(4)
If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).
If in (2) we put, L, M, N = O we get the case of free rotation; thus
(5)
These equations are due to Euler, with whom the conception of moving axes, and the application to the problem of free rotation, originated. If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2+ Bq2+ Cr2and A2p2+ B2q2+ C2r2are both constant. The former is, in fact, equal to 2T, and the latter to Γ2, where T is the kinetic energy and Γ the resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we writeu =∫φ0dφ, Δφ = √(1 − k2sin2φ),Δφwe have, in the notation of elliptic functions, φ = am u. If we assumep = p0cos am (σt + ε), q = q0sin am (σt + ε), r = r0Δ am (σt + ε),(7)we findṗ = −σp0qr, q̇ =σq0rp, ṙ =k2σr0pq.q0r0r0p0p0q0(8)Hence (5) will be satisfied, provided−σp0=B − C,σq0=C − A,−k2σr0=A − B.q0r0Ar0p0Bp0q0C(9)These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have ε = 0, and p0, r0will become identical with the initial values of p, r. The conditions (9) then lead toq02=A (A − C)p02, σ2=(A − C) (B − C)r02, k2=A (A − B)·p02.B (B − C)ABC (B − C)r02(10)For a real solution we must have k2< 1, which is equivalent to 2BT > Γ2. If the initial conditions are such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever φ increases by 2π,i.e.whenever t increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the modulus k.The elliptic functions degenerate into simpler forms when k2= 0 or k2= 1. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2= 1, we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write
we have, in the notation of elliptic functions, φ = am u. If we assume
p = p0cos am (σt + ε), q = q0sin am (σt + ε), r = r0Δ am (σt + ε),
(7)
we find
(8)
Hence (5) will be satisfied, provided
(9)
These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have ε = 0, and p0, r0will become identical with the initial values of p, r. The conditions (9) then lead to
(10)
For a real solution we must have k2< 1, which is equivalent to 2BT > Γ2. If the initial conditions are such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever φ increases by 2π,i.e.whenever t increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the modulus k.
The elliptic functions degenerate into simpler forms when k2= 0 or k2= 1. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2= 1, we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.
When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are
(11)
(12)
To prove these, we may take fixed axes O′x′, O′y′, O′z′ coincident with the moving axes at time t, and compare the linear and angular momenta ξ + δξ, η + δη, ζ + δζ, λ + δλ, μ + δμ, ν + δν relative to the new position of the axes, Ox, Oy, Oz at time t + δt with the original momenta ξ, η, ζ, λ, μ, ν relative to O′x′, O′y′, O′z′ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the mass-centre, we have ξ, η, ζ = M0u, M0v, M0w, where M0is the total mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point O; the angular co-ordinates θ, φ, ψ of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are
p = θ̇ sin φ − sin θ cos φψ̇, q = θ̇ cos φ + sin θ sin φψ̇,r = φ̇ + cos θψ̇,
(13)
by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid, and if A, B, C denote the corresponding moments of inertia, the kinetic energy is given by
2T = A (θ̇ sin φ − sin θ cos φψ̇)2+ B (θ̇ cos φ + sin θ sin θψ)2+ C (φ̇ + cos θψ̇)2.
(14)
If A = B this reduces to
2T = A (θ̇2+ sin2θ ψ̇2) + C (φ̇ + cos θ ψ̇)2;
(15)
cf. § 20 (1).
§ 22.Equations of Motion in Generalized Co-ordinates.—Suppose we have a dynamical system composed of a finite number of material particles or rigid bodies, whether free or constrained in any way, which are subject to mutual forces and also to the action of any given extraneous forces. The configuration of such a system can be completely specified by means of a certain number (n) of independent quantities, called the generalized co-ordinates of the system. These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number ofdegrees of freedomof the system. We denote these co-ordinates by q1, q2, ... qn. It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the q’s, varying in form (of course) from particle to particle. Hence the kinetic energy T is given by
(1)
where
(2)
Thus T is expressed as a homogeneous quadratic function of the quantities q̇1, q̇2, ... q̇n, which are called thegeneralized components of velocity. The coefficients arr, arsare called the coefficients of inertia; they are not in general constants, being functions of the q’s and so variable with the configuration. Again, If (X, Y, Z) be the force on m, the work done in an infinitesimal change of configuration is
Σ (Xδx + Yδy + Zδz) = Q1δq1+ Q2δq2+ ... + Qnδqn,
(3)
where
(4)
The quantities Qrare called thegeneralized components of force.
