(15)which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity ω. (SeeMechanics, § 13.)More generally, let us suppose that we have a certain group of co-ordinates χ, χ′, χ″, ... whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components χ˙, χ˙′, χ˙″, ... are maintained constant. The remaining co-ordinates being denoted by q1, q2, ... qn, we may write2T = ⅋ + T0+ 2(α1q˙1+ α2q˙2+ ...) χ˙ + 2(α′1q˙1+ α′2q˙2+ ...) χ˙′ + ...,(16)where ⅋ is a homogeneous quadratic function of the velocities q˙1, q˙2, ... q˙nof the type § 1 (8), whilst Τ0is a homogeneous quadratic function of the velocities χ˙, χ˙′, χ˙″, ... alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (10) of § 2 give n equations of the typed(∂⅋)−∂⅋+ (r, 1) q˙1+ (r, 2) q˙2+ ... −∂T0= Qrdt∂qr∂qr∂qr(17)where(r, s) =(∂αr−∂αs)χ˙ +(∂α′r−∂α′s)χ˙′ + ....∂qs∂qr∂qs∂qr(18)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 χ, χ′, χ″, ..., yield expressions for the forces which must be applied in order to maintain the velocities χ˙, χ˙′, χ˙″, ... constant; they need not be written down. If we follow the method by which the equation of energy was established in § 2, the equations (17) lead, on taking account of the relations (19), tod(⅋ − T0) = Q1q˙1+ Q2q˙2+ ... + Qnq˙n,dt(20)or, in case the forces Qrdepend only on the co-ordinates q1, q2, ... qnand are conservative,⅋ + V − T0= const.(21)The conditions that the equations (17) should be satisfied by zero values of the velocities q˙1, q˙2, ... q˙nareQr= −∂T0,∂qr(22)or in the case of conservative forces∂(V − T0) = 0,∂qr(23)i.e.the value of V − Τ0must bestationary.We may apply this to the case of a system whose configuration relative to axes rotating with constant angular velocity (ω) is defined by means of the n co-ordinates q1, q2, ... qn.Rotating axes.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, ... qn, of the form § 1 (1), we have, by (15)2⅋ = Σm (ẋ² + ẏ² + z˙²), 2Τ0= ω²Σm (x² + y²),(24)αr= Σm(x∂y− y∂x),∂qr∂qr(25)whence(r, s) = 2ω·Σm∂(x, y).∂(qs, qr)(26)The conditions of relative equilibrium are given by (23).It will be noticed that this expression V − T0, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious “centrifugal forces.” The question of stability of relative equilibrium will be noticed later (§ 6).It should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. From (15) and (14) we finddT=d(⅋ + T0) + ω·Σm (xÿ − yẍ) =d(⅋ − T0) + ω·Σ (xY − yX).dtdtdt(27)This must be equal to the rate at which the forces acting on the system do work, viz. toωΣ (xY − yX) + Q1q˙1+ Q2q˙2+ ... + Qnq˙n,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 q1, q2, ... qnConstrained systems.are not all independently variable. In the first place, we may suppose them connected by a number m (< n) of relations of the typeA (t, q1, q2, ... qn) = 0, B (t, q1, q2, ... qn) = 0, &c.(28)These may be interpreted as introducing partial constraints into a previously free system. The variations δq1, δq2, ... δqnin the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations∂Aδq1+∂Aδq2+ ... = 0,∂Bδq1+∂Bδq2+ ... = 0, &c.∂q1∂q2∂q1∂q2(29)Introducing indeterminate multipliers λ, μ, ..., one for each of these equations, we obtain in the usual manner n equations of the typed∂T−∂T= Qr+ λ∂A+ μ∂B+ ...,dt∂q˙r∂qr∂qr∂qr(30)in place of § 2 (10). These equations, together with (28), serve to determine the n co-ordinates q1, q2, ... qnand the m multipliers λ, μ, ....When t does not occur explicitly in the relations (28) the system is said to beholonomic. 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, ... qn, yet from the circumstances of the problem certain geometrical conditions are imposed on theirvariations, thusA1δq1+ A2δq2+ ... = 0, B1δq1+ B2δq2+ ... = 0, &c.,(31)where the coefficients are functions of q1, q2, ... qnand (possibly) of t. It is assumed that these equations are not integrable as regards the variables q1, q2, ... qn; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. The six 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, thusd∂T−∂T= Qr+ λAr+ μBr+ ...dt∂q˙r∂qr(32)The co-ordinates q1, q2, ... qn, and the indeterminate multipliers λ, μ, ..., 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 formsA1q˙1+ A2q˙2+ ... = 0, B1q˙1+ B2q˙2+ ... = 0, &c.