(1)To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put μ = ν. We then find without difficulty that V + K is a minimum provided ν² ≥ 4AMgh. The method of small oscillations gave us the condition ν² > 4AMgh, and indicated instability in the cases ν² ≤ 4AMgh. The present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable.The question remains, as before, whether it isessentialfor 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 Poincaré, between what we may callordinaryortemporarystability (which is stability in the above sense) andpermanentorsecularstability, which means stability when regard is had to possible dissipative forces called into play whenever the co-ordinates q1, q2, ... qmvary. Since the total energy of the system at any instant is given (in the notation of § 5) by an expression of the form ⅋ + V + K, where ⅋ cannot be negative, the argument of Thomson and Tait, given underMechanics, § 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˙1, q˙2, ... q˙nare zero it is necessary and sufficient that the function V − Τ0should 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 (ω) about the vertical diameter. This position obviously possesses “ordinary” stability. If a be the radius of the bowl, and θ denote angular distance from the lowest point, we haveV − Τ0= mga(1 − cos θ) − ½mω²a² sin² θ;(2)this is a minimum for θ = 0 only so long as ω² < g/a. For greater values of ω the only position of “permanent” stability is that in which the particle rotates with the bowl at an angular distance cos−1(g/ω²a) 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 haveẍ = −p²x − k (ẋ + ωy),ÿ = −p²y − k (ẏ − ωx),(3)where p² = g/a. These combine intoz¨ + kz˙ + (p² − ikω) z = 0,(4)where z = x + iy, i = √−1. Assuming z = Ceλt, we findλ = −½k(1 ∓ ω/p) ± ip,(5)if the square of k be neglected. The complete solution is thenx + iy = C1e−β1teipt+ + C2e−β2te−ipt,(6)whereβ1= ½k (1 − ω/p), β2= ½k (1 + ω/p).(7)This represents two superposed circular vibrations, in opposite directions, of period 2π/p. If ω < p, the amplitude of each of these diminishes asymptotically to zero, and the position x = 0, y = 0 is permanently stable. But if ω > p the amplitude of that circular vibration which agrees in sense with the rotation ω will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. If the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (seeMechanics, § 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 between any two configurations through which it passes,Stationary 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 δ to denote the transition from the actual to any one of the hypothetical motions.The best-known theorem of this class is that ofLeast Action, originated by P.L.M. de Maupertuis, but first put in a definite form by Lagrange. The “action” of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of thevis viva. The action of a dynamical system is the sum of the actions of its constituent particles, and is accordingly given by the formulaA = Σ∫mvds = Σ∫mv²dt = 2∫Τdt.(1)The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the propertyδA = 0,(2)provided the total energy have the same constant value in the varied motion as in the actual motion.If t, t′ be the times of passing through the initial and final configurations respectively, we haveδA = δ∫t′tΣm (ẋ² + ẏ² + z˙²) dt=∫t′tδΤdt + 2Τ′δt′ + 2Τδt,(3)since the upper and lower limits of the integral must both be regarded as variable. This may be writtenδA =∫t′tδΤdt +∫t′tΣm (ẋδẋ + ẏδẏ + z˙δz˙) dt + 2Τ′δt′ − 2Τδt=∫t′tδΤdt +[Σm (ẋδx + ẏδy + z˙δz)]t′t−∫t′tΣm (ẍδx + ÿδy + z¨δz) dt + 2Τ′δt′ − 2Τδt.(4)Now, by d’Alembert’s principle,Σm (ẍδx + ÿδy + z¨δz) = −δV,(5)and by hypothesis we haveδ(Τ + V) = 0.(6)The formula therefore reduces toδA =[Σm (ẋδx + ẏδy + z˙δz)]t′t+ 2Τ′δt′ − 2Τδt.(7)Since the terminal configurations are unaltered, we must have at the lower limitδx + ẋδt = 0, δy + ẏδt = 0, δz + z˙δt = 0,(8)with similar relations at the upper limit. These reduce (7) to the form (2).The equation (2), it is to be noticed, merely expresses that the variation of A vanishesto the first order; the phrasestationary actionhas 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 theleast possiblesubject 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 intoδ∫ds = 0,(9)i.e.the path must be a geodesic line. Now a geodesic is not necessarily theshortestpath between two given points on it; for example, on the sphere a great-circle arc ceases to be the shortest path between its extremities when it exceeds 