ANALYTICAL See also:

• DYNAMICS
• DYNAMICS (from Gr. bbvayts, strength)

For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the See also:

• HISTORICAL

(II) aqr aqr aqr The quantities Qi, Q2, • . are called the generalized components of impulse. Comparing (9) and (1o), we have, since the variations Sqi, Sq2,... are independent, aT=Q'1, aT=Q'2 ... (12) a4i aqz These are the general equations of impulsive motion. It is now usual to write pr=aq (13) The quantities Pi, 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 ~ni(tIx+yay+zaz) =piagi+P2Sg2+... (i4) Also, since T is a homogeneous quadratic See also:

• FUNCTION

(18) The See also:

• INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

The resulpng Velocities equations give us the velocities produced by any given interms ol system of impulses. Further, by substitution in (8), momenta. 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'11p12+A'22p22+•••+2A'12p-P2+... . (i9) where A'11, A'22, ••• A'12,... are certain coefficients depending on the configuration. They have been called by See also:

• MAXWELL
• MAXWELL, JAMES CLERK (1831–1879)

= OT T' aT. aT aT' aT' _ -Sq1+84--Sq2+... +aplplSplM p2Sp2+.... In virtue of (13) this reduces to aT' aT' 41Sp1+42sp2+... =ap1Sp'+ap2Sp2+.... Since Bpi, 432,... may be taken to be independent, we infer that aT' aT' 41=v, 42=dp2' In the very remarkable exposition of the See also:

• MATTER

(24) These equations, when written out in full, determine X, X', i",••• as linear functions of 41, K, K', K", .. We now consider the function — K Y— (25) supposed expressed, by means of the above relations in terms of gl g2i... K, K, K",.... Performing the operation S on both sides of (25), we have a R ag S41+... I aKSK+...=-='ki+...+.=5X+... KaiXSK—..., (26) where, for brevity, only one See also:

• TERM

4m alone, and K the momenta K, K', K",... alone. For in virtue of (29) we have, from (25), aR ,aR 'aR T =R- (KaK +K UK +K lK+...) . (31) and it is evident that the terms in R which are bilinear in respect of the two sets of variables 41, 42,... 4m and K, K",... will dis- appear from the right-hand side. It may be noted that the formula (30) gives immediate See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

Again, suppose that the system is started with prescribed velocity components qi, q2,... 4, , by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have K=o, K'=o, K"=o,... and therefore K=o. The kinetic energy is therefore less than in any other motion consistent with the prescribed velocity-conditions by the value which K assumes when K, K, K",... represent the impulses due to the constraints. See also:

• SIMPLE

Sq,. It is important to See also:

• NOTICE

G. Tait; it has greatly promoted the physical application of the subject. Another proof of the equations (Io), by See also:

• DIRECT

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 any, which exist between the various parts of the system are of the type § I (I), and so do not contain t explicitly. The See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

It is to be re-membered that the coefficients ao, a1, See also:

• A2(z)

(7) The equations of motion will be as in § 2 (TO), with the additional equation and dt{{I+m(x2+y2)}+m(xy—yx)]=I.. . . (13) If we suppose adjusted so as to maintain ¢=o, or (again) if we suppose the moment of inertia I to be infinitely See also:

• GREAT

These quantities (r, s) are subject to the relations (r, s) _ — (s, r), (r, r) =o. (19) 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 z, i",••• 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) See also:

• LEAD
• LEAD (pronounced iced)

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 a =-(6+To)+w.Zm(xy-yx) =dt(6-To)+w.Z(xY-yX). (27) (13) or, again, Case of varying relations. (I) d aT_aT=X, (8) dt az ax where X is the force corresponding to the co-ordinate 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 Xi on the right-hand side; this represents the See also:

