Page:EB1911 - Volume 17.djvu/1010

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

d/dt (T + V) = 0, or T + V = const.,

(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 δr along OP, rδθ 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, rθ. , r sin θ ψ. , and if m be the mass of a moving particle at P we have

2T = m (ṙ² + r ²θ. ² + r ² sin² θ ψ. ²).

(21)

Hence the formula (11) gives

m (r̈ − rθ. 2 − r sin2 θψ. 2)	= R,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$
d/dt (mr 2θ. ) − mr 2 · sin θ cos θψ. 2	= Θ,
d/dt (mr 2 sin2 θψ. )	= Ψ.

(22)

The quantities R, Θ, Ψ are the coefficients in the expression Rδr + Θδθ + Ψδψ 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

θ..  − sin θ cos θψ. 2	= − (g/l) sin θ,
sin2 θψ.	= h,

(23)

where h is a constant. The latter equation expresses that the angular momentum ml² sin² θψ.  about the vertical OZ is constant. By elimination of ψ.  we obtain

θ..  − h2 cos2 θ / sin3 θ = −	g	sin θ.
	l

(24)

If the particle describes a horizontal circle of angular radius α with constant angular velocity Ω, we have ω.  = 0, h = Ω² sin α, and therefore

Ω2 =	g	cos 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,

χ..  + (1 + 3 cos² α) Ω²χ = 0;

(26)

this shows that the variation of χ is simple-harmonic, with the period

2π / √(1 + 3 cos² α)·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 velocity 2/3 abΩ/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

λ = 1/∂T	= Aθ. ,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
μ = 1/∂T	= A sin2 θψ.  + C (φ.  + cos θψ. ) cos θ,
ν = 1/∂T	= C (θ.  + cos θψ. ).

(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

Aθ..  − A sin θ cos θψ. 2 + C (φ.  + cos θψ. ) ψ.  sin θ = Mgh sin θ,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
d/dt · { A sin2 θψ.  + C(φ.  + cos θψ. ) cos θ } = 0,
d/dt · { C (φ.  + cos θψ. ) } = 0,

(28)

of which the last two express the constancy of the momenta μ, ν. Hence

Aθ..  − A sin θ cos θψ. ² + ν sin θψ.  = Mgh sin θ, A sin² θψ.  + ν cosθ = μ.

(29)

If we eliminate ψ.  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

θ..  − θω. 2 = −ν2 − 4AMgh/4A2	θ,	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
θ2ω.  = const.,

(30)

where

ω = ψ −	ν	t.
	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.”

§ 23. Stability of Equilibrium. Theory of Vibrations.—If, in a conservative system, the configuration (q₁, q₂, . . . q_n) be one of equilibrium, the equations (14) of § 22 must be satisfied by q̇₁, q̇₂ . . . q̇_n = 0, whence

∂V / ∂q_r = 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 is necessary. 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, . . . q̇_n differ 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, . . . q_n that 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 + . . . + 22c₁₂q₁q₂ + . . .,

(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 c_rr, c_rs are called coefficients of stability. We may further treat the coefficients of inertia a_rr, a_rs of § 22 (1) as constants. The Lagrangian equations of motion are then of the type

a_1rq̈₁ + a_2rq̈₂ + . . . + a_nrq̈_n + c_1rq₁ + c_2rq₂ + . . . + c_nrq_n = Q_r,

(3)

where Q_r now stands for a component of extraneous force. In a free oscillation we have Q₁, Q₂, . . . Q_n = 0, and if we assume

q_r = A_r eiσ^t,

(4)

we obtain n equations of the type

(c1r − σ2a1r) A1 + (c2r − σ2a2r) A2 + . . . + (cnr − σ2anr) An = 0.

(5)