Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/622

which follow at once from § 1 (23) since V does not involve $p_{1},p_{2},\cdots$ we obtain a complete system of differential equations of the first order for the determination of the motion.

The equation of energy is verified immediately by (5) and (6), since these make

(7)

${\frac {dH}{dt}}={\frac {\partial H}{\partial p_{1}}}{\dot {p}}_{1}+{\frac {\partial H}{\partial p_{2}}}{\dot {p}}_{2}+\dots +{\frac {\partial H}{\partial q_{1}}}{\dot {q}}_{1}+{\frac {\partial H}{\partial q_{2}}}{\dot {q}}_{2}+\dots =0$

The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write

(8)

$H=p_{1}{\dot {q}}_{1}+p_{2}{\dot {q}}_{2}+\dots -T+V$,

and imagine H to be expressed in terms of the momenta the co-ordinates $q_{1},q_{2},\cdots ,$ the time. The internal forces of the system are assumed to be conservative, with the potential energy V. Performing the variation $\delta$ on both sides, we find

(9)

$\delta H={\dot {q}}_{1}\delta p_{1}+\dots -{\frac {\partial T}{\partial q_{1}}}\delta q_{1}+{\frac {\partial V}{\partial q_{1}}}\delta q+\dots$,

terms which cancel in virtue of the definition of $p_{1},p_{2}\cdots ,$ being omitted, Since $\delta p_{1},\delta p_{2}\cdots ,\delta q_{1},\delta q_{2},\cdots$ may be taken to be independent, we infer

(10)

${\dot {q}}_{1}={\frac {\partial H}{\partial p_{1}}},{\dot {q}}_{2}={\frac {\partial H}{\partial p_{2}}},\dots$,

(11)

${\frac {\partial }{\partial q_{1}}}(T-V)=-{\frac {\partial H}{\partial q_{1}}},{\frac {\partial }{\partial q_{2}}}(T-V)=-{\frac {\partial H}{\partial q_{2}}},\dots$

It follows from (11) that

(12)

${\dot {p}}_{1}=-{\frac {\partial H}{\partial q_{1}}},{\dot {p}}_{2}=-{\frac {\partial H}{\partial q_{2}}},\dots$

The equations (10) and (12) have the same form as above, but H is no longer equal to the energy of the system.

Cyclic Systems.

§ 5. A cyclic or gyrostatic system 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 $x,x',x'',\cdots$ Provided the remaining co-ordinates $q_{1},q_{2},\cdots q_{m}$ and the velocities, including of course the velocities $x,x',x'',\cdots$ are unaltered. Secondly, there are no forecs acting on the system of the types $\chi ,\chi ',\chi '',\cdots$ This ease arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates $\chi ,\chi ',\chi '',\cdots$ 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 $\chi ,\chi ',\chi '',\cdots$ then refer to the fluid, and are infinite in number. The same question presents itself in various physical specula- tions where certain phenomena are ascribed to the existence of latent motions in the ultimate constituents of matter. The general theory of such systems has been treated by Routh, Lord Kelvin, and Helmholtz.

Routh's Equations.

If we suppose the Kinetic energy T to be expressed, as in Routh'e Yastange’s method, in terms of the co-ordinates and Routh’s the velocities, the equations of motion corresponding to $\chi ,\chi ',\chi '',\cdots$ reduce, in virtue of the above hypotheses, to the forms

(1)

${\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\chi }}}}=0,{\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\chi }}'}}=0,{\frac {d}{dt}}{\frac {\partial T}{\partial {\dot {\chi }}''}}=0,\dots$,

whence

(2)

${\frac {\partial T}{\partial {\dot {\chi }}}}=\kappa ,{\frac {\partial T}{\partial {\dot {\chi }}'}}=\kappa ',{\frac {\partial T}{\partial {\dot {\chi }}''}}=\kappa '',\dots$,

where $\kappa ,\kappa ',\kappa '',\cdots$ are the constant momenta corresponding to the cyclic co-ordinates $\chi ,\chi ',\chi '',\cdots$ These equations are linear in ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\cdots$; solving them with respect to these quantities and Substituting in the remaining ngian equations, we obtain $m$ differential equations to determine the remaining co-ordinates $q_{1},q_{2},\cdots q_{m}$. The object of the present investigation is to ascertain the general form of the resulting equations: The retained co- ordinates $q_{1},q_{2},\cdots q_{m}$ may be called (for distinction) the palpable co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.

If, as in § 1 (25), we write

(3)

$\Theta =T-\kappa {\dot {\chi }}-\kappa '{\dot {\chi }}'-\kappa ''{\dot {\chi }}'-\dots$,

and imagine $\Theta$ to be expressed by means of (2) as a quadratic function of ${\dot {q}}_{1},{\dot {q}}q_{2},\cdots {\dot {q}}q_{m},\kappa ,\kappa ',\kappa '',\cdots$ With coefficients which are, in general functions of the co-ordinates ${\dot {q}}_{1},{\dot {q}}q_{2},\cdots {\dot {q}}q_{m},$ then, performing the operation $\delta$ on both sides, we find

(4)

${\begin{aligned}{\frac {\partial \Theta }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\dots &+{\frac {\partial \Theta }{\partial \kappa }}\delta \kappa +\dots +{\frac {\partial \Theta }{\partial q_{1}}}\delta q_{1}+\dots ={\frac {\partial T}{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\dots +{\frac {\partial T}{\partial q_{1}}}\delta q_{1}+\dots \\&+{\frac {\partial T}{\partial {\dot {\chi }}}}\delta {\dot {\chi }}+\dots +{\frac {\partial T}{\partial {\dot {\chi }}_{1}}}\delta {\dot {q}}_{1}+\dots -\kappa \delta {\dot {\chi }}-{\dot {\chi }}\delta \kappa -\dots \end{aligned}}$