(33)Systems of this kind, where the relations (31) are not integrable, are callednon-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 § 1 (19). Since the symbol δ now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of § 1 (21) terms of the types∂Tδq1+∂T`δq2+ ....∂q1∂q2(1)Since the variations δp1, δp2, ... δq1, δq2, ... may be taken to be independent, we infer the equations § 1 (23) as before, together with∂T= −∂T`,∂T= −∂T`, ...,∂q1∂q1∂q2∂q2(2)Hence the Lagrangian equations § 2 (14) transform intoṗ1= −∂(T` + V), ṗ2= −∂(T` + V), ...∂q1∂q2(3)If we writeH = T` + V,(4)so that H denotes thetotal energyof the system, supposed expressed in terms of the new variables, we getṗ1= −∂H, ṗ2= −∂H, ...∂q1∂q2(5)If to these we join the equationsq˙1=∂H, q˙2=∂H, ...,∂p1∂p2(6)which follow at once from § 1 (23), since V does not involve p1, p2, ..., we obtain a complete system of differential equationsof the first orderfor the determination of the motion.The equation of energy is verified immediately by (5) and (6), since these makedH=∂Hṗ1+∂Hṗ2+ ... +∂Hq˙1+∂Hq˙2+ ... = 0.dt∂p1∂p2∂q1∂q2(7)The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we writeH = p1q˙1+ p2q˙2+ ... − T + 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 δ on both sides, we findδH = q˙1δp1+ ... −∂Tδq1+∂Vδq + ...,∂q1∂q1(9)terms which cancel in virtue of the definition of p1, p2, ... being omitted. Since δp1, δp2, ..., δq1, δq2, ... may be taken to be independent, we inferq˙1=∂H, q˙2=∂H, ...,∂p1∂p2(10)and∂(T − V) = −∂H,∂(T − V) = −∂H, ....∂q1∂q1∂q2∂q2(11)It follows from (11) thatṗ1= −∂H, ṗ2= −∂H, ....∂q1∂q2(12)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.Acyclicorgyrostaticsystem 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 χ, χ′, χ″, ..., provided the remaining co-ordinates q1, q2, ... qmand the velocities, including of course the velocities χ˙, χ˙′, χ˙″, ..., are unaltered. Secondly, there are no forces acting on the system of the types χ, χ′, χ″, .... This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates χ, χ′, χ″, ... 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 χ, χ′, χ″, ... 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 oflatent motionsin the ultimate constituents of matter. The general theory of such systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F. Helmholtz.If we suppose the kinetic energy Τ to be expressed, as in Lagrange’s method, in terms of the co-ordinates andRouth’s equations.the velocities, the equations of motion corresponding to χ, χ′, χ″, ... reduce, in virtue of the above hypotheses, to the formsd∂Τ= 0,d∂Τ= 0,d∂Τ= 0, ...,dt∂χ˙dt∂χ˙′dt∂χ˙″(1)whence∂Τ= κ,∂Τ= κ′,∂Τ= κ″, ...,∂χ˙∂χ˙′∂χ˙″(2)where κ, κ′, κ″, ... are the constant momenta corresponding to the cyclic co-ordinates χ, χ′, χ″, .... These equations are linear in χ˙, χ˙′, χ˙″, ...; 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, ... qm. The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates q1, q2, ... qmmay be called (for distinction) thepalpableco-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.If, as in § 1 (25), we writeR = T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,(3)and imagine R to be expressed by means of (2) as a quadratic function of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... with coefficients which are in general functions of the co-ordinates q1, q2, ... qm, then, performing the operation δ on both sides, we find∂Rδq˙1+ ... +∂Rδκ + ... +∂Rδq1+ ... =∂Tδq˙1+ ... +∂Tδq1+ ...∂q˙1∂κ∂q1∂q˙1∂q1+∂Tδχ˙ + ... +∂Tδq1+ ... − κδχ˙ − χ˙δκ − ....∂χ˙∂χ1(4)Omitting the terms which cancel by (2), we find∂T=∂R,∂T=∂R, ...,∂q˙1∂q˙1∂q˙2∂q˙2(5)∂T=∂R,∂T=∂R, ...,∂q1∂q1∂q2∂q2(6)χ˙ = −∂R, χ˙′ = −∂R, χ˙″ = −∂R, ...