​

We shall use Hamilton's form of the equations of motion for a system of ${\displaystyle n}$ degrees of freedom, writing ${\displaystyle q_{1},\ldots q_{n}}$ for the (generalized) coördinates, ${\displaystyle {\dot {q}}_{1},\ldots {\dot {q}}_{n}}$ for the (generalized) velocities, and

${\displaystyle F_{1}\,dq_{1}+F_{2}\,dq_{2}\ldots +F_{n}\,dq_{n}}$

(1)

for the moment of the forces. We shall call the quantities ${\displaystyle F_{1},\ldots F_{n}}$, the (generalized) forces, and the quantities ${\displaystyle p_{1}\ldots p_{n}}$, defined by the equations

${\displaystyle p_{1}={\frac {d\epsilon _{p}}{d{\dot {q}}_{1}}},\quad p_{2}={\frac {d\epsilon _{p}}{d{\dot {q}}_{2}}},\quad \mathrm {etc.,} }$

(2)

where ${\displaystyle \epsilon _{p}}$ denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates,^[1] and on this account we denote it by ${\displaystyle \epsilon _{p}}$. This will not prevent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the ${\displaystyle {\dot {q}}}$'s and ${\displaystyle q}$'s. But in expressions like ${\displaystyle d\epsilon _{p}/dq_{1}}$, where the denominator does not determine the question, the kinetic ​energy is always to be treated in the differentiation as function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s.

We have then

${\displaystyle {\dot {q}}_{1}={\frac {d\epsilon _{p}}{dp_{1}}},\quad {\dot {p}}_{1}=-{\frac {d\epsilon _{p}}{dq_{1}}}+F_{1},\quad \mathrm {etc.} }$

(3)

These equations will hold for any forces whatever. If the forces are conservative, in other words, if the expression (1) is an exact differential, we may set

${\displaystyle F_{1}=-{\frac {d\epsilon _{q}}{dq_{1}}},\quad F_{2}=-{\frac {d\epsilon _{q}}{dq_{2}}},\quad \mathrm {etc.} }$

(4)

where ${\displaystyle \epsilon _{q}}$ is a function of the coördinates which we shall call the potential energy of the system. If we write ${\displaystyle \epsilon }$ for the total energy, we shall have

${\displaystyle \epsilon =\epsilon _{p}+\epsilon _{q}}$

(5)

and equations (3) may be written

${\displaystyle {\dot {q}}_{1}={\frac {d\epsilon }{dp_{1}}},\quad {\dot {p}}_{1}=-{\frac {d\epsilon }{dq_{1}}},\quad \mathrm {etc.} }$

(6)

The potential energy (${\displaystyle \epsilon _{q}}$) may depend on other variables beside the coördinates ${\displaystyle q_{1},\ldots q_{n}}$. We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc. We shall then have for the complete value of the differential of the potential energy^[2]

${\displaystyle d\epsilon _{q}=-F_{1}\,dq_{1}\ldots -F_{n}\,dq_{n}-A_{1}\,da_{1}-A_{2}\,da_{2}-\mathrm {etc.} ,}$

(7)

where ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy (${\displaystyle \epsilon }$) we shall have

${\displaystyle {\begin{aligned}d\epsilon ={\dot {q}}_{1}\,dp_{1}\ldots &+{\dot {q}}_{n}\,dp_{n}-{\dot {p}}_{1}\,dq_{1}\ldots \\&-{\dot {p}}_{n}\,dq_{n}-A_{1}\,da_{1}-A_{2}\,da_{2}-\mathrm {etc.} \end{aligned}}}$

(8)

It will be observed that the kinetic energy (${\displaystyle \epsilon _{p}}$) in the most general case is a quadratic function of the ${\displaystyle p}$'s (or ${\displaystyle {\dot {q}}}$'s) ​involving also the ${\displaystyle q}$'s but not the ${\displaystyle a}$'s ; that the potential energy, when it exists, is function of the ${\displaystyle q}$'s and ${\displaystyle a}$'s ; and that the total energy, when it exists, is function of the ${\displaystyle p}$'s (or ${\displaystyle {\dot {q}}}$s), the ${\displaystyle q}$'s, and the ${\displaystyle a}$'s. In expressions like ${\displaystyle d\epsilon /dq_{1}}$ the ${\displaystyle p}$'s, and not the ${\displaystyle {\dot {q}}}$'s, are to be taken as independent variables, as has already been stated with respect to the kinetic energy.