The equations of motion of m being
mẍ = X, mÿ = Y, mz̈ = Z,
(5)
we have
(6)
Now
(7)
whence
(8)
Also
(9)
Hence
(10)
By these and the similar transformations relating to y and z the equation (6) takes the form
(11)
If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange’s own proof will be found underDynamics, §Analytical. In a conservative system free from extraneous force we have
Σ (X δx + Y δy + Z δz) = −δV,
(12)
where V is the potential energy. Hence
(13)
and
(14)
If we imagine any given state of motion (q̇1, q̇2... q̇n) through the configuration (q1, q2, ... qn) to be generated instantaneously from rest by the action of suitable impulsive forces, we find on integrating (11) with respect to t over the infinitely short duration of the impulse
(15)
where Qr′ is the time integral of Qrand so represents ageneralized component of impulse. By an obvious analogy, the expressions ∂T/∂q̇rmay be called thegeneralized components of momentum; they are usually denoted by prthus
pr= ∂T / ∂q̇r= a1rq̇1+ a2rq̇2+ ... + anrq̇n.
(16)
Since T is a homogeneous quadratic function of the velocities q̇1, q̇2, ... q̇n, we have
(17)
Hence
(18)
or
(19)
This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy
(20)
in virtue of (13).
As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving particle at P we have2T = m (ṙ2+ r2θ;̇2+ r2sin2θψ;̇2).(21)Hence the formula (11) givesm (r̈ − rθ̇2− r sin2θψ̇2)= R,d/dt (mr2θ̇) − mr2· sin θ cos θψ̇2= Θ,d/dt (mr2sin2θψ̇)= Ψ.(22)The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ + Ψ δψ for the work done in an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and thereforeθ̈ − sin θ cos θψ̇2= − (g/l) sin θ,sin2θψ̇= h,(23)where h is a constant. The latter equation expresses that the angular momentum ml2sin2θψ̇ about the vertical OZ is constant. By elimination of ψ̇ we obtainθ̈ − h2cos2θ / sin3θ = −gsin θ.l(24)If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we have ω̇ = 0, h = Ω2sin α, and thereforeΩ2=gcos α,l(25)as is otherwise evident from the elementary theory of uniform circular motion. To investigate the small oscillations about this state of steady motion we write θ = α + χ in (24) and neglect terms of the second order in χ. We find, after some reductions,χ̈ + (1 + 3 cos2α) Ω2χ = 0;(26)this shows that the variation of χ is simple-harmonic, with the period2π / √(1 + 3 cos2α)·ΩAs regards the most general motion of a spherical pendulum, it is obvious that a particle moving under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a vertical plane passing through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path. Hence the path will be confined between two horizontal circles which it touches alternately, and the direction of motion is never horizontal except at these circles. In the case of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a closer investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity2⁄3abΩ/l2, where a, b are the semi-axes.To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle. If λ, μ, ν be the components of momentum, we haveλ = ∂T / ∂θ̇= Aθ̇,μ = ∂T / ∂ψ̇= A sin2θψ̇ + C (φ̇ + cos θψ̇) cos θ,ν = ∂T / ∂φ̇= C (θ̇ + cos θψ̇).(27)The meaning of these quantities is easily recognized; thus λ is the angular momentum about a horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V = Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) becomeAθ̇ − A sin θ cos θψ̇2+ C (φ̇ + cos θψ̇) ψ̇ sin θ = Mgh sin θ,d/dt · { A sin2θψ̇ + C(φ̇ + cos θψ̇) cos θ } = 0,d/dt · { C (φ̇ + cos θψ̇) } = 0,(28)of which the last two express the constancy of the momenta μ, ν. HenceAθ̈ − A sin θ cos θψ̇2+ ν sin θψ̇ = Mgh sin θ,A sin2θψ̇ + ν cosθ = μ.(29)If we eliminate ψ̇ we obtain the equation (7) of § 20. The theory of disturbed precessional motion there outlined does not give a convenient view of the oscillations of the axis about the vertical position. If θ be small the equations (29) may be writtenθ̈ − θω̇2= −ν2− 4AMghθ,4A2θ2ω̇ = const.,(30)whereω = ψ −νt.2A(31)Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14 (15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation ν/2A, provided ν2> 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”
As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving particle at P we have
2T = m (ṙ2+ r2θ;̇2+ r2sin2θψ;̇2).