∂κ∂κ′∂κ″(7)Substituting in § 2 (10), we haved∂R−∂R= Q1,d∂R−∂R= Q2, ...dt∂q˙1∂q1dt∂q˙2∂q2(8)These are Routh’s forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.The function R is made up of three parts, thusR = R2, 0+ R1, 1+ R0, 2, ...(9)where R2, 0is a homogeneous quadratic function of q˙1, q˙2, ... q˙m, R0, 2isKelvin’s equations.a homogeneous quadratic function of κ, κ′, κ″, ..., whilst R1, 1consists of products of the velocities q˙1, q˙2, ... q˙minto the momenta κ, κ′, κ″.... Hence from (3) and (7) we haveT = R −(κ∂R+ κ′∂R+ κ″∂R+ ...)= R2, 0− R0, 2.∂κ∂κ′∂κ″(10)If, as in § 1 (30), we write this in the formΤ = ⅋ + K,(11)then (3) may be writtenR = ⅋ − K + β1q˙1+ β2q˙2+ ...,(12)where β1, β2, ... are linear functions of κ, κ′, κ″, ..., sayβr= αrκ + α′rκ′ + α″rκ″ + ...,(13)the coefficients αr, α′r, α″r, ... being in general functions of the co-ordinates q1, q2, ... qm. Evidently βrdenotes that part of the momentum-component ∂R / ∂q˙rwhich is due to the cyclic motions. Nowd∂R=d(∂⅋+ βr)=d∂⅋+∂βrq˙1+∂βrq˙2+ ...,dt∂q˙rdt∂q˙rdt∂q˙r∂q1∂q2(14)∂R=∂⅋−∂K+∂β1q˙1+∂β2q˙2+ ....∂qr∂qr∂qr∂qr∂qr(15)Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the formd∂⅋−∂⅋+ (r, 1) q˙1+ (r, 2) q˙2+ ... + (r, s) q˙s+ ... +∂K= Qr,dt∂q˙r∂qr∂qr(16)where(r, s) =∂βr−∂βs.∂qs∂qr(17)This form is due to Lord Kelvin. When q1, q2, ... qmhave 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 writtenχ˙ =∂K− α1q˙1− α2q˙2− ...,∂κ(18)χ˙′ =∂K− α′1q˙1− α′2q˙2− ...,∂κ′&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˙mrespectively, and add, we findd(⅋ + K) = Q1q˙1+ Q2q˙2+ ...,dt(20)or, in the case of a conservative system⅋ + V + K = const.,(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 κ, κ′, κ″, ... are all zero, (16) reduces tod∂⅋−∂⅋= Qr.dt∂q˙r∂qr(22)The form is the same as in § 2, and the system now behaves, as regards the co-ordinates q1, q2, ... qm, 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, ..., qm), after performing any evolutions, the ignored co-ordinates χ, χ′, χ″, ... will not in general return to their original values.If in Lagrange’s equations § 2 (10) we reverse the sign of the time-element dt, the equations are unaltered. The motion is therefore reversible; that is to say, if as the system is passing through any configuration its velocities q˙1, q˙2, ..., q˙mbe all reversed, it will (if the forces be the same in the same configuration) retrace its former path. But it is important to observe that the statement does not in general hold of a gyrostatic system; the terms of (16), which are linear in q˙1, q˙2, ..., q˙m, change sign with dt, whilst the others do not. Hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities q˙1, q˙2, ..., q˙m. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.Theconditions of equilibriumof a system with latent cyclic motionsKineto-statics.are obtained by putting q˙1= 0, q˙2= 0, ... q˙m= 0 in (16); viz. they areQ1=∂K, Q2=∂K, ...∂q1∂q2(23)These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration (q1, q2, ... qm) to rest in the configuration q1+ δq1, q2+ δq2, ..., qm+ δqm, the work done by the forces must be equal to the increment of the kinetic energy. HenceQ1δq1+ Q2δq2+ ... = δK,(24)which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy 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 toχ˙ =∂K, χ˙′ =∂K, χ˙″ =∂K, ...,∂κ∂κ′∂κ″(25)K may also be expressed as a homogeneous quadratic function of the cyclic velocities χ˙, χ˙′, χ˙″,... Denoting it in this form by Τ0, we haveδ (T0+ K) = 2δK = δ (κχ˙ + κ′χ˙′ + κ″χ˙″ + ...)(26)Performing the variations, and omitting the terms which cancel by (2) and (25), we find∂Τ0= −∂K,∂Τ0= −∂K, ...,∂q1∂q1∂q2∂q2(27)so that the formulae (23) becomeQ1= −∂Τ0, Q2= −∂Τ0, ...