180°. More generally, taking any surface, let a point P, starting from O, move along a geodesic; this geodesic will be a minimum path from O to P until P passes through a point O′ (if such exist), which is the intersection with a consecutive geodesic through O. After this point the minimum property ceases. On an anticlastic surface two geodesics cannot intersect more than once, and each geodesic is therefore a minimum path between any two of its points. These illustrations are due to K.G.J. Jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:—Let O and P denote any two configurations on a natural path of the system. If this be the sole free path from O to P with the prescribed amount of energy, the action from O to P is a minimum. But ifthere be several distinct paths, let P vary from coincidence with O along the first-named path; the action will then cease to be a minimum when a configuration O′ is reached such that two of the possible paths from O to O′ coincide. For instance, if O and P be positions on the parabolic path of a projectile under gravity, there will be a second path (with the same energy and therefore the same velocity of projection from O), these two paths coinciding when P is at the other extremity (O′, say) of the focal chord through O. The action from O to P will therefore be a minimum for all positions of P short of O′. Two configurations such as O and O′ in the general statement are called conjugatekinetic foci. Cf.Variations, Calculus of.Before leaving this topic the connexion of the principle of stationary action with a well-known theorem of optics may be noticed. For the motion of a particle in a conservative field of force the principle takes the formδ∫vds = 0.(10)On the corpuscular theory of light v is proportional to the refractive index μ of the medium, whenceδ∫μds = 0.(11)In the formula (2) the energy in the hypothetical motion is prescribed, whilst the time of transit from the initial to the final configurationHamiltonian principle.is variable. In another and generally more convenient theorem, due to Hamilton, the time of transit is prescribed to be the same as in the actual motion, whilst the energy may be different and need not (indeed) be constant. Under these conditions we haveδ∫t′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 haveδ∫t′t(T − V)dt =∫t′t(δΤ − δV)dt =∫t′t{Σm (ẋδẋ + ẏδẏ + z˙δz˙) − δV} dt=[Σm (ẋδx + ẏδy + z˙δz)]t′t−∫t′t{Σm (ẍδx + ÿδy + z¨δz) + δV} dt.(13)The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d’Alembert’s principle.The fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange’s equations, we have∫t′t(δΤ − δV) dt =∫t′t{∂Tδq˙1+∂Tδq1+ ... −∂Vδq1− ...}dt∂q˙1∂q1∂q1=[p1δq1+ p2δq2+ ...]t′t−∫t′t{ [ṗ1−∂T+∂V)δq1+(ṗ2−∂T+∂V)δq2+ ...}dt.∂q1∂q1∂q2∂q2(14)The integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of δq1, δq2, ... 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 toExtension to cyclic systems.the case of cyclic systems has been given by J. Larmor. If we write, as in § 1 (25),R = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,(15)we shall haveδ∫t′t(R − V) dt = 0,(16)provided that the variation does not affect the cyclic momenta κ, κ′, κ″, ..., and that the configurations at times t and t′ are unaltered, so far as they depend on the palpable co-ordinates q1, q2, ... qm. The initial and final values of the ignored co-ordinates will in general be affected.To prove (16) we have, on the above understandings,δ∫t′t(R − V) dt =∫t′t(δT − κδχ˙ − ... − δV) dt=∫t′t(∂Tδq˙1+ ... +∂Tδq1+ ... − δV)dt,∂q˙1∂q1(17)where terms have been cancelled in virtue of § 5 (2). The last member of (17) represents a variation of the integral∫t′t(T − V) dton the supposition that δX = 0, δX′ = 0, δX″ = 0, ... throughout, whilst δq1, δq2, δqmvanish 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 writeA =∫(2T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...) dt,(18)thenδA = 0,(19)provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these being regarded as defined by the palpable co-ordinates alone.§ 8.Hamilton’s Principal and Characteristic Functions.In the investigations next to be described a more extended meaning is given to the symbol δ. We will, in the first instance, denote by it an infinitesimal variation of the mostPrincipal function.general kind, affecting not merely the values of the co-ordinates at any instant, but also the initial and final configurations and the times of passing through them. If we putS =∫t′t(Τ − V) dt,(1)we have, then,δS = (T′ − V′) δt′ − (T − V) δt +∫t′t(δΤ − δV) dt= (T′ − V′) δt′ − (T − V) δt +[Σm (ẋδx + ẏδy + z˙δz)]t′t.