which follow at once from § 1 (23) since V does not involve 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)
The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write
(8)
,
and imagine H to be expressed in terms of the momenta
the co-ordinates the time. The internal forces of
the system are assumed to be conservative, with the potential
energy V. Performing the variation on both sides, we find
(9)
,
terms which cancel in virtue of the definition of being omitted, Since may be taken to be independent, we infer
(10)
,
(11)
It follows from (11) that
(12)
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 Provided the remaining co-ordinates and the velocities, including of course the velocities
are unaltered. Secondly, there are no forecs acting
on the system of the types This ease arises, for
example, when the system includes gyrostats which are free to
rotate about their axes, the co-ordinates 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
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 reduce, in virtue of the above hypotheses, to the forms
(1)
,
whence
(2)
,
where are the constant momenta corresponding to the cyclic co-ordinates These equations are linear in ; solving them with respect to these quantities and
Substituting in the remaining ngian equations, we obtain
differential equations to determine the remaining co-ordinates
. The object of the present investigation is to ascertain the general form of the resulting equations: The retained co-
ordinates 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)
,
and imagine to be expressed by means of (2) as a quadratic function of With coefficients which are, in
general functions of the co-ordinates then, performing the operation on both sides, we find
(4)
Omitting the terms which cancel by (2), we find
(5)
(6)
(7)
Substituting in § 2 (10), we have
(8)
These are Routh’s forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.
The function is made up of three parts, thus
(9)
where is a homogeneous quadratic function of
is
a homogeneous quadratic function of whilst
Kelvin's equations. consists of products of the velocities
into the momenta Hence from (3) and (7)
we have
(10)
If, as in § 1 (30), we write this in the form
(11)
then (3) may be written
(12)
where are linear functions of Say
(13)
the coefficients being in general functions of the coordinates Evident denotes that part of the momenutum-component , which is due to the cyclic motions
Now
(14)
(15)
Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form
(16)
where
(17)
This form is due to Lord Kelvin. When 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)
It is to be particularly noticed that
(19)
,
Hence, if in (16) we put , and multiply by
respectively, and add, we find
(20)
or, in the case of a conservative system
(21)
,
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 all zero, (16) reduces to
(22)
,
The form is the same as in § 2, and the system now behaves,
as regards the co-ordinates exactly like the acyclic
type there contemplated. These co-ordinates do not, however,