∂q1∂q2(28)A simple example is furnished by the top (Mechanics, § 22). The cyclic co-ordinates being ψ, φ, we find2⅋ = Aθ˙², 2K =(μ − ν cos θ)²+ν²,A sin² θC2Τ0= A sin² θψ˙² + C (φ˙ + ψ cos θ)²,(29)whence we may verify that ∂Τ0/ ∂θ = −∂K / ∂θ in accordance with (27). And the condition of equilibrium∂K= −∂V∂θ∂θ(30)gives the condition of steady precession.6.Stability of Steady Motion.The small oscillations of a conservative system about a configuration of equilibrium, and the criterion of stability, are discussed inMechanics, § 23. The question of the stability of given types of motion is more difficult, owing to the want of a sufficiently general, and at the same time precise, definition of what we mean by “stability.” A number of definitions which have been propounded by different writers are examined by F. Klein and A. Sommerfeld in their workÜber die Theorie des Kreisels(1897-1903). Rejecting previous definitions, they base their criterion of stability on the character of the changes produced in thepathof the system by small arbitrary disturbing impulses. If the undisturbed path be thelimiting formof the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for agivenimpulsive disturbance, however small, the deviation of the particle’s position at any time t from the position which it would have occupied in the original motion increases indefinitely with t. Even this criterion, as the writers quoted themselves recognize, is not free from ambiguity unless the phrase “limiting form,” as applied to a path, be strictly defined. It appears, moreover, that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions. Thus a particle moving in a circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number ofconvolutions. Each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable.A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates q1, q2, ... qmall vanish (see § 5). This has been discussed by Routh, Lord Kelvin and Tait, and Poincaré. These writers treat the question, by an extension of Lagrange’s method, as a problem of small oscillations. Whether we adopt the notion of stability which this implies, or take up the position of Klein and Sommerfeld, there is no difficulty in showing that stability is ensured if V + K be a minimum as regards variations of q1, q2, ... qm. The proof is the same as that of Dirichlet for the case of statical stability.We can illustrate this condition from the case of the top, where, in our previous notation,V + K = Mgh cos θ +(μ − νcos θ)²+ν².2A sin² θ2C
(15)
which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity ω. (SeeMechanics, § 13.)
More generally, let us suppose that we have a certain group of co-ordinates χ, χ′, χ″, ... whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components χ˙, χ˙′, χ˙″, ... are maintained constant. The remaining co-ordinates being denoted by q1, q2, ... qn, we may write
2T = ⅋ + T0+ 2(α1q˙1+ α2q˙2+ ...) χ˙ + 2(α′1q˙1+ α′2q˙2+ ...) χ˙′ + ...,
(16)
where ⅋ is a homogeneous quadratic function of the velocities q˙1, q˙2, ... q˙nof the type § 1 (8), whilst Τ0is a homogeneous quadratic function of the velocities χ˙, χ˙′, χ˙″, ... alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (10) of § 2 give n equations of the type
(17)
where
(18)
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 χ, χ′, χ″, ..., yield expressions for the forces which must be applied in order to maintain the velocities χ˙, χ˙′, χ˙″, ... constant; they need not be written down. If we follow the method by which the equation of energy was established in § 2, the equations (17) lead, on taking account of the relations (19), to
(20)
or, in case the forces Qrdepend only on the co-ordinates q1, q2, ... qnand are conservative,
⅋ + V − T0= const.
(21)
The conditions that the equations (17) should be satisfied by zero values of the velocities q˙1, q˙2, ... q˙nare
(22)
or in the case of conservative forces
(23)
i.e.the value of V − Τ0must bestationary.
We may apply this to the case of a system whose configuration relative to axes rotating with constant angular velocity (ω) is defined by means of the n co-ordinates q1, q2, ... qn.Rotating axes.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, ... qn, of the form § 1 (1), we have, by (15)
2⅋ = Σm (ẋ² + ẏ² + z˙²), 2Τ0= ω²Σm (x² + y²),
(24)
(25)
whence
(26)
The conditions of relative equilibrium are given by (23).