(2)Let us now denote by x′ + δx′, y′ + δy′, z′ + δz′, the final co-ordinates (i.e.at time t′ + δt′) of a particle m. In the terms in (2) which relate to the upper limit we must therefore write δx′ − ẋ′δt′, δy′ − ẏ′δt′, δz′ − z˙′δt′ for δx, δy, δz. With a similar modification at the lower limit, we obtainδS = −Hδτ + Σm (ẋ′δx′ + ẏ′δy′ + z˙′δz′) − Σm (ẋδx + ẏδy + z˙δz),(3)where H (= T + V) is the constant value of the energy in the free motion of the system, and τ (= t′ − t) is the time of transit. In generalized co-ordinates this takes the formδS = −Hδτ + p′1δq′1+ p′2δq′2+ ... − p1δq1− p2δq2− ....(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 τ. On this view of the matter, S will be a function of the initial and final co-ordinates (q1, q2, ... and q′1, q′2, ...) and the time τ, as independent variables. And we obtain at once from (4)p′1=∂S, p′2=∂S, ... ,∂q′1∂q′2(5)p1= −∂S, p2= −∂S, ... ,∂q1∂q2andH = −∂S.∂τ(6)S is called by Hamilton theprincipal function; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. If we substitute the values of p1, p2, ... and H from (5) and (6) in the expression for the kinetic energy in the form T′ (see § 1), the equationT¹ + V = H(7)becomes a partial differential equation to be satisfied by S. It has been shown by Jacobi that the dynamical problem resolves itself into obtaining a “complete” solution of this equation, involving n + 1 arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here.There is a similar theoryCharacteristic function.for the functionA = 2∫Tdt = S + Hτ(8)It follows from (4) thatδA = τδH + p′1δq′1+ p′2δq′2+ ... − p1δq1− p2δq2− ....(9)This formula (it may be remarked) contains the principle of “leastaction” as a particular case. Selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy H, so that it shall pass through the other. Hence, regarding A as a function of the initial and final co-ordinates and the energy, we findp′1=∂A, p′2=∂A, ... ,∂q′1∂q′2(10)p1= −∂A, p2= −∂A, ... ,∂q1∂q2andτ =∂A.∂H(11)A is called by Hamilton thecharacteristic function; it represents, of course, the “action” of the system in the free motion (with prescribed energy) between the two configurations. Like S, it satisfies a partial differential equation, obtained by substitution from (10) in (7).The preceding theorems are easily adapted to the case of cyclic systems. We have only to writeS =∫t′t(R − V) dt =∫t′t(T − κχ˙ − κ′χ˙′ − ... − V) dt(12)in place of (1), andA =∫(2T − κχ˙ − κ′χ˙′ − ...) dt,(13)in place of (8); cf. § 7ad fin. It is understood, of course, that in (12) S is regarded as a function of the initial and final values of the palpable co-ordinates q1, q2, ... qm, and of the time of transit τ, the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of q1, q2, ... qm, 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 δq1, δq2, ... be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols.9.Reciprocal Properties of Direct and Reversed Motions.We may employ Hamilton’s principal function to prove a very remarkable formula connecting anytwoslightly disturbedLagrange’s formula.natural motions of the system. If we use the symbols δ and Δ to denote the corresponding variations, the theorem isdΣ (δpr·Δqr− Δpr·δqr) = 0;dt(1)or integrating from t to t′,Σ (δp′r·Δq′r− Δq′r·δq′r) = Σ (δpr·Δqr− Δpr·δqr).(2)If for shortness we write(r, s) =∂²S, (r, s′) =∂²S,∂qr∂qs∂qr∂q′s(3)we have∂pr= −Σs(r, s) δqs− Σs(r, s′) δq′s(4)with a similar expression for Δpr. Hence the right-hand side of (2) becomes− Σr{Σs(r, s) δqs+ Σs(r, s′) δq′s} Δqr+ Σr{Σs(r, s)Δqs+ Σs(r, s′) Δq′s} δqr= ΣrΣs(r, s′) {δqr·Δq′s− Δqr·δq′s}.(5)The same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to Lagrange, and was employed by him as the basis of his method of treating the dynamical theory ofVariation 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 conservative system between two configurations O and O′Helmholtz’s reciprocal theorems.through which it passes at times t and t′ respectively, and let t′ − t = τ. As the system is passing through O let a small impulse δprbe given to it, and let the consequent alteration in the co-ordinate qsafter the time τ be δq′s. Next consider thereversedmotion of the system, in which it would, if undisturbed, pass from O′ to O in the same time τ. Let a small impulse δp′sbe applied as the system is passing through O′, and let the consequent change in the co-ordinate qrafter a time τ be δqr. Helmholtz’s first theorem is to the effect thatδqr: δp′s= δq′s: δpr.