• RATE

whence This must be equal to the rate at which the forces acting on the system do work, viz. to wE(xY-yX)+Q1g1+Q2g2+ ••• +Q4., where the first term represents the work done in virtue of the rotation. We have still to notice the modifications which Lagrange's equations undergo when the co-ordinates qi, q2,•••q, are not all independently variable. In the first place, we may suppose them connected by a number m (<a) of relations of the type A(t, q1, q2, ...q,.) =0, B(t, q1, q2, ... qn) =0, &c. (28) These may be interpreted as introducing partial constraints into a previously free system. The variations Sq1, 3g2r...6gn in the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations OA Sq1+a4 bq2+... =0, aq q1+a4 SQ2+ ... =0, &c. (29) Introducing indeterminate multipliers A, µ,..., one for each of these equations, we obtain in the usual manner n equations of the type d aT d aq - OT = Qr+aa4 + aq +... , . (30) in place of § 2 (to). These equations, together with (28), serve to determine the n co-ordinates q1, q2, ... 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 See also:

• DIFFERENTIAL

Again, it may happen that although there are no prescribed relations between the co-ordinates q1, g2,••.Qn, yet from the circumstances of the problem certain geometrical conditions are imposed on their variations, thus Aiog1+A2bg2+ ... =o, Bibgi+B2Sg2+ ... =o, &c., (31) where the coefficients are functions of q1, q2,...q, and (possibly) of t. It is assumed that these equations are not integrable as regards the variables 0, g2,..•q,.; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

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 qi, q2,.... as in § I (19). Since the See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

—T+V, (8) and imagine H to be expressed in terms of the momenta P1, p2, ..., the co-ordinates qi, q2, ..., and the time. The See also:

• INTERNAL

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 co-ordinates x, x'> x",••.then being the angular co-ordinates of the gyrostats relatively to their frames. Again, in theoretical See also:

• HYDRODYNAMICS (Gr. Scot)

The retained co-ordinates q1, Q2, ... q+n may be called (for distinction) the palpable co-ordinates of the system; in many See also:

• PRACTICAL

Substituting in § 2 (to), we have See also:

• DAR

(17) This form is due to Lord Kelvin. When q1, q2,... gm 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 aK x = aK -49141-49242-..., aK hh X'=aK -49'141-a'212 -..., &c., &c. It is to be particularly noticed that (r, r) = o, (r, s) = — (s, r). . . (19) Hence, if in (16) we put r=1, 2, 3,... m, and multiply by 41, 42,... 4m respectively, and add, we find dt(6+K) =Q141+Q242+..., or,'in the case of a conservative system 6+V+K=cont., . . (21) which is the equation of energy. The equation (16) includes § 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 K, IC', K",... are all zero, (16) reduces to See also:

• DAM

If in Lagrange's equations § 2 (to) we See also:

• REVERSE

The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy K. This is important from a physical point of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic. By means of the formulae (18), which now reduce to aK , aK " aK X = .aK , X =49K , x =See also:

• ALE

See also:

• KLEIN, JULIUS LEOPOLD (1810–1876)

(22) Kinetostatics. . (23) • (27) . (30) convolutions. Each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable. A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates qi, q2, ... q,0 all vanish (see § 5). This has been discussed by Routh, Lord Kelvin and Tait, and See also:

• POINCARE, RAYMOND (186o— )

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 See also:

• THOMSON, JAMES (1822-1892)
• THOMSON, JAMES (1834-1882)
• THOMSON, JAMES (r700-1748)
• THOMSON, JOHN (1778-184o)
• THOMSON, JOSEPH (1858-1895)
• THOMSON, SIR CHARLES WYVILLE (1830-1882)
• THOMSON, THOMAS (1773-1852)
• THOMSON, WILLIAM (1819-189o)

If a be the See also:

• RADIUS

If the bowl be not spherical but ellipsoidal, the vertical diameter being a See also:

• PRINCIPAL

The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property 3A =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 con-figurations respectively, we have SA = a f` Em(2+5,2+2)dt =2f STdt+2T'St'—2TSt,. . (3) t since the upper and See also:

• LOWER

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 See also:

• AIDS

If this be the See also:

• SOLE (Solea)