(21)
Hence the formula (11) gives
(22)
The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ + Ψ δψ for the work done in an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and therefore
(23)
where h is a constant. The latter equation expresses that the angular momentum ml2sin2θψ̇ about the vertical OZ is constant. By elimination of ψ̇ we obtain
(24)
If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we have ω̇ = 0, h = Ω2sin α, and therefore
(25)
as is otherwise evident from the elementary theory of uniform circular motion. To investigate the small oscillations about this state of steady motion we write θ = α + χ in (24) and neglect terms of the second order in χ. We find, after some reductions,
χ̈ + (1 + 3 cos2α) Ω2χ = 0;
(26)
this shows that the variation of χ is simple-harmonic, with the period
2π / √(1 + 3 cos2α)·Ω
As regards the most general motion of a spherical pendulum, it is obvious that a particle moving under gravity on a smooth sphere cannot pass through the highest or lowest point unless it describes a vertical circle. In all other cases there must be an upper and a lower limit to the altitude. Again, a vertical plane passing through O and a point where the motion is horizontal is evidently a plane of symmetry as regards the path. Hence the path will be confined between two horizontal circles which it touches alternately, and the direction of motion is never horizontal except at these circles. In the case of disturbed steady motion, just considered, these circles are nearly coincident. When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in § 13; a closer investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity2⁄3abΩ/l2, where a, b are the semi-axes.
To apply the equations (11) to the case of the top we start with the expression (15) of § 21 for the kinetic energy, the simplified form (1) of § 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle. If λ, μ, ν be the components of momentum, we have
(27)
The meaning of these quantities is easily recognized; thus λ is the angular momentum about a horizontal axis normal to the plane of θ, μ is the angular momentum about the vertical OZ, and ν is the angular momentum about the axis of symmetry. If M be the total mass, the potential energy is V = Mgh cos θ, if OZ be drawn vertically upwards. Hence the equations (11) become
(28)
of which the last two express the constancy of the momenta μ, ν. Hence
Aθ̈ − A sin θ cos θψ̇2+ ν sin θψ̇ = Mgh sin θ,A sin2θψ̇ + ν cosθ = μ.
(29)
If we eliminate ψ̇ we obtain the equation (7) of § 20. The theory of disturbed precessional motion there outlined does not give a convenient view of the oscillations of the axis about the vertical position. If θ be small the equations (29) may be written
θ2ω̇ = const.,
(30)
where
(31)
Since θ, ω are the polar co-ordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves with constant angular velocity ν/2A, we see by comparison with § 14 (15) (16) that the motion of such a point will be elliptic-harmonic superposed on a uniform rotation ν/2A, provided ν2> 4AMgh. This gives (in essentials) the theory of the “gyroscopic pendulum.”
§ 23.Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q1, q2, ... qn) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇1, q̇2... q̇n= 0, whence
∂V / ∂qr= 0.
(1)
A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates. If, further, V be a minimum, the equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet (1846). In the motion consequent on any slight disturbance the total energy T + V is constant, and since T is essentially positive it follows that V can never exceed its equilibrium value by more than a slight amount, depending on the energy of the disturbance. This implies, on the present hypothesis, that there is an upper limit to the deviation of each co-ordinate from its equilibrium value; moreover, this limit diminishes indefinitely with the energy of the original disturbance. No such simple proof is available to show without qualification that the above condition isnecessary. If, however, we recognize the existence of dissipative forces called into play by any motion whatever of the system, the conclusion can be drawn as follows. However slight these forces may be, the total energy T + V must continually diminish so long as the velocities q̇1, q̇2, ... q̇ndiffer from zero. Hence if the system be started from rest in a configuration for which V is less than in the equilibrium configuration considered, this quantity must still further decrease (since T cannot be negative), and it is evident that either the system will finally come to rest in some other equilibrium configuration, or V will in the long run diminish indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).