(6)To prove this, suppose, in (2), that all the δq vanish, and likewise all the δp with the exception of δpr. Further, suppose all the Δq′ to vanish, and likewise all the Δp′ except Δp′s, the formula then givesδpr·Δqr= −Δp′s·δq′s,(7)which is equivalent to Helmholtz’s result, since we may suppose the symbol Δ to refer to the reversed motion, provided we change the signs of the Δp. In the most general motion of a top (Mechanics, § 22), suppose that a small impulsive couple about the vertical produces after a time τ a change δθ in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of θ will produce after a time τ a change δψ, in the azimuth of the axis, which is equal to δθ. It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. Again, in the corpuscular theory of light let O, O′ be any two points on the axis of a symmetrical optical combination, and let V, V′ be the corresponding velocities of light. At O let a small impulse be applied perpendicular to the axis so as to produce an angular deflection δθ, and let β′ be the corresponding lateral deviation at O′. In like manner in the reversed motion, let a small deflection δθ′ at O′ produce a lateral deviation β at O. The theorem (6) asserts thatβ=β′β′,V′δθ′Vδθ(8)or, in optical language, the “apparent distance” of O from O′ is to that of O′ from O in the ratio of the refractive indices at O′ and O respectively.In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change δqrin one of the co-ordinates, the momenta being all unaltered, and δq′sisHelmholtz’s second reciprocal theorem.the consequent variation in one of the momenta after time τ. Similarly in the reversed motion a change δp′sproduces after time τ a change of momentum δpr. The theorem asserts thatδp′s: δqr= δpr: δq′s(9)This follows at once from (2) if we imagine all the δp to vanish, and likewise all the δq save δqr, and if (further) we imagine all the Δp′ to vanish, and all the Δq′ save Δq′s. Reverting to the optical illustration, if F, F′, be principal foci, we can infer that the convergence at F′ of a parallel beam from F is to the convergence at F of a parallel beam from F′ in the inverse ratio of the refractive indices at F′ and F. This is equivalent to Gauss’s relation between the two principal focal lengths of an optical instrument. It may be obtained otherwise as a particular case of (8).We have by no means exhausted the inferences to be drawn from Lagrange’s formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and Tait, and Lord Rayleigh. In applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium.It may be worth while to point out, however, that there is no such limitation to the use of Lagrange’s formula (1). In applying it to cyclic systems, it is convenient to introduce conditions already laid down, viz. that the co-ordinates qrare the palpable co-ordinates and that the cyclic momenta are invariable. Special inference can then be drawn as before, but the interpretation cannot be expressed so neatly owing to the non-reversibility of the motion.Authorities.—The most important and most accessible early authorities are J.L. Lagrange,Mécanique analytique(1st ed. Paris, 1788, 2nd ed. Paris, 1811; reprinted inŒuvres, vols. xi., xii., Paris, 1888-89); Hamilton, “On a General Method in Dynamics,”Phil. Trans.1834 and 1835; C.G.J. Jacobi,Vorlesungen über Dynamik(Berlin, 1866, reprinted inWerke, Supp.-Bd., Berlin, 1884). An account of the extensive literature on the differential equations of dynamics and on the theory of variation of parameters is given by A. Cayley, “Report on Theoretical Dynamics,”Brit. Assn. Rep.(1857),Mathematical Papers, vol. iii. (Cambridge, 1890). For the modern developments reference may be made to Thomson and Tait,Natural Philosophy(1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord Rayleigh,Theory of Sound, vol. i. (1st ed. London, 1877; 2nd ed. London, 1894); E.J. Routh,Stability of Motion(London, 1877), andRigid Dynamics(4th ed. London, 1884); H. Helmholtz, “Über die physikalische Bedeutung des Prinzips der kleinsten Action,”Crelle, vol. c., 1886, reprinted (with other cognate papers) inWiss. 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. Poincaré, “Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation”Acta math.vol. vii. (1885); F. Klein and A. Sommerfeld,Theorie des Kreisels, pts. 1, 2 (Leipzig, 1897-1898); A. Lioupanoff and J. Hadamard,Liouville, 5me série, 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 workDie Prinzipien der Mechanik(Leipzig, 1894), of which an English translation appeared in 1900.
