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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1999.
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\centerline{\Largebf ON THE APPLICATION TO DYNAMICS}
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\centerline{\Largebf OF A GENERAL MATHEMATICAL METHOD}
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\centerline{\Largebf PREVIOUSLY APPLIED TO OPTICS}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (British Association Report, Edinburgh 1834,
pp.\ 513--518.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 1999}
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\noindent
{\largeit On the Application to Dynamics of a General
Mathematical Method previously applied to Optics.
By {\largesc W.~R. Hamilton}, M.R.I.A., Astronomer Royal for
Ireland.\par}
\bigskip
\centerline{[{\it Report of the Fourth Meeting of the
British Association for the Advancement of}}
\centerline{{\it Science; held at Edinburgh in 1834.}
(John Murray, London, 1835), pp.\ 513--518.]}
\bigbreak
The method is founded on a combination of the principles of
variations with those of partial differentials, and may suggest
to analysts a separate branch of algebra, which may be called,
perhaps, the {\it Calculus of Principal Functions\/}; because, in
all the chief applications of algebra to physics, and in a very
extensive class of purely mathematical questions, it reduces the
determination of many mutually connected functions to the search
and study of one principal or central relation. In applying this
method to Dynamics, (having previously applied it to Optics,)
Professor Hamilton has discovered the existence of a principal
function, which, if its form were fully known, would give, by its
partial differential coefficients, all the intermediate and all
the final integrals of the known equations of motion.
Professor Hamilton is of opinion that the mathematical
explanation of all the ph{\ae}nomena of matter distinct from the
ph{\ae}nomena of life, will ultimately be found to depend on the
properties of systems of attracting and repelling points. And he
thinks that those who do not adopt this opinion in all its extent,
must yet admit the properties of such systems to be more highly
important in the present state of science, than any other part of
the application of mathematics to physics. He therefore accounts
it the capital problem of Dynamics ``to determine the $3n$
rectangular coordinates, or other marks of position, of a free
system of $n$ attracting or repelling points as functions of the
time,'' involving also $6n$ initial constants, which depend on the
initial circumstances of the motion, and involving besides, $n$
other constants called the masses, which measure, for a standard
distance, the attractive or repulsive energies.
Denoting these $n$ masses by $m_1 \, m_2 \,\ldots\, m_n$ and their
$3n$ rectangular coordinates by
$x_1 \, y_1 \, z_1 \,\ldots\, x_n \, y_n \, z_n$,
and also the $3n$ component accelerations, or second differential
coefficients of these coordinates, taken with respect to the time, by
$x_1'' \, y_1'' \, z_1'' \,\ldots\, x_n'' \, y_n'' \, z_n''$,
he adopts {\sc Lagrange}'s statement of this problem; namely, a
formula of the following kind,
$$\Sigma . m ( x'' \, \delta x + y'' \, \delta y
+ z'' \, \delta z )
= \delta U,
\eqno {\rm (1.)}$$
in which $U$ is the sum of the products of the masses, taken two
by two, and then multiplied by each other and by certain
functions of their mutual distances, such that their first derived
functions express the laws of their mutual repulsion, being
negative in the case of attraction. Thus, for the solar system,
each product of two masses is to be multiplied by the reciprocal
of their distance and the results are to be added in order to
compose the function $U$.
Mr.~Hamilton next multiplies this formula of Lagrange by the
element of the time $dt$, and integrates from the time $0$ to the
time $t$, considering the time and its element as not subject at
present to the variation $\delta$. He denotes the initial
values, or values at the time $0$, of the coordinates $x$ $y$ $z$,
and of their first differential coefficients $x'$ $y'$ $z'$,
by $a$ $b$ $c$ and $a'$ $b'$ $c'$; and thus he obtains, from
Lagrange's formula (1.), this other important formula,
$$\Sigma . m ( x' \, \delta x - a' \, \delta a
+ y' \, \delta y - b' \, \delta b
+ z' \, \delta z - c' \, \delta c )
= \delta S,
\eqno {\rm (2.)}$$
$S$ being the definite integral
$$S = \int_0^t \left\{ U + \Sigma . {m \over 2}
( x'^2 + y'^2 + z'^2 ) \right\} \, dt.