Let us imagine a great number of independent systems, identical in nature, but differing in phase, that is, in their condition with respect to configuration and velocity. The forces are supposed to be determined for every system by the same law, being functions of the coördinates of the system ${\displaystyle q_{1},\ldots q_{n}}$, either alone or with the coördinates ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coördinates ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coördinates ${\displaystyle q_{1},\ldots q_{n}}$, which at the same time have different values in the different systems considered.

Let us especially consider the number of systems which at a given instant fall within specified limits of phase, viz., those for which

${\displaystyle {\begin{array}{ccc}p_{1}'<p_{1}<p_{1}'',&\qquad &q_{1}'<q_{1}<q_{1}''\\p_{2}'<p_{2}<p_{2}'',&\qquad &q_{2}'<q_{2}<q_{2}''\\\ldots &&\ldots \\p_{n}'<p_{n}<p_{n}'',&\qquad &q_{n}'<q_{n}<q_{n}''\end{array}}}$

(9)

the accented letters denoting constants. We shall suppose the differences ${\displaystyle p_{1}''-p_{1}'}$, ${\displaystyle q_{1}''-q_{1}'}$, etc. to be infinitesimal, and that the systems are distributed in phase in some continuous manner,^[3] so that the number having phases within the limits specified may be represented by

${\displaystyle D(p_{1}''-p_{1}')\ldots (p_{n}''-p_{n}')(q_{1}''-q_{1}')\ldots (q_{n}''-q_{n}')}$

(10)

​or more briefly by

${\displaystyle D\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},}$

(11)

where ${\displaystyle D}$ is a function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s and in general of ${\displaystyle t}$ also, for as time goes on, and the individual systems change their phases, the distribution of the ensemble in phase will in general vary. In special cases, the distribution in phase will remain unchanged. These are cases of statistical equilibrium.

If we regard all possible phases as forming a sort of extension of ${\displaystyle 2n}$ dimensions, we may regard the product of differentials in (11) as expressing an element of this extension, and ${\displaystyle D}$ as expressing the density of the systems in that element. We shall call the product

${\displaystyle dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}}$

(12)

an element of extension-in-phase, and ${\displaystyle D}$ the density-in-phase of the systems.

It is evident that the changes which take place in the density of the systems in any given element of extension-in-phase will depend on the dynamical nature of the systems and their distribution in phase at the time considered.

In the case of conservative systems, with which we shall be principally concerned, their dynamical nature is completely determined by the function which expresses the energy (${\displaystyle \epsilon }$) in terms of the ${\displaystyle p}$'s, ${\displaystyle q}$'s, and ${\displaystyle a}$'s (a function supposed identical for all the systems); in the more general case which we are considering, the dynamical nature of the systems is determined by the functions which express the kinetic energy (${\displaystyle \epsilon _{p}}$) in terms of the ${\displaystyle p}$'s and ${\displaystyle q}$'s, and the forces in terms of the ${\displaystyle q}$'s and ${\displaystyle a}$'s. The distribution in phase is expressed for the time considered by ${\displaystyle D}$ as function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s. To find the value of ${\displaystyle dD/dt}$ for the specified element of extension-in-phase, we observe that the number of systems within the limits can only be varied by systems passing the limits, which may take place in ${\displaystyle 4n}$ different ways, viz., by the ${\displaystyle p_{1}}$ of a system passing the limit ${\displaystyle p_{1}'}$, or the limit ${\displaystyle p_{1}''}$, or by the ${\displaystyle q_{1}}$ of a system passing the limit ${\displaystyle q_{1}'}$ or the limit ${\displaystyle q_{1}''}$, etc. Let us consider these cases separately.