(1)
To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put μ = ν. We then find without difficulty that V + K is a minimum provided ν² ≥ 4AMgh. The method of small oscillations gave us the condition ν² > 4AMgh, and indicated instability in the cases ν² ≤ 4AMgh. The present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable.
The question remains, as before, whether it isessentialfor 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 Poincaré, between what we may callordinaryortemporarystability (which is stability in the above sense) andpermanentorsecularstability, which means stability when regard is had to possible dissipative forces called into play whenever the co-ordinates q1, q2, ... qmvary. Since the total energy of the system at any instant is given (in the notation of § 5) by an expression of the form ⅋ + V + K, where ⅋ cannot be negative, the argument of Thomson and Tait, given underMechanics, § 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˙1, q˙2, ... q˙nare zero it is necessary and sufficient that the function V − Τ0should 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 (ω) about the vertical diameter. This position obviously possesses “ordinary” stability. If a be the radius of the bowl, and θ denote angular distance from the lowest point, we have
V − Τ0= mga(1 − cos θ) − ½mω²a² sin² θ;
(2)
this is a minimum for θ = 0 only so long as ω² < g/a. For greater values of ω the only position of “permanent” stability is that in which the particle rotates with the bowl at an angular distance cos−1(g/ω²a) 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
ẍ = −p²x − k (ẋ + ωy),ÿ = −p²y − k (ẏ − ωx),
(3)
where p² = g/a. These combine into
z¨ + kz˙ + (p² − ikω) z = 0,
(4)
where z = x + iy, i = √−1. Assuming z = Ceλt, we find
λ = −½k(1 ∓ ω/p) ± ip,
(5)
if the square of k be neglected. The complete solution is then
x + iy = C1e−β1teipt+ + C2e−β2te−ipt,
(6)
where
β1= ½k (1 − ω/p), β2= ½k (1 + ω/p).
(7)
This represents two superposed circular vibrations, in opposite directions, of period 2π/p. If ω < p, the amplitude of each of these diminishes asymptotically to zero, and the position x = 0, y = 0 is permanently stable. But if ω > p the amplitude of that circular vibration which agrees in sense with the rotation ω will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. If the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (seeMechanics, § 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 between any two configurations through which it passes,Stationary 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 δ to denote the transition from the actual to any one of the hypothetical motions.
The best-known theorem of this class is that ofLeast Action, originated by P.L.M. de Maupertuis, but first put in a definite form by Lagrange. The “action” of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of thevis 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 = Σ∫mvds = Σ∫mv²dt = 2∫Τdt.
(1)
The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property
δA = 0,
(2)
provided the total energy have the same constant value in the varied motion as in the actual motion.
If t, t′ be the times of passing through the initial and final configurations respectively, we have
δA = δ∫t′tΣm (ẋ² + ẏ² + z˙²) dt
=∫t′tδΤdt + 2Τ′δt′ + 2Τδt,
(3)
since the upper and lower limits of the integral must both be regarded as variable. This may be written
δA =∫t′tδΤdt +∫t′tΣm (ẋδẋ + ẏδẏ + z˙δz˙) dt + 2Τ′δt′ − 2Τδt
=∫t′tδΤdt +[Σm (ẋδx + ẏδy + z˙δz)]t′t
−∫t′tΣm (ẍδx + ÿδy + z¨δz) dt + 2Τ′δt′ − 2Τδt.
(4)
Now, by d’Alembert’s principle,
Σm (ẍδx + ÿδy + z¨δz) = −δV,
(5)
and by hypothesis we have
δ(Τ + V) = 0.
(6)
The formula therefore reduces to
δA =[Σm (ẋδx + ẏδy + z˙δz)]t′t+ 2Τ′δt′ − 2Τδt.
(7)
Since the terminal configurations are unaltered, we must have at the lower limit
δx + ẋδt = 0, δy + ẏδt = 0, δz + z˙δt = 0,
(8)
with similar relations at the upper limit. These reduce (7) to the form (2).
