _|

If we suppose [Phi] adjusted so as to maintain [phi]" = 0, or (again)
if we suppose the moment of inertia I to be infinitely great, we
obtain the familiar equations of motion relative to moving axes, viz.

m(x" - 2[omega]y` - [omega] squaredx) = X, m(y" + 2[omega]x` - [omega] squaredy) = Y, mz" = Z, (14)

where [omega] has been written for [phi]. These are the equations
which we should have obtained by applying Lagrange's rule at once to
the formula

2[Tau] = m(x` squared + y` squared + z` squared) + 2m[omega](xy` - yx`) + m[omega] squared(x squared + y squared), (15)

which gives the kinetic energy of the particle referred to axes
rotating with the constant angular velocity [omega]. (See MECHANICS, Sec.
13.)

More generally, let us suppose that we have a certain group of
co-ordinates [chi], [chi]', [chi]", ... whose absolute values do not
affect the expression for the kinetic energy, and that by suitable
forces of the corresponding types the velocity-components [chi]`,
[chi]`', [chi]`", ... are maintained constant. The remaining
co-ordinates being denoted by q1, q2, ... q_n, we may write

2[Tau] = [@] + [Tau]0 + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ 2([alpha]'1q`1 + [alpha]'2q`2 + ...)[chi]`' + ..., (16)

where [@] is a homogeneous quadratic function of the velocities q`1,
q`2, ... q`_n of the type Sec.1 (8), whilst [Tau]0 is a homogeneous
quadratic function of the velocities [chi]`,[chi]`', [chi]`", ...
alone. The remaining terms, which are bilinear in respect of the two
sets of velocities, are indicated more fully. The formulae (10) of Sec. 2
give n equations of the type

d /dP[@]\ /dP[@]\ dP[Tau]0
--( ----- ) - ( ----- ) + (r, 1)q`1 + (r, 2)q`2 + ... - -------- = Q_r (17)
dt \dPq_r/ \dPq_r/ dPq_r

where

/dPa_r dPa_s\ /dPa'_r dPa'_s\
(r, s) = ( ----- - ----- )[chi]` + ( ------ - ------ )[chi]`' + .... (18)
\dPq_s dPq_r/ \dPq_s dPq_r/

These quantities (r, s) are subject to the relations

(r, s) = -(s, r), (r, r) = 0 (19)

The remaining dynamical equations, equal in number to the co-ordinates
[chi], [chi]', [chi]", ..., yield expressions for the forces which
must be applied in order to maintain the velocities [chi]`, [chi]`',
[chi]`", ... constant; they need not be written down. If we follow the
method by which the equation of energy was established in Sec. 2, the
equations (17) lead, on taking account of the relations (19), to

d
--([@] - [Tau]0) = Q1q`1 + Q2q`2 + ... + Q_nq`_n, (20)
dt

or, in case the forces Q_r depend only on the co-ordinates q1, q2, ...
q_n and are conservative,

[@] + V - [Tau]0 = const. (21)

The conditions that the equations (17) should be satisfied by zero
values of the velocities q`1, q`2, ... q`_n are

dP[Tau]0
Q_r = - --------, (22)
dPq_r

or in the case of conservative forces

dP
------ (V - [Tau]0) = 0, (23)
dPq_r

i.e. the value of V - [Tau]0 must be _stationary_.

Rotating axes.

We may apply this to the case of a system whose configuration relative
to axes rotating with constant angular velocity ([omega]) is defined
by means of the n co-ordinates q1, q2, ... q_n. This is important on
account of its bearing on the kinetic theory of the tides. Since the
Cartesian co-ordinates x, y, z of any particle m of the system
relative to the moving axes are functions of q1, q2, ... q_n, of the
form Sec. 1 (1), we have, by (15)

2[@] = [Sigma]m(x` squared + y` squared + z` squared), 2[Tau]0 = [omega] squared[Sigma]m(x squared + y squared), (24)

/ dPy dPx \
a_r= [Sigma]m( x----- - y----- ), (25)
\ dPq_r dPq_r/

whence

dP(x, y)
(r, s) = 2[omega].[Sigma]m ------------. (26)
dP(q_s, q_r)

The conditions of relative equilibrium are given by (23).