​In the first place, let us consider the number of systems which in the time ${\displaystyle dt}$ pass into or out of the specified element by ${\displaystyle p_{1}}$ passing the limit ${\displaystyle p_{1}'}$. It will be convenient, and it is evidently allowable, to suppose ${\displaystyle dt}$ so small that the quantities ${\displaystyle {\dot {p}}_{1}\,dt}$, ${\displaystyle {\dot {q}}_{1}\,dt}$, etc., which represent the increments of ${\displaystyle p_{1}}$, ${\displaystyle q_{1}}$, etc., in the time ${\displaystyle dt}$ shall be infinitely small in comparison with the infinitesimal differences ${\displaystyle p_{1}''-p_{1}'}$, ${\displaystyle q_{1}''-q_{1}'}$, etc., which determine the magnitude of the element of extension-in-phase. The systems for which ${\displaystyle p_{1}}$ passes the limit ${\displaystyle p_{1}'}$ in the interval ${\displaystyle dt}$ are those for which at the commencement of this interval the value of ${\displaystyle p_{1}}$ lies between ${\displaystyle p_{1}'}$ and ${\displaystyle p_{1}'-{\dot {p}}_{1}\,dt}$, as is evident if we consider separately the cases in which ${\displaystyle {\dot {p}}_{1}}$ is positive and negative. Those systems for which ${\displaystyle p_{1}}$ lies between these limits, and the other ${\displaystyle p}$'s and ${\displaystyle q}$'s between the limits specified in (9), will therefore pass into or out of the element considered according as ${\displaystyle {\dot {p}}}$ is positive or negative, unless indeed they also pass some other limit specified in (9) during the same interval of time. But the number which pass any two of these limits will be represented by an expression containing the square of ${\displaystyle dt}$ as a factor, and is evidently negligible, when ${\displaystyle dt}$ is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms containing ${\displaystyle dt^{2}}$) may be found by substituting ${\displaystyle {\dot {p}}_{1}\,dt}$ for ${\displaystyle p_{1}''-p_{1}'}$ in (10) or for ${\displaystyle dp_{1}}$ in (11).

The expression

${\displaystyle D\,{\dot {p}}_{1}\,dt\,dp_{2}\ldots dp_{n}\,dq_{1}\ldots dq_{n}}$

(13)

will therefore represent, according as it is positive or negative, the increase or decrease of the number of systems within the given limits which is due to systems passing the limit ${\displaystyle p_{1}'}$. A similar expression, in which however ${\displaystyle D}$ and ${\displaystyle {\dot {p}}}$ will have slightly different values (being determined for ${\displaystyle p_{1}''}$ instead of ${\displaystyle p_{1}'}$), will represent the decrease or increase of the number of systems due to the passing of the limit ${\displaystyle p_{1}''}$. The difference of the two expressions, or

${\displaystyle {\frac {d(D\,{\dot {p}}_{1})}{dp_{1}}}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt}$

(14)

​will represent algebraically the decrease of the number of systems within the limits due to systems passing the limits ${\displaystyle p_{1}'}$ and ${\displaystyle p_{1}''}$.

The decrease in the number of systems within the limits due to systems passing the limits ${\displaystyle q_{1}'}$ and ${\displaystyle q_{1}''}$ may be found in the same way. This will give

${\displaystyle {\bigg (}{\frac {d(D\,{\dot {p}}_{1})}{dp_{1}}}+{\frac {d(D\,{\dot {q}}_{1})}{dq_{1}}}{\bigg )}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt}$

(15)

for the decrease due to passing the four limits ${\displaystyle p_{1}'}$, ${\displaystyle p_{1}''}$, ${\displaystyle q_{1}'}$, ${\displaystyle q_{1}''}$. But since the equations of motion (3) give

${\displaystyle {\frac {d{\dot {p}}_{1}}{dp_{1}}}+{\frac {d{\dot {q}}_{1}}{dq_{1}}}=0,}$

(16)

the expression reduces to

${\displaystyle {\bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\bigg )}\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}\,dt}$

(17)

If we prefix ${\displaystyle \Sigma }$ to denote summation relative to the suffixes ${\displaystyle 1\ldots n}$, we get the total decrease in the number of systems within the limits in the time ${\displaystyle dt}$. That is,