The equation (2), it is to be noticed, merely expresses that the variation of A vanishesto the first order; the phrasestationary actionhas 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 theleast possiblesubject 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
δ∫ds = 0,
(9)
i.e.the path must be a geodesic line. Now a geodesic is not necessarily theshortestpath between two given points on it; for example, on the sphere a great-circle arc ceases to be the shortest path between its extremities when it exceeds 180°. More generally, taking any surface, let a point P, starting from O, move along a geodesic; this geodesic will be a minimum path from O to P until P passes through a point O′ (if such exist), which is the intersection with a consecutive geodesic through O. After this point the minimum property ceases. On an anticlastic surface two geodesics cannot intersect more than once, and each geodesic is therefore a minimum path between any two of its points. These illustrations are due to K.G.J. Jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:—Let O and P denote any two configurations on a natural path of the system. If this be the sole free path from O to P with the prescribed amount of energy, the action from O to P is a minimum. But ifthere be several distinct paths, let P vary from coincidence with O along the first-named path; the action will then cease to be a minimum when a configuration O′ is reached such that two of the possible paths from O to O′ coincide. For instance, if O and P be positions on the parabolic path of a projectile under gravity, there will be a second path (with the same energy and therefore the same velocity of projection from O), these two paths coinciding when P is at the other extremity (O′, say) of the focal chord through O. The action from O to P will therefore be a minimum for all positions of P short of O′. Two configurations such as O and O′ in the general statement are called conjugatekinetic foci. Cf.Variations, Calculus of.
Before leaving this topic the connexion of the principle of stationary action with a well-known theorem of optics may be noticed. For the motion of a particle in a conservative field of force the principle takes the form
δ∫vds = 0.
(10)
On the corpuscular theory of light v is proportional to the refractive index μ of the medium, whence
δ∫μds = 0.
(11)
In the formula (2) the energy in the hypothetical motion is prescribed, whilst the time of transit from the initial to the final configurationHamiltonian principle.is variable. In another and generally more convenient theorem, due to Hamilton, the time of transit is prescribed to be the same as in the actual motion, whilst the energy may be different and need not (indeed) be constant. Under these conditions we have
δ∫t′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
δ∫t′t(T − V)dt =∫t′t(δΤ − δV)dt =∫t′t{Σm (ẋδẋ + ẏδẏ + z˙δz˙) − δV} dt
=[Σm (ẋδx + ẏδy + z˙δz)]t′t−∫t′t{Σm (ẍδx + ÿδy + z¨δz) + δV} dt.
(13)
The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d’Alembert’s principle.
The fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange’s equations, we have
(14)
The integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of δq1, δq2, ... 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 toExtension to cyclic systems.the case of cyclic systems has been given by J. Larmor. If we write, as in § 1 (25),
R = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,
(15)
we shall have
δ∫t′t(R − V) dt = 0,
(16)
provided that the variation does not affect the cyclic momenta κ, κ′, κ″, ..., and that the configurations at times t and t′ are unaltered, so far as they depend on the palpable co-ordinates q1, q2, ... qm. The initial and final values of the ignored co-ordinates will in general be affected.
To prove (16) we have, on the above understandings,
δ∫t′t(R − V) dt =∫t′t(δT − κδχ˙ − ... − δV) dt
(17)
where terms have been cancelled in virtue of § 5 (2). The last member of (17) represents a variation of the integral
∫t′t(T − V) dt
on the supposition that δX = 0, δX′ = 0, δX″ = 0, ... throughout, whilst δq1, δq2, δqmvanish 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 =∫(2T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...) dt,
(18)
then
δA = 0,
(19)
provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these being regarded as defined by the palpable co-ordinates alone.
§ 8.Hamilton’s Principal and Characteristic Functions.
In the investigations next to be described a more extended meaning is given to the symbol δ. We will, in the first instance, denote by it an infinitesimal variation of the mostPrincipal function.general kind, affecting not merely the values of the co-ordinates at any instant, but also the initial and final configurations and the times of passing through them. If we put
S =∫t′t(Τ − V) dt,
(1)
we have, then,
δS = (T′ − V′) δt′ − (T − V) δt +∫t′t(δΤ − δV) dt
= (T′ − V′) δt′ − (T − V) δt +[Σm (ẋδx + ẏδy + z˙δz)]t′t.