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

(5)

${\frac {\partial T}{\partial {\dot {q}}_{1}}}={\frac {\partial \Theta }{\partial {\dot {q}}_{1}}},{\frac {\partial T}{\partial {\dot {q}}_{2}}}={\frac {\partial \Theta }{\partial {\dot {q}}_{2}}},\dots ,$

(6)

${\frac {\partial T}{\partial q_{1}}}={\frac {\partial \Theta }{\partial q_{1}}},{\frac {\partial T}{\partial q_{2}}}={\frac {\partial \Theta }{\partial q_{2}}},\dots ,$

(7)

${\dot {\chi }}=-{\frac {\partial \Theta }{\partial \delta }},{\dot {\chi }}'=-{\frac {\partial \Theta }{\partial \kappa '}},{\dot {\chi }}''=-{\frac {\partial \Theta }{\partial \kappa ''}},\dots$

Substituting in § 2 (10), we have

(8)

${\frac {d}{dt}}{\frac {\partial \Theta }{\partial {\dot {q}}_{1}}}-{\frac {\partial \Theta }{\partial q_{1}}}=\mathrm {Q} _{1},{\frac {d}{dt}}{\frac {\partial \Theta }{\partial {\dot {q}}_{2}}}-{\frac {\partial \Theta }{\partial q_{2}}}=\mathrm {Q} _{2},\dots$

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

The function $\Theta$ is made up of three parts, thus

(9)

$\Theta =\Theta _{2,0}+\Theta _{1,1}+\Theta _{0,2},$

where $\Theta _{2,0}$ is a homogeneous quadratic function of ${\dot {q}}_{1},{\dot {q}}_{2}\cdots \,{\dot {q}}_{m},\Theta _{2,0}$ is a homogeneous quadratic function of $\kappa ,\kappa ',\kappa '',\cdots ,$ whilst Kelvin's equations. $\Theta _{1,1}$ consists of products of the velocities ${\dot {q}}_{1},{\dot {q}}_{2}\cdots \,{\dot {q}}_{m}$ into the momenta $\kappa ,\kappa ',\kappa '',\cdots$ Hence from (3) and (7) we have

(10)

${\begin{aligned}T&=\Theta -\left(\kappa {\frac {\partial \Theta }{\partial \kappa }}+\kappa '{\frac {\partial \Theta }{\partial \kappa '}}+\kappa ''{\frac {\partial \Theta }{\partial \kappa ''}}+\dots \right)\\&=\Theta _{2,0}-\Theta _{0,2}\end{aligned}}$

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

(11)

$T=\xi +K,$

then (3) may be written

(12)

$\Theta =\xi -K+\beta _{1}{\dot {q}}_{1}+\beta _{2}{\dot {q}}_{2}+\dots$

where $\beta _{1},\beta _{2},\cdots$ are linear functions of $\kappa ,\kappa ',\kappa '',\cdots ,$ Say

(13)

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

the coefficients $a_{r},a'_{r},a''_{r},\cdots$ being in general functions of the coordinates ${\dot {q}}_{1},{\dot {q}}_{2}\cdots \,{\dot {q}}_{m}$ Evident $\beta$ denotes that part of the momenutum-component $\partial \Theta /\partial {\dot {q}}_{r}$, which is due to the cyclic motions

Now

(14)

${\frac {d}{dt}}{\frac {\partial \Theta }{\partial {\dot {q}}_{r}}}={\frac {d}{dt}}\left({\frac {\partial \xi }{\partial {\dot {q}}_{r}}}+\beta _{r}\right)={\frac {d}{dt}}{\frac {\partial \xi }{\partial {\dot {q}}_{r}}}+{\frac {\partial \beta _{r}}{\partial q_{1}}}{\dot {q}}_{1}+{\frac {\partial \beta _{r}}{\partial q_{2}}}{\dot {q}}_{2}+\dots$

(15)

${\frac {\partial \Theta }{\partial q_{r}}}={\frac {\partial \xi }{\partial q_{r}}}-{\frac {\partial K}{\partial q_{r}}}+{\frac {\partial \beta _{1}}{\partial q_{r}}}{\dot {q}}_{1}+{\frac {\partial \beta _{2}}{\partial q_{r}}}{\dot {q}}_{2}+\dots$

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

(16)

${\frac {d}{dt}}{\frac {\partial \xi }{\partial {\dot {q}}_{r}}}-{\frac {\partial \xi }{\partial q_{r}}}+(r,1){\dot {q}}_{1}+(r,2){\dot {q}}_{2}+\dots +(r,s){\dot {q}}_{s}+\dots +{\frac {\partial K}{\partial q_{r}}}=Q_{r},$

where

(17)

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

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

(18)

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

It is to be particularly noticed that

(19)

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

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

(20)

${\frac {d}{dt}}(\xi +\mathrm {K} )=Q_{1}{\dot {q}}_{1}+Q_{2}{\dot {q}}_{2}+\dots$

or, in the case of a conservative system

(21)

$\xi +V+K={\text{const.}}$,

which is the equation of energy.

The equations (16) include § 3 (17) as a eliminated co-ordinate being the angular co-o solid having an infinite moment of inertia.

In the particular case where the cyclic momenta $\kappa ,\kappa ',\kappa ''$ all zero, (16) reduces to

(22)

${\frac {d}{dt}}{\frac {\partial \xi }{\partial {\dot {q}}_{r}}}-{\frac {\partial \xi }{\partial {\dot {q}}_{r}}}=Q_{r}$,

The form is the same as in § 2, and the system now behaves, as regards the co-ordinates $q_{1},q_{2},\cdots q_{m}$ exactly like the acyclic type there contemplated. These co-ordinates do not, however,