Page:EB1911 - Volume 17.djvu/1009

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 ≷ Γ² determines 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

dξ	− r η + qζ = X,	dη	− pζ + r χ = Y,	dζ	− qχ + pη = Z,
dt		dt		dt

(11)

dλ	− r μ + qν − wη + vζ = L,	dμ	− pν + r λ- uζ + wξ = M,	dν	− qλ + pμ − vξ + uη = N.
dt		dt		dt

(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 M₀ is 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 = θ.  sin φ − sin θ cos φψ. ,   q = θ.  cos φ + sin θ sin φψ. , r = φ.  + cos θψ. ,

(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 (θ.  sin φ − sin θ cos φψ. )² + B (θ.  cos φ + sin θ sin θψ)² + C (φ.  + cos θψ. )².

(14)

If A = B this reduces to

2T = A (θ. ² + sin² θ ψ. ²) + C (φ.  + cos θ ψ. )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 of degrees of freedom of the system. We denote these co-ordinates by q₁, q₂, . . . q_n. 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

2T	= Σ {m (ẋ2 + ẏ2 + ż2) }
	= a11q̇12 + a22q̇22 + . . . + 2a12q̇1q̇2 + . . .,

(1)

where

arr = Σ [ m { (	∂x	${\Big )}$2 + ${\Big (}$	∂y	${\Big )}$2 + ${\Big (}$	∂z	${\Big )}$2 } ],
	∂qr		∂qr		∂qr

ars = Σ { m ${\Big (}$

∂x

∂x

+

∂y

∂y

+

∂z

∂z

) } = asr.

∂qr

∂qs

∂qr

∂qs

∂qr

∂qs

(2)

Thus T is expressed as a homogeneous quadratic function of the quantities q̇₁, q̇₂, . . . q̇_n, which are called the generalized components of velocity. The coefficients arr, ars are 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 + . . . + Q_nδq_n,

(3)

where

Qr = Σ ${\Big (}$ X	∂x	+ Y	∂y	+ Z	∂z	${\Big )}$.
	∂qr		∂qr		∂qr

(4)

The quantities Q_r are called the generalized components of force.

The equations of motion of m being

mẍ = X,   mÿ = Y,   mz̈ = Z,

(5)

we have

Σ { m ${\Big (}$ ẍ	∂x	+ ÿ	∂y	+ z̈	∂z	) } = Qr.
	∂qr		∂qr		∂qr

(6)

Now

ẋ =	∂x	q̇1 +	∂x	q̇2 + . . . +	∂x	q̇n,
	∂q1		∂q2		∂qn

(7)

whence

∂ẋ/∂q̇r = ∂x/∂qr

(8)

Also

d	${\Big (}$	∂x	${\Big )}$ =	∂2x	q̇1 +	∂2x	q̇2 + . . . +	∂2x	q̇r =	∂ẋ	.
dt		∂qr		∂q1∂qr		∂q2∂qr		∂qn∂qr		∂qr

(9)

Hence

ẍ

∂x

=

d

${\Big (}$ ẋ

∂x

${\Big )}$ − ẋ

d

${\Big (}$

∂x

${\Big )}$ =

d

${\Big (}$ ẋ

∂ẋ

${\Big )}$ − ẋ

∂ẋ

.

∂qr

dt

∂qr

dt

∂qr

dt

∂q̇r

∂qr

(10)

By these and the similar transformations relating to y and z the equation (6) takes the form

d	${\Big (}$	∂T	${\Big )}$ −	∂T	= Qr.
dt		∂q̇r		∂qr

(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 under Dynamics, § 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

Qr = −	∂V	,
	∂qr

(13)

and

d	${\Big (}$	∂T	${\Big )}$ −	∂T	= −	∂V	.
dt		∂q̇r		∂qr		∂qr

(14)

If we imagine any given state of motion (q̇₁, q̇₂ . . . q̇_n) through the configuration (q₁, q₂, . . . q_n) 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

∂T/∂q̇r = Qr′,

(15)

where Q_r′ is the time integral of Q_r and so represents a generalized component of impulse. By an obvious analogy, the expressions ∂T/∂q̇_r may be called the generalized components of momentum; they are usually denoted by p_r thus

p_r = ∂T / ∂q̇_r = a_1rq̇₁ + a_2rq̇₂ + . . . + a_nrq̇_n.

(16)

Since T is a homogeneous quadratic function of the velocities q̇₁, q̇₂, . . . q̇_n, we have

2T =	∂T	q̇1 +	∂T	q̇2 + . . . +	∂T	q̇n = p1q̇2 + p2q̇2 + . . . + pnq̇n.
	∂q̇1		∂q̇2		∂q̇n

(17)

Hence

2dT/dt

= ṗ1q̇1 + ṗ2q̇2 + . . . + ṗnq̇n + ṗ1q̈1 + ṗ2q̈2 + . . . + ṗnq̈n

$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$

= ${\Big (}$

∂T

+ Q1 ${\Big )}$ q̇1 + ${\Big (}$

∂T

+ Q2 ${\Big )}$ q̇2 + . . . + ${\Big (}$

∂T

+ Qn ${\Big )}$ q̇n +

∂T

q̈1 +

∂T

q̈2 + . . . +

∂T

q̈n

∂q̇1

∂q̇2

∂q̇n

∂q̇1

∂q̇2

∂q̇n

=dT/dt + Q1q̇1 + Q2q̇2 + . . . + Qnq̇n,

(18)

or

dT/dt = Q1q̇1 + Q2q̇2 + . . . + Qnq̇n.

(19)