(2)
Let us now denote by x′ + δx′, y′ + δy′, z′ + δz′, the final co-ordinates (i.e.at time t′ + δt′) of a particle m. In the terms in (2) which relate to the upper limit we must therefore write δx′ − ẋ′δt′, δy′ − ẏ′δt′, δz′ − z˙′δt′ for δx, δy, δz. With a similar modification at the lower limit, we obtain
δS = −Hδτ + Σm (ẋ′δx′ + ẏ′δy′ + z˙′δz′) − Σm (ẋδx + ẏδy + z˙δz),
(3)
where H (= T + V) is the constant value of the energy in the free motion of the system, and τ (= t′ − t) is the time of transit. In generalized co-ordinates this takes the form
δS = −Hδτ + p′1δq′1+ p′2δq′2+ ... − p1δq1− p2δq2− ....
(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 τ. On this view of the matter, S will be a function of the initial and final co-ordinates (q1, q2, ... and q′1, q′2, ...) and the time τ, as independent variables. And we obtain at once from (4)
(5)
and
(6)
S is called by Hamilton theprincipal function; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. If we substitute the values of p1, p2, ... and H from (5) and (6) in the expression for the kinetic energy in the form T′ (see § 1), the equation
T¹ + V = H
(7)
becomes a partial differential equation to be satisfied by S. It has been shown by Jacobi that the dynamical problem resolves itself into obtaining a “complete” solution of this equation, involving n + 1 arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here.
There is a similar theoryCharacteristic function.for the function
A = 2∫Tdt = S + Hτ
(8)
It follows from (4) that
δA = τδH + p′1δq′1+ p′2δq′2+ ... − p1δq1− p2δq2− ....
(9)
This formula (it may be remarked) contains the principle of “leastaction” as a particular case. Selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy H, so that it shall pass through the other. Hence, regarding A as a function of the initial and final co-ordinates and the energy, we find
(10)
and
(11)
A is called by Hamilton thecharacteristic function; it represents, of course, the “action” of the system in the free motion (with prescribed energy) between the two configurations. Like S, it satisfies a partial differential equation, obtained by substitution from (10) in (7).
The preceding theorems are easily adapted to the case of cyclic systems. We have only to write
S =∫t′t(R − V) dt =∫t′t(T − κχ˙ − κ′χ˙′ − ... − V) dt
(12)
in place of (1), and
A =∫(2T − κχ˙ − κ′χ˙′ − ...) dt,
(13)
in place of (8); cf. § 7ad fin. It is understood, of course, that in (12) S is regarded as a function of the initial and final values of the palpable co-ordinates q1, q2, ... qm, and of the time of transit τ, the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of q1, q2, ... qm, 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 δq1, δq2, ... be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols.
9.Reciprocal Properties of Direct and Reversed Motions.
We may employ Hamilton’s principal function to prove a very remarkable formula connecting anytwoslightly disturbedLagrange’s formula.natural motions of the system. If we use the symbols δ and Δ to denote the corresponding variations, the theorem is
(1)
or integrating from t to t′,
Σ (δp′r·Δq′r− Δq′r·δq′r) = Σ (δpr·Δqr− Δpr·δqr).
(2)
If for shortness we write
(3)
we have
∂pr= −Σs(r, s) δqs− Σs(r, s′) δq′s
(4)
with a similar expression for Δpr. Hence the right-hand side of (2) becomes
− Σr{Σs(r, s) δqs+ Σs(r, s′) δq′s} Δqr+ Σr{Σs(r, s)Δqs+ Σs(r, s′) Δq′s} δqr
= ΣrΣs(r, s′) {δqr·Δq′s− Δqr·δq′s}.
(5)
The same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to Lagrange, and was employed by him as the basis of his method of treating the dynamical theory ofVariation 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 conservative system between two configurations O and O′Helmholtz’s reciprocal theorems.through which it passes at times t and t′ respectively, and let t′ − t = τ. As the system is passing through O let a small impulse δprbe given to it, and let the consequent alteration in the co-ordinate qsafter the time τ be δq′s. Next consider thereversedmotion of the system, in which it would, if undisturbed, pass from O′ to O in the same time τ. Let a small impulse δp′sbe applied as the system is passing through O′, and let the consequent change in the co-ordinate qrafter a time τ be δqr. Helmholtz’s first theorem is to the effect that
δqr: δp′s= δq′s: δpr.