It will be noticed that this expression V − T0, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious “centrifugal forces.” The question of stability of relative equilibrium will be noticed later (§ 6).
It should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. From (15) and (14) we find
(27)
This must be equal to the rate at which the forces acting on the system do work, viz. to
ωΣ (xY − yX) + Q1q˙1+ Q2q˙2+ ... + Qnq˙n,
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 q1, q2, ... qnConstrained systems.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, ... qn) = 0, B (t, q1, q2, ... qn) = 0, &c.
(28)
These may be interpreted as introducing partial constraints into a previously free system. The variations δq1, δq2, ... δqnin the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations
(29)
Introducing indeterminate multipliers λ, μ, ..., one for each of these equations, we obtain in the usual manner n equations of the type
(30)
in place of § 2 (10). These equations, together with (28), serve to determine the n co-ordinates q1, q2, ... qnand the m multipliers λ, μ, ....
When t does not occur explicitly in the relations (28) the system is said to beholonomic. 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, ... qn, yet from the circumstances of the problem certain geometrical conditions are imposed on theirvariations, thus
A1δq1+ A2δq2+ ... = 0, B1δq1+ B2δq2+ ... = 0, &c.,
(31)
where the coefficients are functions of q1, q2, ... qnand (possibly) of t. It is assumed that these equations are not integrable as regards the variables q1, q2, ... qn; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. The six 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
(32)
The co-ordinates q1, q2, ... qn, and the indeterminate multipliers λ, μ, ..., 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 callednon-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 § 1 (19). Since the symbol δ now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of § 1 (21) terms of the types
(1)
Since the variations δp1, δp2, ... δq1, δq2, ... may be taken to be independent, we infer the equations § 1 (23) as before, together with
(2)
Hence the Lagrangian equations § 2 (14) transform into
(3)
If we write
H = T` + V,
(4)
so that H denotes thetotal energyof the system, supposed expressed in terms of the new variables, we get
(5)
If to these we join the equations
(6)
which follow at once from § 1 (23), since V does not involve p1, p2, ..., we obtain a complete system of differential equationsof the first orderfor the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since these make
(7)
The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write
H = p1q˙1+ p2q˙2+ ... − T + 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 δ on both sides, we find
(9)
terms which cancel in virtue of the definition of p1, p2, ... being omitted. Since δp1, δp2, ..., δq1, δq2, ... may be taken to be independent, we infer
(10)
and
(11)
It follows from (11) that
(12)
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.
Acyclicorgyrostaticsystem 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 χ, χ′, χ″, ..., provided the remaining co-ordinates q1, q2, ... qmand the velocities, including of course the velocities χ˙, χ˙′, χ˙″, ..., are unaltered. Secondly, there are no forces acting on the system of the types χ, χ′, χ″, .... This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates χ, χ′, χ″, ... 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 χ, χ′, χ″, ... 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 oflatent motionsin the ultimate constituents of matter. The general theory of such systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F. Helmholtz.
If we suppose the kinetic energy Τ to be expressed, as in Lagrange’s method, in terms of the co-ordinates andRouth’s equations.the velocities, the equations of motion corresponding to χ, χ′, χ″, ... reduce, in virtue of the above hypotheses, to the forms
(1)
whence
(2)
where κ, κ′, κ″, ... are the constant momenta corresponding to the cyclic co-ordinates χ, χ′, χ″, .... These equations are linear in χ˙, χ˙′, χ˙″, ...; 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, ... qm. The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates q1, q2, ... qmmay be called (for distinction) thepalpableco-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.
If, as in § 1 (25), we write
R = T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,
(3)
and imagine R to be expressed by means of (2) as a quadratic function of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... with coefficients which are in general functions of the co-ordinates q1, q2, ... qm, then, performing the operation δ on both sides, we find
(4)
Omitting the terms which cancel by (2), we find
(5)
(6)
(7)
Substituting in § 2 (10), we have
(8)
These are Routh’s forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.
The function R is made up of three parts, thus
R = R2, 0+ R1, 1+ R0, 2, ...