\eqno {\rm (3.)}$$
If the known equations of motion, of the forms
$$m_i x_i'' = {\delta U \over \delta x_i},\quad
m_i y_i'' = {\delta U \over \delta y_i},\quad
m_i z_i'' = {\delta U \over \delta z_i},
\eqno {\rm (4.)}$$
had been completely integrated, they would give the $3n$
coordinates $x$ $y$ $z$, and therefore also $S$, as a function of
the time $t$, the masses $m_1 \,\ldots \, m_n$ and the $6n$ initial
constants $a$ $b$ $c$ $a'$ $b'$ $c'$; so that, by
eliminating the $3n$ initial components of velocities $a'$ $b'$ $c'$
we should in general obtain a relation between the $7n + 2$
quantities $S$, $t$, $m$, $x$, $y$, $z$, $a$, $b$, $c$, which
would give $S$ as a function of the time, the masses, and the final
and initial coordinates. We do not yet know the form of this
last function, but we know its variation (2.), taken with respect
to the $6n$ coordinates; and on account of the independence of
their $6n$ variations, we can resolve this expression (2.) into
two groups, containing each $3n$ equations: namely
$${\delta S \over \delta x_i} = m_i x_i',\quad
{\delta S \over \delta y_i} = m_i y_i',\quad
{\delta S \over \delta z_i} = m_i z_i',
\eqno {\rm (5.)}$$
and
$${\delta S \over \delta a_i} = - m_i a_i',\quad
{\delta S \over \delta b_i} = - m_i b_i',\quad
{\delta S \over \delta c_i} = - m_i c_i';
\eqno {\rm (6.)}$$
the first members being partial differential coefficients of the
function $S$, which Mr.~Hamilton calls the {\it Principal
Function\/} of motion of the attracting or repelling system. He
thinks that if analysts had perceived this principal function
$S$, and these groups of equations (5.) and (6.), they must have
preceived their importance. For the group (5.) expresses the
$3n$ intermediate integrals of the known equations of motion (4.)
under the form of $3n$ relations between the time~$t$, the
masses~$m$, the varying coordinates $x$,$ y$, $z$, the varying
components of velocities $x'$ $y'$ $z'$, and the $3n$
initial constants $a$ $b$ $c$; while the group (6.) expresses
the $3n$ final integrals of the same known differential equations,
as $3n$ relations, with $6n$ initial and arbitrary constants
$a$ $b$ $c$ $a'$ $b'$ $c'$,
between the time, the masses, and the $3n$ varying coordinates.
These $3n$ intermediate and $3n$ final integrals, it was the
problem of dynamics to discover. Mathematicians had found seven
intermediate, and none of the final integrals.