${\displaystyle {\begin{aligned}\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )}\,dp_{1}\ldots &dp_{n}\,dq_{1}\ldots dq_{n}\,dt=\\&-dD\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},\end{aligned}}}$

(18)

or

${\displaystyle {\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{p,q}=-\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )},}$

(19)

where the suffix applied to the differential coefficient indicates that the ${\displaystyle p}$'s and ${\displaystyle q}$'s are to be regarded as constant in the differentiation. The condition of statistical equilibrium is therefore

${\displaystyle \Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}{\Bigg )}=0.}$

(20)

If at any instant this condition is fulfilled for all values of the ${\displaystyle p}$'s and ${\displaystyle q}$'s, ${\displaystyle (dD/dt)_{p,q}}$ vanishes, and therefore the condition will continue to hold, and the distribution in phase will be permanent, so long as the external coördinates remain constant. But the statistical equilibrium would in general be disturbed by a change in the values of the external coördinates, which ​would alter the values of the ${\displaystyle {\dot {p}}}$'s as determined by equations (3), and thus disturb the relation expressed in the last equation.

If we write equation (19) in the form

${\displaystyle {\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{p,q}dt+\Sigma {\Bigg (}{\frac {dD}{dp_{1}}}{\dot {p}}_{1}dt+{\frac {dD}{dq_{1}}}{\dot {q}}_{1}dt{\Bigg )}=0}$

(21)

it will be seen to express a theorem of remarkable simplicity. Since ${\displaystyle D}$ is a function of ${\displaystyle t}$, ${\displaystyle p_{1},\ldots p_{n}}$, ${\displaystyle q_{1}\ldots q_{n}}$, its complete differential will consist of parts due to the variations of all these quantities. Now the first term of the equation represents the increment of ${\displaystyle D}$ due to an increment of ${\displaystyle t}$ (with constant values of the ${\displaystyle p}$'s and ${\displaystyle q}$'s), and the rest of the first member represents the increments of ${\displaystyle D}$ due to increments of the ${\displaystyle p}$'s and ${\displaystyle q}$'s, expressed by ${\displaystyle {\dot {p}}_{1}\,dt}$, ${\displaystyle {\dot {q}}_{1}\,dt}$, etc. But these are precisely the increments which the ${\displaystyle p}$'s and ${\displaystyle q}$'s receive in the movement of a system in the time ${\displaystyle dt}$. The whole expression represents the total increment of ${\displaystyle D}$ for the varying phase of a moving system. We have therefore the theorem:—

In an ensemble of mechanical systems identical in nature and subject to forces determined by identical laws, but distributed in phase in any continuous manner, the density-in-phase is constant in time for the varying phases of a moving system; provided, that the forces of a system are functions of its coördinates, either alone or with the time.^[4]

This may be called the principle of conservation of density-in-phase. It may also be written

${\displaystyle {\Bigg (}{\frac {dD}{dt}}{\Bigg )}_{a,\ldots h}=0,}$

(22)

where ${\displaystyle a,\ldots h}$ represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential ​coefficient to indicate that they are to be regarded as constant in the differentiation.

We may give to this principle a slightly different expression. Let us call the value of the integral

${\displaystyle \int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}}$

(23)

taken within any limits the extension-in-phase within those limits.

When the phases bounding an extension-in-phase vary in the course of time according to the dynamical laws of a system subject to forces which are functions of the coördinates either alone or with the time, the value of the extension-in-phase thus bounded remains constant. In this form the principle may be called the principle of conservation of extension-in-phase. In some respects this may be regarded as the most simple statement of the principle, since it contains no explicit reference to an ensemble of systems.