It will be noticed that this expression V - [Tau]0, 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 (Sec. 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

d[Tau] d
------ = --([@] + [Tau]0) + [omega].[Sigma]m(xy" - yx")
dt dt

d
= --([@] - [Tau]0) + [omega].[Sigma](xY - yX). (27)
dt

This must be equal to the rate at which the forces acting on the
system do work, viz. to

[omega][Sigma](xY - yX) + Q1q`1 + Q2q`2 + ... + Q_nq`_n,

where the first term represents the work done in virtue of the
rotation.

Constrained systems.

We have still to notice the modifications which Lagrange's equations
undergo when the co-ordinates q1, q2, ... q_n 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, q1, q2, ... q_n) = 0, B(t, q1, q2, ... q_n) = 0, &c. (28)

These may be interpreted as introducing partial constraints into a
previously free system. The variations [delta]q1, [delta]q2, ...
[delta]q_n in the expressions (6) and (7) of Sec. 2 which are to be
equated are no longer independent, but are subject to the relations

dPA dPA dPB dPB
---- [delta]q1 + ---- [delta]q2 + ... = 0, ---- [delta]q1 + ---- [delta]q2 + ... = 0, &c. (29)
dPq1 dPq2 dPq1 dPq2

Introducing indeterminate multipliers [lambda], mu, ..., one for each
of these equations, we obtain in the usual manner n equations of the
type

d dP[Tau] dP[Tau] dPA dPB
-- ------- - ------- = Q_r + [lambda] ----- + mu ----- + ..., (30)
dt dPq`_r dPq_r dPq_r dPq_r

in place of Sec. 2 (10). These equations, together with (28), serve to
determine the n co-ordinates q1, q2, ... q_n and the m multipliers
[lambda], mu, ....

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 co-ordinates,
independent of the time.

Again, it may happen that although there are no prescribed relations
between the co-ordinates q1, q2, ... q_n, yet from the circumstances
of the problem certain geometrical conditions are imposed on their
_variations_, thus

A1[delta]q1 + A2[delta]q2 + ... = 0, B1[delta]q1 + B2[delta]q2 + ... = 0, &c., (31)

where the coefficients are functions of q1, q2, ... q_n and (possibly)
of t. It is assumed that these equations are not integrable as regards
the variables q1, q2, ... q_n; 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 co-ordinates 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

d dP[Tau] dP[Tau]
-- ------- - ------- = Q_r + [lambda]A_r + muB_r + ... (32)
dt dPq`_r dPq_r

The co-ordinates q1, q2, ... q_n, and the indeterminate multipliers
[lambda], mu, ..., are determined by these equations and by the
velocity-conditions corresponding to (31). When t does not appear
explicitly in the coefficients, these velocity-conditions take the
forms

A1q`1 + A2q`2 + ... = 0, B1q`1 + B2q`2 + ... = 0, &c. (33)

Systems of this kind, where the relations (31) are not integrable, are
called _non-holonomic_.

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_ p1, p2, ... and the co-ordinates
q1, q2, ..., as in Sec. 1 (19). Since the symbol [delta] now denotes a
variation extending to the co-ordinates as well as to the momenta, we
must add to the last member of Sec. 1 (21) terms of the types

dP[Tau] dP[Tau]'
------- [delta]q1 + -------- [delta]q1 + ... (1)
dPq1 dPq1

Since the variations [delta]p1, [delta]p2, ... [delta]q1, [delta]q2,
... may be taken to be independent, we infer the equations Sec. 1 (23) as
before, together with

dP[Tau] dP[Tau]' dP[Tau] dP[Tau]'
------ = - --------, ------- = - --------, ..., (2)
dPq1 dPq1 dPq2 dPq2

Hence the Lagrangian equations Sec. 2 (14) transform into

dP dP
p`1 = - ----([Tau]' + V), p`2 = ---- ([Tau]' + V), ... (3)
dPq1 dPq2

If we write

H = [Tau]' + V, (4)

so that H denotes the _total energy_ of the system, supposed expressed
in terms of the new variables, we get

dPH dPH
p`1 = - ----, p`2 = - ----, ... (5)
dPq1 dPq2

If to these we join the equations

dPH dPH
q`1 = ----, q`2 = ----, ..., (6)
dPp1 dPp2

which follow at once from Sec. 1 (23), since V does not involve p1, p2,
..., 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 dPH dPH dPH dPH
-- = ---- p`1 + ---- p`2 + ... + ---- q`1 + ---- q`2 + ... = 0. (7)
dt dPp1 dPp2 dPq1 dPq2

The Hamiltonian transformation is extended to the case of varying
relations as follows. Instead of (4) we write