Professor Hamilton's solution of this long celebrated problem
contains, indeed, one unknown function, namely, the
{\it principal function\/} $S$, to the search and study of which
he has reduced mathematical dynamics. This function must not be
confounded with that so beautifully conceived by Lagrange for the
more simple and elegant expression of the known differential
equations. Lagrange's function {\it states}, Mr.~Hamilton's
function would {\it solve\/} the problem. The one serves to form
the {\it differential\/} equations of motion, the other would
give their {\it integrals}. To assist in pursuing this new track,
and in discovering the form of this new function, Mr.~Hamilton
remarks that it must satisfy the following partial differential
equation of the first order and second degree, (the time being
now made to vary,)
$${\delta S \over \delta t} + \Sigma . {1 \over 2m}
\left\{ \left( {\delta S \over \delta x} \right)^2
+ \left( {\delta S \over \delta y} \right)^2
+ \left( {\delta S \over \delta z} \right)^2
\right\}
= U;
\eqno {\rm (7.)}$$
which may rigorously be thus transformed, by the help of the
equations (5.),
$$\eqalignno{
S = S_1
&+ \int_0^t \left( U - {\delta S_1 \over \delta t}
- \Sigma . {1 \over 2m}
\left\{ \left( {\delta S_1 \over \delta x} \right)^2
+ \left( {\delta S_1 \over \delta y} \right)^2
+ \left( {\delta S_1 \over \delta z} \right)^2
\right\}
\right) \, dt \cr
&+ \int_0^t \Sigma . {1 \over 2m} \left\{
\left( {\delta S \over \delta x}
- {\delta S_1 \over \delta x} \right)^2
+ \left( {\delta S \over \delta y}
- {\delta S_1 \over \delta y} \right)^2
+ \left( {\delta S \over \delta z}
- {\delta S_1 \over \delta z} \right)^2
\right\} \, dt,
&{\rm (8.)}\cr}$$
$S_1$ being any arbitrary function of the same quantities
$t$, $m$, $x$, $y$, $z$, $a$, $b$, $c$,
supposed only to vanish (like $S$) at the origin of time. If
this arbitrary function $S_1$ be so chosen as to be an
approximate value of the sought function $S$, (and it is always
easy so to choose it,) then the two definite integrals in the
formula (8.) are small, but the second is in general much smaller
than the first; it may, therefore, be neglected in passing to a
second approximation, and in calculating the first definite
integral, the following appproximate forms of the equations (6.)
may be used,
$${\delta S_1 \over \delta a} = - m a',\quad
{\delta S_1 \over \delta b} = - m b',\quad
{\delta S_1 \over \delta c} = - m c'.
\eqno {\rm (9.)}$$
In this manner, a first approximation may be successively and
indefinitely corrected. And for the practical perfection of the
method nothing further seems to be required, except to make this
process of correction more easy and rapid in its applications.
Professor Hamilton has written two Essays on this new method in
Dynamics, and one of them is already printed in the second part
of the {\it Philosophical Transactions\/} (of London) for 1834.
The method did not at first present itself to him under quite so
simple a form. He used at first a {\it Characteristic Function}
$V$, more closely analogous to that optical function which he had
discovered, and denoted by the same letter, in his {\it Theory of
Systems of Rays}. In both optics and dynamics this function was
the quantity called {\it Action}, considered as depending
(chiefly) on the final and initial coordinates. But when this
{\it Action-Function\/} was employed in dynamics, it involved an
auxiliary quantity $H$, namely the known constant part in the
expression of half the living force of a system; and many
troublesome eliminations were required in consequence, which are
avoided by the new form of the method.