(6)
To prove this, suppose, in (2), that all the δq vanish, and likewise all the δp with the exception of δpr. Further, suppose all the Δq′ to vanish, and likewise all the Δp′ except Δp′s, the formula then gives
δpr·Δqr= −Δp′s·δq′s,
(7)
which is equivalent to Helmholtz’s result, since we may suppose the symbol Δ to refer to the reversed motion, provided we change the signs of the Δp. In the most general motion of a top (Mechanics, § 22), suppose that a small impulsive couple about the vertical produces after a time τ a change δθ in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of θ will produce after a time τ a change δψ, in the azimuth of the axis, which is equal to δθ. It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. Again, in the corpuscular theory of light let O, O′ be any two points on the axis of a symmetrical optical combination, and let V, V′ be the corresponding velocities of light. At O let a small impulse be applied perpendicular to the axis so as to produce an angular deflection δθ, and let β′ be the corresponding lateral deviation at O′. In like manner in the reversed motion, let a small deflection δθ′ at O′ produce a lateral deviation β at O. The theorem (6) asserts that
(8)
or, in optical language, the “apparent distance” of O from O′ is to that of O′ from O in the ratio of the refractive indices at O′ and O respectively.
In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change δqrin one of the co-ordinates, the momenta being all unaltered, and δq′sisHelmholtz’s second reciprocal theorem.the consequent variation in one of the momenta after time τ. Similarly in the reversed motion a change δp′sproduces after time τ a change of momentum δpr. The theorem asserts that
δp′s: δqr= δpr: δq′s
(9)
This follows at once from (2) if we imagine all the δp to vanish, and likewise all the δq save δqr, and if (further) we imagine all the Δp′ to vanish, and all the Δq′ save Δq′s. Reverting to the optical illustration, if F, F′, be principal foci, we can infer that the convergence at F′ of a parallel beam from F is to the convergence at F of a parallel beam from F′ in the inverse ratio of the refractive indices at F′ and F. This is equivalent to Gauss’s relation between the two principal focal lengths of an optical instrument. It may be obtained otherwise as a particular case of (8).
We have by no means exhausted the inferences to be drawn from Lagrange’s formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and Tait, and Lord Rayleigh. In applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium.
It may be worth while to point out, however, that there is no such limitation to the use of Lagrange’s formula (1). In applying it to cyclic systems, it is convenient to introduce conditions already laid down, viz. that the co-ordinates qrare the palpable co-ordinates and that the cyclic momenta are invariable. Special inference can then be drawn as before, but the interpretation cannot be expressed so neatly owing to the non-reversibility of the motion.
Authorities.—The most important and most accessible early authorities are J.L. Lagrange,Mécanique analytique(1st ed. Paris, 1788, 2nd ed. Paris, 1811; reprinted inŒuvres, vols. xi., xii., Paris, 1888-89); Hamilton, “On a General Method in Dynamics,”Phil. Trans.1834 and 1835; C.G.J. Jacobi,Vorlesungen über Dynamik(Berlin, 1866, reprinted inWerke, Supp.-Bd., Berlin, 1884). An account of the extensive literature on the differential equations of dynamics and on the theory of variation of parameters is given by A. Cayley, “Report on Theoretical Dynamics,”Brit. Assn. Rep.(1857),Mathematical Papers, vol. iii. (Cambridge, 1890). For the modern developments reference may be made to Thomson and Tait,Natural Philosophy(1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord Rayleigh,Theory of Sound, vol. i. (1st ed. London, 1877; 2nd ed. London, 1894); E.J. Routh,Stability of Motion(London, 1877), andRigid Dynamics(4th ed. London, 1884); H. Helmholtz, “Über die physikalische Bedeutung des Prinzips der kleinsten Action,”Crelle, vol. c., 1886, reprinted (with other cognate papers) inWiss. 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. Poincaré, “Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation”Acta math.vol. vii. (1885); F. Klein and A. Sommerfeld,Theorie des Kreisels, pts. 1, 2 (Leipzig, 1897-1898); A. Lioupanoff and J. Hadamard,Liouville, 5me série, 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 workDie Prinzipien der Mechanik(Leipzig, 1894), of which an English translation appeared in 1900.