Since any extension-in-phase may be divided into infinitesimal portions, it is only necessary to prove the principle for an infinitely small extension. The number of systems of an ensemble which fall within the extension will be represented by the integral

${\displaystyle \int \ldots \int D\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}.}$

If the extension is infinitely small, we may regard ${\displaystyle D}$ as constant in the extension and write

${\displaystyle D\int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}}$

for the number of systems. The value of this expression must be constant in time, since no systems are supposed to be created or destroyed, and none can pass the limits, because the motion of the limits is identical with that of the systems. But we have seen that ${\displaystyle D}$ is constant in time, and therefore the integral

${\displaystyle \int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n},}$

​which we have called the extension-in-phase, is also constant in time.^[5]

Since the system of coördinates employed in the foregoing discussion is entirely arbitrary, the values of the coördinates relating to any configuration and its immediate vicinity do not impose any restriction upon the values relating to other configurations. The fact that the quantity which we have called density-in-phase is constant in time for any given system, implies therefore that its value is independent of the coördinates which are used in its evaluation. For let the density-in-phase as evaluated for the same time and phase by one system of coördinates be ${\displaystyle D_{1}'}$, and by another system ${\displaystyle D_{2}'}$. A system which at that time has that phase will at another time have another phase. Let the density as calculated for this second time and phase by a third system of coördinates be ${\displaystyle D_{3}''}$. Now we may imagine a system of coördinates which at and near the first configuration will coincide with the first system of coördinates, and at and near the second configuration will coincide with the third system of coördinates. This will give ${\displaystyle D_{1}'=D_{3}''}$. Again we may imagine a system of coördinates which at and near the first configuration will coincide with the second system of coördinates, and at and near the ​second configuration will coincide with the third system of coördinates. This will give ${\displaystyle D_{2}'=D_{3}''}$. We have therefore ${\displaystyle D_{1}'=D_{2}'}$.

It follows, or it may be proved in the same way, that the value of an extension-in-phase is independent of the system of coördinates which is used in its evaluation. This may easily be verified directly. If ${\displaystyle q_{1},\ldots q_{n}}$, ${\displaystyle Q_{1},\ldots Q_{n}}$ are two systems of coördinates, and ${\displaystyle p_{1},\ldots p_{n}}$, ${\displaystyle P_{1},\ldots P_{n}}$ the corresponding momenta, we have to prove that

${\displaystyle \int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}=\int \ldots \int dP_{1}\ldots dP_{n}\,dQ_{1}\ldots dQ_{n},}$

(24)

when the multiple integrals are taken within limits consisting of the same phases. And this will be evident from the principle on which we change the variables in a multiple integral, if we prove that

${\displaystyle {\frac {d(P_{1},\ldots P_{n},Q_{1},\ldots Q_{n})}{d(p_{1},\ldots p_{n},q_{1},\ldots q_{n})}}=1,}$

(25)

where the first member of the equation represents a Jacobian or functional determinant. Since all its elements of the form ${\displaystyle dQ/dp}$ are equal to zero, the determinant reduces to a product of two, and we have to prove that

${\displaystyle {\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}{\frac {d(Q_{1},\ldots Q_{n})}{d(q_{1},\ldots q_{n})}}=1.}$

(26)

We may transform any element of the first of these determinants as follows. By equations (2) and (3), and in view of the fact that the ${\displaystyle {\dot {Q}}}$'s are linear functions of the ${\displaystyle {\dot {q}}}$'s and therefore of the ${\displaystyle p}$'s, with coefficients involving the ${\displaystyle q}$'s, so that a differential coefficient of the form ${\displaystyle d{\dot {Q}}_{r}/dp_{y}}$ is function of the ${\displaystyle q}$'s alone, we get^[6] ​

${\displaystyle {\begin{aligned}{\frac {dP_{x}}{dp_{y}}}={\frac {d}{dp_{y}}}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{x}}}&=\sum _{r=1}^{r=n}{\bigg (}{\frac {d^{2}\epsilon _{p}}{d{\dot {Q}}_{r}d{\dot {Q}}_{x}}}{\frac {d{\dot {Q}}_{r}}{dp_{y}}}{\bigg )}=\\&{\frac {d}{d{\dot {Q}}_{x}}}\sum _{r=1}^{r=n}{\bigg (}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{r}}}{\frac {d{\dot {Q}}_{r}}{dp_{y}}}{\bigg )}={\frac {d}{d{\dot {Q}}_{x}}}{\frac {d\epsilon _{p}}{dp_{y}}}={\frac {d{\dot {q}}_{y}}{d{\dot {Q}}_{x}}}.\end{aligned}}}$

(27)

