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570 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

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

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

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

It follows from (11) that

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

whence

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

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

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

Substituting in § 2 (10), we have

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

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 $$\Theta_{1,1}$$ 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

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

then (3) may be written

where $$\beta_1, \beta_2, \cdots$$ are linear functions of $$\kappa, \kappa', \kappa'', \cdots,$$ Say

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

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

where

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

{{c|$$\left. \begin{aligned}\dot{\chi} = \frac{\partial \Kappa}{\partial \kappa} - \alpha_1 \dot{q}_1 - \alpha_2 \dot{q}_2 - \dots, \\ \dot{\chi}' = \frac{\partial \Kappa}{\partial \kappa'} - \alpha'_1 \dot{q}_1 - \alpha'_2 \dot{q}_2 - \dots, \\ \&c., \&c.\end{aligned} \right \} $$}}

It is to be particularly noticed that

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

or, in the case of a conservative system

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

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,