Page:EB1911 - Volume 08.djvu/787

Omitting the terms which cancel by (2), we find

${\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}={\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{1}}},\qquad {\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}={\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{2}}},\ldots ,$

(5)

${\frac {\partial \mathrm {T} }{\partial q_{1}}}={\frac {\partial \mathrm {R} }{\partial q_{1}}},\qquad {\frac {\partial \mathrm {T} }{\partial q_{2}}}={\frac {\partial \mathrm {R} }{\partial q_{2}}},\ldots ,$

(6)

${\dot {\chi }}=-{\frac {\partial \mathrm {R} }{\partial \kappa }},\qquad {\dot {\chi }}'=-{\frac {\partial \mathrm {R} }{\partial \kappa '}},\qquad {\dot {\chi }}''=-{\frac {\partial \mathrm {R} }{\partial \kappa ''}},\ldots$

(7)

Substituting in § 2 (10), we have

${\frac {d}{dt}}\,{\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{1}}}-{\frac {\partial \mathrm {R} }{\partial q_{1}}}=\mathrm {Q} _{1},\qquad {\frac {d}{dt}}\,{\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{2}}}-{\frac {\partial \mathrm {R} }{\partial q_{2}}}=\mathrm {Q} _{2},\ldots$

(8)

These are Routh’s forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.

The function $\mathrm {R} \!$ is made up of three parts, thus

$\mathrm {R} =\mathrm {R} _{2,0}+\mathrm {R} _{1,1}+\mathrm {R} _{0,2},\ldots$

(9)

where $\mathrm {R} _{2,0}\!$ is a homogeneous quadratic function of ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m},\mathrm {R} _{0,2}\!$ is a homogeneous quadratic function of $\kappa ,\kappa ',\kappa '',\ldots \!$, whilst $\mathrm {R} _{1,1}\!$ consists of products of the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$ into the momenta $\kappa ,\kappa ',\kappa ''\ldots \!$. Hence from (3) and (7) we haveKelvin’s equations.

$\mathrm {T} =\mathrm {R} -\left(\kappa {\frac {\partial \mathrm {R} }{\partial \kappa }}+\kappa '{\frac {\partial \mathrm {R} }{\partial \kappa '}}+\kappa ''{\frac {\partial \mathrm {R} }{\partial \kappa ''}}+\ldots \right)=\mathrm {R} _{2,0}-\mathrm {R} _{0,2}.$

(10)

If, as in § 1 (30), we write this in the form

$\mathrm {T} ={\mathfrak {T}}+\mathrm {K} ,$

(11)

then (3) may be written

$\mathrm {R} ={\mathfrak {T}}-\mathrm {K} +\beta _{1}{\dot {q}}_{1}+\beta _{2}{\dot {q}}_{2}+\ldots ,$

(12)

where $\beta _{1},\beta _{2},\ldots \!$ are linear functions of $\kappa ,\kappa ',\kappa '',\ldots \!$, say

$\beta _{r}=\alpha _{r}\kappa +\alpha '_{r}\kappa '+\alpha ''_{r}\kappa ''+\ldots ,$

(13)

the coefficients $\alpha _{r},\alpha '_{r},\alpha ''_{r},\ldots \!$ being in general functions of the co-ordinates $q_{1},q_{2},\ldots q_{m}\!$. Evidently $\beta _{r}\!$ denotes that part of the momentum-component $\partial \mathrm {R} /\partial {\dot {q}}_{r}\!$ which is due to the cyclic motions. Now

${\frac {d}{dt}}\,{\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{r}}}={\frac {d}{dt}}\left({\frac {\partial {\mathfrak {T}}}{\partial {\dot {q}}_{r}}}+\beta _{r}\right)={\frac {d}{dt}}\,{\frac {\partial {\mathfrak {T}}}{\partial {\dot {q}}_{r}}}+{\frac {\partial \beta _{r}}{\partial q_{1}}}{\dot {q}}_{1}+{\frac {\partial \beta _{r}}{\partial q_{2}}}{\dot {q}}_{2}+\ldots ,$

(14)

${\frac {\partial \mathrm {R} }{\partial q_{r}}}={\frac {\partial {\mathfrak {T}}}{\partial q_{r}}}-{\frac {\partial \mathrm {K} }{\partial q_{r}}}+{\frac {\partial \beta _{1}}{\partial q_{r}}}{\dot {q}}_{1}+{\frac {\partial \beta _{2}}{\partial q_{r}}}{\dot {q}}_{2}+\ldots .$

(15)

Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form

${\frac {d}{dt}}\,{\frac {\partial {\mathfrak {T}}}{\partial {\dot {q}}_{r}}}-{\frac {\partial {\mathfrak {T}}}{\partial q_{r}}}+(r,1){\dot {q}}_{1}+(r,2){\dot {q}}_{2}+\ldots +(r,s){\dot {q}}_{s}+\ldots +{\frac {\partial \mathrm {K} }{\partial q_{r}}}=\mathrm {Q} _{r},$

(16)

where

$(r,s)={\frac {\partial \beta _{r}}{\partial q_{s}}}-{\frac {\partial \beta _{s}}{\partial q_{r}}}.$

(17)

This form is due to Lord Kelvin. When $q_{1},q_{2},\ldots q_{m}\!$ have 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

$\left.{\begin{aligned}{\dot {\chi }}&={\frac {\partial \mathrm {K} }{\partial \kappa }}-\alpha _{1}{\dot {q}}_{1}-\alpha _{2}{\dot {q}}_{2}-\ldots ,\\{\dot {\chi }}'&={\frac {\partial \mathrm {K} }{\partial \kappa '}}-\alpha '_{1}{\dot {q}}_{1}-\alpha '_{2}{\dot {q}}_{2}-\ldots ,\\&\qquad \And \!\!\!{\text{c.}},\And \!\!\!{\text{c.}}\end{aligned}}\right\}$