But since

${\displaystyle {\dot {q}}_{y}=\sum _{r=1}^{r=n}{\bigg (}{\frac {dq_{y}}{dQ_{r}}}{\dot {Q}}_{r}{\bigg )}}$,

${\displaystyle {\frac {d{\dot {q}}_{y}}{d{\dot {Q}}_{x}}}={\frac {dq_{y}}{dQ_{x}}}.}$

(28)

Therefore,

${\displaystyle {\frac {d(P_{1},\ldots P_{n})}{d(p_{1},\ldots p_{n})}}={\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d({\dot {Q}}_{1},\ldots {\dot {Q}}_{n})}}={\frac {d(q_{1},\ldots q_{n})}{d(Q_{1},\ldots Q_{n})}}.}$

(29)

The equation to be proved is thus reduced to

${\displaystyle {\frac {d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}{d(Q_{1},\ldots Q_{n})}}{\frac {d(Q_{1},\ldots Q_{n})}{d({\dot {q}}_{1},\ldots {\dot {q}}_{n})}}=1,}$

(30)

which is easily proved by the ordinary rule for the multiplication of determinants.

The numerical value of an extension-in-phase will however depend on the units in which we measure energy and time. For a product of the form ${\displaystyle dp\,dq}$ has the dimensions of energy multiplied by time, as appears from equation (2), by which the momenta are defined. Hence an extension-in-phase has the dimensions of the ${\displaystyle n}$th power of the product of energy and time. In other words, it has the dimensions of the ${\displaystyle n}$th power of action, as the term is used in the `principle of Least Action.'

If we distinguish by accents the values of the momenta and coördinates which belong to a time ${\displaystyle t'}$, the unaccented letters relating to the time ${\displaystyle t}$, the principle of the conservation of extension-in-phase may be written

${\displaystyle \int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}=\int \ldots \int dp_{1}'\ldots dp_{n}'\,dq_{1}'\ldots dq_{n}',}$

(31)

or more briefly

${\displaystyle \int \ldots \int dp_{1}\ldots dq_{n}=\int \ldots \int dp_{1}'\ldots dq_{n}',}$

(32)

​the limiting phases being those which belong to the same systems at the times ${\displaystyle t}$ and ${\displaystyle t'}$ respectively. But we have identically

${\displaystyle \int \ldots \int dp_{1}\ldots dq_{n}=\int \ldots \int {\frac {d(p_{1}\ldots q_{n})}{d(p_{1}'\ldots q_{n}')}}dp_{1}'\ldots dq_{n}'}$

for such limits. The principle of conservation of extension-in-phase may therefore be expressed in the form

${\displaystyle {\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'\ldots q_{n}')}}=1.}$

(33)

This equation is easily proved directly. For we have identically

${\displaystyle {\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}={\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}{\frac {d(p_{1}'',\ldots q_{n}'')}{d(p_{1}',\ldots q_{n}')}},}$

where the double accents distinguish the values of the momenta and coördinates for a time ${\displaystyle t''}$. If we vary ${\displaystyle t}$, while ${\displaystyle t'}$ and ${\displaystyle t''}$ remain constant, we have

${\displaystyle {\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}={\frac {d(p_{1}'',\ldots q_{n}'')}{d(p_{1}',\ldots q_{n}')}}{\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}.}$

(34)

Now since the time ${\displaystyle t''}$ is entirely arbitrary, nothing prevents us from making if ${\displaystyle t''}$ identical with ${\displaystyle t}$ at the moment considered. Then the determinant

${\displaystyle {\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}}$

will have unity for each of the elements on the principal diagonal, and zero for all the other elements. Since every term of the determinant except the product of the elements on the principal diagonal will have two zero factors, the differential of the determinant will reduce to that of the product of these elements, i. e., to the sum of the differentials of these elements. This gives the equation

${\displaystyle {\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}={\frac {d{\dot {p}}_{1}}{dp_{1}''}}\ldots +{\frac {d{\dot {p}}_{n}}{dp_{n}''}}+{\frac {d{\dot {q}}_{1}}{dq_{1}''}}\ldots +{\frac {d{\dot {q}}_{n}}{dq_{n}''}}.}$

Now since ${\displaystyle t=t''}$, the double accents in the second member of this equation may evidently be neglected. This will give, in virtue of such relations as (16), ​

${\displaystyle {\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}'',\ldots q_{n}'')}}=0,}$

which substituted in (34) will give

${\displaystyle {\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}=0.}$

The determinant in this equation is therefore a constant, the value of which may be determined at the instant when ${\displaystyle t=t'}$, when it is evidently unity. Equation (33) is therefore demonstrated.