Mr.~Hamilton thinks it worth while, however, to point out briefly
a new property of this constant $H$, which suggests a new manner
of expressing the differential and integral equations of motion
of an attracting or repelling system. It is often useful to
express the $3n$ rectangular coordinates
$x_1 \, y_1 \, z_1 \,\ldots \, x_n \, y_n \, z_n$,
as functions of $3n$ other marks of position, which may be thus
denoted, $\eta_1 \, \eta_2 \,\ldots\, \eta_{3n}$; and if $3n$ other
new variables $\varpi_1 \, \varpi_2 \,\ldots\, \varpi_{3n}$, be
introduced, and defined as follows,
$$\varpi_i
= \Sigma . m \left(
x' {\delta x \over \delta \eta_i}
+ y' {\delta y \over \delta \eta_i}
+ z' {\delta z \over \delta \eta_i}
\right),
\eqno {\rm (10.)}$$
it is, in general, possible to express, reciprocally, the $6n$
variables $x$ $y$ $z$ $x'$ $y'$ $z'$ as functions of these
$6n$ new variables $\eta$ $\varpi$; it is, therefore, possible
to express, as such a function, the quantity
$$H = \Sigma . {m \over 2}
( x'^2 + y'^2 + z'^2 ) - U
\eqno {\rm (11.)}$$
under the form
$$H = F(\varpi_1,\ldots\, \varpi_{3n}, \eta_1,\ldots\, \eta_{3n})
- U(\eta_1, \ldots, \eta_{3n}),
\eqno {\rm (12.)}$$
in which the part $F$ is rational, integer and homogeneous of the
second dimension with respect to the variables $\varpi$. Now
Mr.~Hamilton has found that when the quantity $H$ is expressed in
this last way as a function of these $6n$ new variables,
$\eta$ $\varpi$, its variation may be put under this form,
$$\delta H = \Sigma
( \eta' \, \delta \varpi - \varpi' \, \delta \eta),
\eqno {\rm (13.)}$$
$\eta'$ $\varpi'$ denoting the first differential coefficients
of these new variables $\eta$ $\varpi$, considered as functions
of the time. The $3n$ differential equations of motion of the
second order, (4.), between the rectangular coordinates and the
time, for any attracting or repelling system, may therefore be
generally transformed into twice that number of equations of the
first order between these $6n$ variables and the time, of the
forms
$$\eta_i' = {\delta H \over \delta \varpi_i},\quad
\varpi_i' = - {\delta H \over \delta \eta_i}.
\eqno {\rm (14.)}$$
To integrate this system of equations is to assign, from them, $6n$
relations between the time $t$, the $6n$ variables
$\eta_i$ $\varpi_i$, and their $6n$ initial values which may be
called $e_i$ $p_i$. Mr.~Hamilton resolves the problem, under this
more general form, by the same {\it principal function\/} $S$ as
before, regarding it, however, as depending now on the new marks
$\eta$ $e$ of final and initial positions of the various points
of the system. For, putting in this new notation,
$$S = \int_0^t \left( \Sigma .
\varpi {\delta H \over \delta \varpi} - H
\right) \, dt,
\eqno {\rm (15.)}$$
and considering the time as given, he finds now the formula of
variation
$$\delta S = \Sigma
( \omega \, \delta \eta - p \, \delta e),
\eqno {\rm (16.)}$$
and therefore the $6n$ separate equations
$$\varpi_i = {\delta S \over \delta \eta_i},\quad
p_i = - {\delta S \over \delta e_i},\quad
\eqno {\rm (17.)}$$
which are forms for the sought relations.
Professor Hamilton thinks that these two formul{\ae} of
variation, (13.) and (16.), namely
$$\delta H = \Sigma
( \eta' \, \delta \varpi - \varpi' \, \delta \eta),
\eqno {\rm (A.)}$$
and
$$\delta S = \Sigma
( \omega \, \delta \eta - p \, \delta e),
\eqno {\rm (B.)}$$
are worthy of attention, as expressing, under concise and simple
forms, the one the differential and the other the integral
equations of motion, of an attracting or repelling system. They
may be extended to other problems of dynamics, besides this
capital problem. The expression $H$ can always easily be found,
and the function $S$ can be determined with indefinite accuracy
by a method of successive approximation of the kind already
explained.
The properties of his {\it Principal Function\/} are treated of
more fully in his ``Second Essay on a General Method in
Dynamics\footnote*{This essay will be found in the {\it
Philosophical Transactions\/} for 1835.}''; in which he has
introduced several forms of a certain {\it Function of Elements},
connected with the Principal Function, and with each other, and
adapted to questions of perturbation; and has shown that for the
perturbations of a ternary or multiple system with any laws of
attraction or repulsion and with one predominant mass, the
differential equations of the varying elements of {\it all\/} the
smaller masses may be expressed together, and as simply as in the
usual way, by the coefficients of {\it one\/} disturbing
function, (namely, the disturbing part of the whole expression
$H$,) and may be integrated rigorously by a corollary of his
general method.
\bye