(9)
where R2, 0is a homogeneous quadratic function of q˙1, q˙2, ... q˙m, R0, 2isKelvin’s equations.a homogeneous quadratic function of κ, κ′, κ″, ..., whilst R1, 1consists of products of the velocities q˙1, q˙2, ... q˙minto the momenta κ, κ′, κ″.... Hence from (3) and (7) we have
(10)
If, as in § 1 (30), we write this in the form
Τ = ⅋ + K,
(11)
then (3) may be written
R = ⅋ − K + β1q˙1+ β2q˙2+ ...,
(12)
where β1, β2, ... are linear functions of κ, κ′, κ″, ..., say
βr= αrκ + α′rκ′ + α″rκ″ + ...,
(13)
the coefficients αr, α′r, α″r, ... being in general functions of the co-ordinates q1, q2, ... qm. Evidently βrdenotes that part of the momentum-component ∂R / ∂q˙rwhich is due to the cyclic motions. Now
(14)
(15)
Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form
(16)
where
(17)
This form is due to Lord Kelvin. When q1, q2, ... qmhave 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
(18)
&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˙mrespectively, and add, we find
(20)
or, in the case of a conservative system
⅋ + V + K = const.,
(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 κ, κ′, κ″, ... are all zero, (16) reduces to
(22)
The form is the same as in § 2, and the system now behaves, as regards the co-ordinates q1, q2, ... qm, 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, ..., qm), after performing any evolutions, the ignored co-ordinates χ, χ′, χ″, ... will not in general return to their original values.
If in Lagrange’s equations § 2 (10) we reverse the sign of the time-element dt, the equations are unaltered. The motion is therefore reversible; that is to say, if as the system is passing through any configuration its velocities q˙1, q˙2, ..., q˙mbe all reversed, it will (if the forces be the same in the same configuration) retrace its former path. But it is important to observe that the statement does not in general hold of a gyrostatic system; the terms of (16), which are linear in q˙1, q˙2, ..., q˙m, change sign with dt, whilst the others do not. Hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities q˙1, q˙2, ..., q˙m. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.
Theconditions of equilibriumof a system with latent cyclic motionsKineto-statics.are obtained by putting q˙1= 0, q˙2= 0, ... q˙m= 0 in (16); viz. they are
(23)
These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration (q1, q2, ... qm) to rest in the configuration q1+ δq1, q2+ δq2, ..., qm+ δqm, the work done by the forces must be equal to the increment of the kinetic energy. Hence
Q1δq1+ Q2δq2+ ... = δK,
(24)
which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy 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
(25)
K may also be expressed as a homogeneous quadratic function of the cyclic velocities χ˙, χ˙′, χ˙″,... Denoting it in this form by Τ0, we have
δ (T0+ K) = 2δK = δ (κχ˙ + κ′χ˙′ + κ″χ˙″ + ...)
(26)
Performing the variations, and omitting the terms which cancel by (2) and (25), we find
(27)
so that the formulae (23) become
(28)
A simple example is furnished by the top (Mechanics, § 22). The cyclic co-ordinates being ψ, φ, we find
2Τ0= A sin² θψ˙² + C (φ˙ + ψ cos θ)²,
(29)
whence we may verify that ∂Τ0/ ∂θ = −∂K / ∂θ in accordance with (27). And the condition of equilibrium
(30)
gives the condition of steady precession.
6.Stability of Steady Motion.
The small oscillations of a conservative system about a configuration of equilibrium, and the criterion of stability, are discussed inMechanics, § 23. The question of the stability of given types of motion is more difficult, owing to the want of a sufficiently general, and at the same time precise, definition of what we mean by “stability.” A number of definitions which have been propounded by different writers are examined by F. Klein and A. Sommerfeld in their workÜber die Theorie des Kreisels(1897-1903). Rejecting previous definitions, they base their criterion of stability on the character of the changes produced in thepathof the system by small arbitrary disturbing impulses. If the undisturbed path be thelimiting formof the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for agivenimpulsive disturbance, however small, the deviation of the particle’s position at any time t from the position which it would have occupied in the original motion increases indefinitely with t. Even this criterion, as the writers quoted themselves recognize, is not free from ambiguity unless the phrase “limiting form,” as applied to a path, be strictly defined. It appears, moreover, that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions. Thus a particle moving in a circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number ofconvolutions. Each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable.
A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates q1, q2, ... qmall vanish (see § 5). This has been discussed by Routh, Lord Kelvin and Tait, and Poincaré. These writers treat the question, by an extension of Lagrange’s method, as a problem of small oscillations. Whether we adopt the notion of stability which this implies, or take up the position of Klein and Sommerfeld, there is no difficulty in showing that stability is ensured if V + K be a minimum as regards variations of q1, q2, ... qm. The proof is the same as that of Dirichlet for the case of statical stability.
We can illustrate this condition from the case of the top, where, in our previous notation,