H = p1q`1 + p2q`2 + ... - [Tau] + V, (8)

and imagine H to be expressed in terms of the momenta p1, p2, ..., the
co-ordinates q1, q2, ..., and the time. The internal forces of the
system are assumed to be conservative, with the potential energy V.
Performing the variation [delta] on both sides, we find

dP[Tau] dPV
[delta]H = q`1[delta]p1 + ... - ------- [delta]q1 + ---- [delta]q + ..., (9)
dPq1 dPq1

terms which cancel in virtue of the definition of p1, p2, ... being
omitted. Since [delta]p1, [delta]p2, ..., [delta]q1, [delta]q2, ...
may be taken to be independent, we infer

dPH dPH
q`1 = ----, q`2 = ----, ..., (10)
dPp1 dPp2

and

dP dPH dP dPH
---- ([Tau] - V) = - ----, ----([Tau] - V) = - ----, .... (11)
dPq1 dPq1 dPq2 dPq2

It follows from (11) that

dPH dPH
p`1 = - ----, p`2 = - ----, .... (12)
dPq1 dPq2

The equations (10) 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 co-ordinates, which we
will denote by [chi], [chi]', [chi]", ..., provided the remaining
co-ordinates q1, q2, ... q_m and the velocities, including of course
the velocities [.[chi]], [.[chi]]', [.[chi]]", ..., are unaltered.
Secondly, there are no forces acting on the system of the types [chi],
[chi]', [chi]", .... This case arises, for example, when the system
includes gyrostats which are free to rotate about their axes, the
co-ordinates [chi], [chi]', [chi]", ... then being the angular
co-ordinates of the gyrostats relatively to their frames. Again, in
theoretical hydrodynamics we have the problem of moving solids in a
frictionless liquid; the ignored co-ordinates [chi], [chi]', [chi]",
... 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.

Routh's equations.

If we suppose the kinetic energy [Tau] to be expressed, as in
Lagrange's method, in terms of the co-ordinates and the velocities,
the equations of motion corresponding to [chi], [chi]', [chi]'', ...
reduce, in virtue of the above hypotheses, to the forms

d dP[Tau] d dP[Tau] d dP[Tau]
-- --------- = 0, -- --------- = 0, -- --------- = 0, ..., (1)
dt dP[chi]` dt dP[chi]`' dt dP[chi]`"

whence

dP[Tau] dP[Tau] dP[Tau]
-------- = [kappa], --------- = [kappa]', --------- = [kappa]", ..., (2)
dP[chi]` dP[chi]`' dP[chi]`"

where [kappa], [kappa]', [kappa]", ... are the constant momenta
corresponding to the cyclic co-ordinates [chi], [chi]', [chi]", ....
These equations are linear in [.[chi]], [.[chi]]', [.[chi]]", ...;
solving them with respect to these quantities and substituting in the
remaining Lagrangian equations, we obtain m differential equations to
determine the remaining co-ordinates q1, q2, ... q_m. The object of
the present investigation is to ascertain the general form of the
resulting equations. The retained co-ordinates q1, q2, ... q_m may be
called (for distinction) the _palpable_ co-ordinates of the system; in
many practical questions they are the only co-ordinates directly in
evidence.

If, as in Sec. 1 (25), we write

R = [Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ..., (3)

and imagine R to be expressed by means of (2) as a quadratic function
of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]", ... with
coefficients which are in general functions of the co-ordinates q1,
q2, ... q_m, then, performing the operation [delta] on both sides, we
find

dPR dPR dPR dP[Tau] dP[Tau]
-----[delta]q`1 + ... + ---------[delta][kappa] + ... + ----[delta]q1 + ... = -------[delta]q`1 + ... + -------[delta]q1 + ...
dPq`1 dP[kappa] dPq1 dPq`1 dPq1

dP[Tau] dP[Tau]
+ --------[delta][chi]` + ... + --------[delta]q1 + ... - [kappa][delta][chi]` - [chi]`[delta][kappa] - .... (4)
dP[chi]` dP[chi]1

Omitting the terms which cancel by (2), we find

dP[Tau] dPR dP[Tau] dPR
------- = -----, ------- = -----, ..., (5)
dPq`1 dPq`1 dPq`2 dPq`2

dP[Tau] dPR dP[Tau] dPR
------- = ----, ------- = ----, ..., (6)
dPq1 dPq1 dPq2 dPq2

dPR dPR dPR
[chi]` = - ---------, [chi]`' = - ----------, [chi]`" = - ----------, ... (7)
dP[kappa] dP[kappa]' dP[kappa]"

Substituting in Sec. 2 (10), we have

d dPR dPR d dPR dPR
-- ----- - ----- = Q1, -- ----- - ---- = Q2, ... (8)
dt dPq`1 dPq1 dt dPq`2 dPq2

These are Routh's forms of the modified Lagrangian equations.
Equivalent forms were obtained independently by Helmholtz at a later
date.