Again, if we write ${\displaystyle a,\ldots h}$ for a system of ${\displaystyle 2n}$ arbitrary constants of the integral equations of motion, ${\displaystyle p_{1}}$, ${\displaystyle q_{1}}$, etc. will be functions of ${\displaystyle a,\ldots h}$, and ${\displaystyle t}$, and we may express an extension-in-phase in the form

${\displaystyle \int \ldots \int {\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}da\ldots dh.}$

(35)

If we suppose the limits specified by values of ${\displaystyle a,\ldots h}$, a system initially at the limits will remain at the limits. The principle of conservation of extension-in-phase requires that an extension thus bounded shall have a constant value. This requires that the determinant under the integral sign shall be constant, which may be written

${\displaystyle {\frac {d}{dt}}{\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}=0.}$

(36)

This equation, which may be regarded as expressing the principle of conservation of extension-in-phase, may be derived directly from the identity

${\displaystyle {\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}={\frac {d(p_{1},\ldots q_{n})}{d(p_{1}',\ldots q_{n}')}}{\frac {d(p_{1}',\ldots q_{n}')}{d(a,\ldots h)}}}$

in connection with equation (33).

Since the coördinates and momenta are functions of ${\displaystyle a,\ldots h}$, and ${\displaystyle t}$, the determinant in (36) must be a function of the same variables, and since it does not vary with the time, it must be a function of ${\displaystyle a,\ldots h}$ alone. We have therefore

${\displaystyle {\frac {d(p_{1},\ldots q_{n})}{d(a,\ldots h)}}=\operatorname {func.} (a,\ldots h).}$

(37)

​It is the relative numbers of systems which fall within different limits, rather than the absolute numbers, with which we are most concerned. It is indeed only with regard to relative numbers that such discussions as the preceding will apply with literal precision, since the nature of our reasoning implies that the number of systems in the smallest element of space which we consider is very great. This is evidently inconsistent with a finite value of the total number of systems, or of the density-in-phase. Now if the value of ${\displaystyle D}$ is infinite, we cannot speak of any definite number of systems within any finite limits, since all such numbers are infinite. But the ratios of these infinite numbers may be perfectly definite. If we write ${\displaystyle N}$ for the total number of systems, and set

${\displaystyle P={\frac {D}{N}},}$

(38)

${\displaystyle P}$ may remain finite, when ${\displaystyle N}$ and ${\displaystyle D}$ become infinite. The integral

${\displaystyle \int \ldots \int P\,dp_{1}\ldots dq_{n}}$

(39)

taken within any given limits, will evidently express the ratio of the number of systems falling within those limits to the whole number of systems. This is the same thing as the probability that an unspecified system of the ensemble (i. e. one of which we only know that it belongs to the ensemble) will lie within the given limits. The product

${\displaystyle P\,dp_{1}\ldots dq_{n}}$

(40)

expresses the probability that an unspecified system of the ensemble will be found in the element of extension-in-phase ${\displaystyle dp_{1}\ldots dq_{n}}$. We shall call ${\displaystyle P}$ the coefficient of probability of the phase considered. Its natural logarithm we shall call the index of probability of the phase, and denote it by the letter ${\displaystyle \eta }$.

If we substitute ${\displaystyle NP}$ and ${\displaystyle Ne^{\eta }}$ for ${\displaystyle D}$ in equation (19), we get

${\displaystyle {\bigg (}{\frac {dP}{dt}}{\bigg )}_{p,q}=-\Sigma {\bigg (}{\frac {dP}{dp_{1}}}{\dot {p}}_{1}+{\frac {dP}{dq_{1}}}d{\dot {q}}_{1}{\bigg )},}$

(41)

and

${\displaystyle {\bigg (}{\frac {d\eta }{dt}}{\bigg )}_{p,q}=-\Sigma {\bigg (}{\frac {d\eta }{dp_{1}}}{\dot {p}}_{1}+{\frac {d\eta }{dq_{1}}}d{\dot {q}}_{1}{\bigg )}.}$

(42)

​The condition of statistical equilibrium may be expressed by equating to zero the second member of either of these equations.