In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized cc-ordinates q1, q2, ... qnthat they shall vanish in the configuration in question. The potential energy is then given with sufficient approximation by an expression of the form
2V = c11q12+ c22q22+ ... + 2c12q1q2+ ...,
(2)
a constant term being irrelevant, and the terms of the first order being absent since the equilibrium value of V is stationary. The coefficients crr, crsare calledcoefficients of stability. We may further treat the coefficients of inertia arr, arsof § 22 (1) as constants. The Lagrangian equations of motion are then of the type
a1rq̈1+ a2rq̈2+ ... + anrq̈n+ c1rq1+ c2rq2+ ... + cnrqn= Qr,
(3)
where Qrnow stands for a component of extraneous force. In afree oscillationwe have Q1, Q2, ... Qn= 0, and if we assume
qr= Areiσt,
(4)
we obtain n equations of the type
(c1r− σ2a1r) A1+ (c2r− σ2a2r) A2+ ... + (cnr− σ2anr) An= 0.
(5)
Eliminating the n − 1 ratios A1: A2: ... : Anwe obtain the determinantal equation
Δ (σ2) = 0,
(6)
where
(7)
The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability. It may be shown algebraically that under these conditions the n roots of the above equation in σ2are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities A1, A2, ... An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate Δ(σ2). By combining the solutions corresponding to a pair of equal and opposite values of σ we obtain a solution in real form:
qr= Carcos (σt + ε),
(8)
where a1, a2... arare a determinate series ofquantitieshaving to one another the above-mentioned ratios, whilst the constants C, ε are arbitrary. This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the q’s) executes a simple vibration of period 2π/σ. The amplitudes of oscillation of the various particles have definite ratios to one another, and the phases are in agreement, the absolute amplitude (depending on C) and the phase-constant (ε) being alone arbitrary. A vibration of this character is called anormal modeof vibration of the system; the number n of such modes is equal to that of the degrees of freedom possessed by the system. These statements require some modification when two or more of the roots of the equation (6) are equal. In the case of a multiple root the minors of Δ(σ2) all vanish, and the basis for the determination of the quantities ardisappears. Two or more normal modes then become to some extent indeterminate, and elliptic vibrations of the individual particles are possible. An example is furnished by the spherical pendulum (§ 13).
As an example of the method of determination of the normal modes we may take the “double pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs from M by a string of length b. For simplicity we will suppose that the motion is confined to one vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,2T = Ma2θ̇2+ m (aθ̇ + bψ̇)22V = Mgaθ2+ mg (aθ2+ bψ2).(9)The equations (3) take the formsaθ ̈ + μbφ̈ + gθ = 0,aθ ̈ + bφ̈ + gφ = 0.(10)where μ = m/(M + m). Hence(σ2− g/a) aθ + μσ2bφ = 0,σ2aθ + (σ2− g/b) bφ = 0.(11)The frequency equation is therefore(σ2− g/a) (σ2− g/b) − μσ4= 0.(12)The roots of this quadratic in σ2are easily seen to be real and positive. If M be large compared with m, μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the former root, M swings almost like the bob of a simple pendulum of length a, being comparatively uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of σ2can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ is very small. A curious phenomenon is then to be observed; the motion of each particle, being made up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats” in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals. If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are σ2= g/(a + b) and σ2= mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode aθ + bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends and subject to a tension mg.
As an example of the method of determination of the normal modes we may take the “double pendulum.” A mass M hangs from a fixed point by a string of length a, and a second mass m hangs from M by a string of length b. For simplicity we will suppose that the motion is confined to one vertical plane. If θ, φ be the inclinations of the two strings to the vertical, we have, approximately,
(9)
The equations (3) take the forms
(10)
where μ = m/(M + m). Hence
(11)
The frequency equation is therefore
(σ2− g/a) (σ2− g/b) − μσ4= 0.
(12)
The roots of this quadratic in σ2are easily seen to be real and positive. If M be large compared with m, μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the former root, M swings almost like the bob of a simple pendulum of length a, being comparatively uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of σ2can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ is very small. A curious phenomenon is then to be observed; the motion of each particle, being made up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats” in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals. If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are σ2= g/(a + b) and σ2= mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode aθ + bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends and subject to a tension mg.
The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of the n normal modes with suitable amplitudes and phases. We have then
qr= αrθ + αr′θ′ + αr″θ″ + ...,
(13)
where
θ = C cos (σt + ε), θ′ = C′ cos (σ′t + ε), θ″ = C″ cos (σ″t + ε), ...