Kelvin's equations.

The function R is made up of three parts, thus

R = R(2,0) + R(1,1) + R(0,2), ... (9)

where R(2,0) is a homogeneous quadratic function of q`1, q`2, ...
q`_m, R(0,2) is a homogeneous quadratic function of [kappa], [kappa]',
[kappa]", ..., whilst R(1,1) consists of products of the velocities
q`1, q`2, ... q`_m into the momenta [kappa], [kappa]', [kappa]"....
Hence from (3) and (7) we have

/ dPR dPR dPR \
[Tau] = R - ( [kappa] --------- + [kappa]'---------- + [kappa]" ---------- + ...)
\ dP[kappa] dP[kappa]' dP[kappa]" /

= R(2,0) - R(0,2). (10)

If, as in Sec. 1 (30), we write this in the form

[Tau] = [@] + [Kappa], (11)

then (3) may be written

R = [@] - [Kappa] + ss1q`1 + ss2q`2 + ..., (12)

where ss1, ss2, ... are linear functions of [kappa], [kappa]', [kappa]",
..., say

ss_r = [alpha]_r[kappa] + [alpha]'_r[kappa]' + [alpha]"_r[kappa]" + ..., (13)

the coefficients [alpha]_r, [alpha]'_r, [alpha]"_r, ... being in
general functions of the co-ordinates q1, q2, ... q_m. Evidently ss_r
denotes that part of the momentum-component dPR/dPq`_r which is due to
the cyclic motions. Now

d dPR d / dP[@] \ d dP[@] dPss_r dPss_r
-- ------ = -- ( ------ + ss_r) = -- ------ + -----q`1 + -----q`2 + ..., (14)
dt dPq`_r dt \dPq`_r / dt dPq`_r dPq1 dPq2

dPR dP[@] dP[Kappa] dPss1 dPss2
----- = ----- - --------- + -----q`1 + -----q`2 + .... (15)
dPq_r dPq_r dPq_r dPq_r dPq_r

Hence, substituting in (8), we obtain the typical equation of motion
of a gyrostatic system in the form

d dP[@] dP[@] dP[Kappa]
-- ------ - ----- + (r, 1)q`1 + (r, 2)q`2 + ... + (r, s)q`_s + ... + --------- = Q_r, (16)
dt dPq`_r dPq_r dPq_r

where

dPss_r dPss_s
(r, s) = ----- - -----. (17)
dPq_s dPq_r

This form is due to Lord Kelvin. When q1, q2, ... q_m have been
determined, as functions of the time, the velocities corresponding to
the cyclic co-ordinates can be found, if required, from the relations
(7), which may be written

dP[Kappa] \
[Chi]` = --------- - [alpha]1q`1 - [alpha]2q`2 - ..., |
dP[kappa] |
|
dP[Kappa] > (18)
[Chi]`' = ---------- - [alpha]'1q`1 - [alpha]'2q`2 - ..., |
dP[kappa]' |
|
&c., &c. /

It is to be particularly noticed that

(r, r) = 0, (r, s) = -(s, r). (19)

Hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q`1,
q`2, ... q`_m respectively, and add, we find

d
--([@] + [Kappa]) = Q1q`1 + Q2q`2 + ..., (20)
dt

or, in the case of a conservative system

[@] + V + [Kappa] = const., (21)

which is the equation of energy.

The equation (16) includes Sec. 3 (17) as a particular case, the
eliminated co-ordinate being the angular co-ordinate of a rotating
solid having an infinite moment of inertia.

In the particular case where the cyclic momenta [kappa], [kappa]',
[kappa]", ... are all zero, (16) reduces to

d dP[@] dP[@]
-- ------ - ----- = Q_r. (22)
dt dPq`_r dPq_r

The form is the same as in Sec. 2, and the system now behaves, as regards
the co-ordinates q1, q2, ... q_m, exactly like the acyclic type there
contemplated. These co-ordinates 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 configuration (so far as
this is defined by q1, q2, ..., q_m), after performing any evolutions,
the ignored co-ordinates [chi], [chi]', [chi]", ... will not in
general return to their original values.