The same substitutions in (22) give

${\displaystyle {\bigg (}{\frac {dP}{dt}}{\bigg )}_{a,\ldots h}=0,}$

(43)

and

${\displaystyle {\bigg (}{\frac {d\eta }{dt}}{\bigg )}_{a,\ldots h}=0.}$

(44)

That is, the values of ${\displaystyle P}$ and ${\displaystyle \eta }$, like those of ${\displaystyle D}$, are constant in time for moving systems of the ensemble. From this point of view, the principle which otherwise regarded has been called the principle of conservation of density-in-phase or conservation of extension-in-phase, may be called the principle of conservation of the coefficient (or index) of probability of a phase varying according to dynamical laws, or more briefly, the principle of conservation of probability of phase. It is subject to the limitation that the forces must be functions of the coördinates of the system either alone or with the time.

The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact customary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described. But if we prefer to avoid any reference to an ensemble of systems, we may observe that the probability that the phase of a system falls within certain limits at a certain time, is equal to the probability that at some other time the phase will fall within the limits formed by phases corresponding to the first. For either occurrence necessitates the other. That is, if we write ${\displaystyle P'}$ for the coefficient of probability of the phase ${\displaystyle p_{1}',\ldots q_{n}'}$ at the time ${\displaystyle t'}$, and ${\displaystyle P''}$ for that of the phase ${\displaystyle p_{1}'',\ldots q_{n}''}$ at the time ${\displaystyle t''}$, ​

${\displaystyle \int \ldots \int P'\,dq_{1}'\ldots dq_{n}'=\int \ldots \int P''\,dp_{1}''\ldots dq_{n}'',}$

(45)

where the limits in the two cases are formed by corresponding phases. When the integrations cover infinitely small variations of the momenta and coördinates, we may regard ${\displaystyle P'}$ and ${\displaystyle P''}$ as constant in the integrations and write

${\displaystyle P'\int \ldots \int \,dp_{1}'\ldots dq_{n}''=P'\int \ldots \int \,dp_{1}''\ldots dq_{n}''.}$

Now the principle of the conservation of extension-in-phase, which has been proved (viz., in the second demonstration given above) independently of any reference to an ensemble of systems, requires that the values of the multiple integrals in this equation shall be equal. This gives

${\displaystyle P''=P'.}$

With reference to an important class of cases this principle may be enunciated as follows.

When the differential equations of motion are exactly known, but the constants of the integral equations imperfectly determined, the coefficient of probability of any phase at any time is equal to the coefficient of probability of the corresponding phase at any other time. By corresponding phases are meant those which are calculated for different times from the same values of the arbitrary constants of the integral equations.

Since the sum of the probabilities of all possible cases is necessarily unity, it is evident that we must have

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\,dp_{1}\ldots dq_{n}=1,}$

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where the integration extends over all phases. This is indeed only a different form of the equation

${\displaystyle N=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}D\,dp_{1}\ldots dq_{n},}$

which we may regard as defining ${\displaystyle N}$.

​The values of the coefficient and index of probability of phase, like that of the density-in-phase, are independent of the system of coördinates which is employed to express the distribution in phase of a given ensemble.

In dimensions, the coefficient of probability is the reciprocal of an extension-in-phase, that is, the reciprocal of the ${\displaystyle n}$th power of the product of time and energy. The index of probability is therefore affected by an additive constant when we change our units of time and energy. If the unit of time is multiplied by ${\displaystyle c_{t}}$ and the unit of energy is multiplied by ${\displaystyle c_{e}}$, all indices of probability relating to systems of ${\displaystyle n}$ degrees of freedom will be increased by the addition of

${\displaystyle n\log c_{t}+n\log c_{e}.}$

(47)