(18)

It is to be particularly noticed that

$(r,r)=0,(r,s)=-(s,r).$

(19)

Hence, if in (16) we put $r=1,2,3,\ldots m\!$, and multiply by ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$ respectively, and add, we find

${\frac {d}{dt}}({\mathfrak {\mathrm {T} }}+\mathrm {K} )=\mathrm {Q} _{1}{\dot {q}}_{1}+\mathrm {Q} _{2}{\dot {q}}_{2}+\ldots ,$

(20)

or, in the case of a conservative system

${\mathfrak {T}}+\mathrm {V} +\mathrm {K} ={\text{const.}},$

(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 $\kappa ,\kappa ',\kappa '',\ldots \!$ are all zero, (16) reduces to

${\frac {d}{dt}}\,{\frac {\partial {\mathfrak {T}}}{\partial {\dot {q}}_{r}}}-{\frac {\partial {\mathfrak {T}}}{\partial q_{r}}}=\mathrm {Q} _{r}.$

(22)

The form is the same as in § 2, and the system now behaves, as regards the co-ordinates $q_{1},q_{2},\ldots q_{m}\!$, 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 $q_{1},q_{2},\ldots ,q_{m})\!$, after performing any evolutions, the ignored co-ordinates $\chi ,\chi ',\chi '',\ldots \!$ 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 ${\dot {q}}_{1},{\dot {q}}_{2},\ldots ,{\dot {q}}_{m}\!$ be 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 ${\dot {q}}_{1},{\dot {q}}_{2},\ldots ,{\dot {q}}_{m}\!$, 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 ${\dot {q}}_{1},{\dot {q}}_{2},\ldots ,{\dot {q}}_{m}\!$. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.

The conditions of equilibrium of a system with latent cyclic motions are obtained by putting ${\dot {q}}_{1}=0,{\dot {q}}_{2}=0,\ldots {\dot {q}}_{m}=0$ in (16); viz. they areKineto-statics.

$\mathrm {Q} _{1}={\frac {\partial \mathrm {K} }{\partial q_{1}}},\qquad \mathrm {Q} _{2}={\frac {\partial \mathrm {K} }{\partial q_{2}}},\ldots$

(23)

These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration $(q_{1},q_{2},\ldots q_{m})$ to rest in the configuration $q_{1}+\delta q_{1},q_{2}+\delta q_{2},\ldots ,q_{m}+\delta q_{m}$, the work done by the forces must be equal to the increment of the kinetic energy. Hence

$\mathrm {Q} _{1}\delta q_{1}+\mathrm {Q} _{2}\delta q_{2}+\ldots =\delta \mathrm {K} ,$

(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 $\mathrm {K} \!$. 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

${\dot {\chi }}={\frac {\partial \mathrm {K} }{\partial \kappa }},\qquad {\dot {\chi }}'={\frac {\partial \mathrm {K} }{\partial \kappa '}},\qquad {\dot {\chi }}''={\frac {\partial \mathrm {K} }{\partial \kappa ''}},\ldots ,$

(25)

$\mathrm {K} \!$ may also be expressed as a homogeneous quadratic function of the cyclic velocities ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$. Denoting it in this form by $\mathrm {T} _{0}\!$, we have

$\delta (\mathrm {T} _{0}+\mathrm {K} )=2\delta \mathrm {K} =\delta (\kappa {\dot {\chi }}+\kappa '{\dot {\chi }}'+\kappa ''{\dot {\chi }}''+\ldots$

(26)

Performing the variations, and omitting the terms which cancel by (2) and (25), we find

${\frac {\partial \mathrm {T} _{0}}{\partial q_{1}}}=-{\frac {\partial \mathrm {K} }{\partial q_{1}}},\qquad {\frac {\partial \mathrm {T} _{0}}{\partial q_{2}}}=-{\frac {\partial \mathrm {K} }{\partial q_{2}}},\ldots ,$

(27)

so that the formulae (23) become

$\mathrm {Q} _{1}=-{\frac {\partial \mathrm {T} _{0}}{\partial q_{1}}},\qquad \mathrm {Q} _{2}=-{\frac {\partial \mathrm {T} _{0}}{\partial q_{2}}},\ldots$

(28)

A simple example is furnished by the top (Mechanics, § 22). The cyclic co-ordinates being $\psi ,\phi \!$, we find

$2{\mathfrak {T}}=\mathrm {A} {\dot {\theta }}^{2},\qquad 2\mathrm {K} ={\frac {(\mu -\nu \cos \theta )^{2}}{\mathrm {A} \sin ^{2}\theta }}+{\frac {\nu ^{2}}{\mathrm {C} }},$

$2\mathrm {T} _{0}=\mathrm {A} \sin ^{2}\theta {\dot {\psi }}^{2}+\mathrm {C} ({\dot {\phi }}+\psi \cos \theta )^{2},$

(29)

whence we may verify that $\partial \mathrm {T} _{0}/\partial \theta =-\partial \mathrm {K} /\partial \theta$ in accordance with (27). And the condition of equilibrium

${\frac {\partial \mathrm {K} }{\partial \theta }}=-{\frac {\partial \mathrm {V} }{\partial \theta }}$

(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 in Mechanics, § 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 work Über die Theorie des Kreisels (1897–1903). Rejecting previous definitions, they base their criterion of stability on the character of the changes produced in the path of the system by small arbitrary disturbing impulses. If the undisturbed path be the limiting form of 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 a given impulsive 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 of