Elementary Principles in Statistical Mechanics/Chapter III

by Josiah Willard Gibbs

CHAPTER III.

APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE INTEGRATION OF THE DIFFERENTIAL EQUATIONS OF MOTION.^[1]

We have seen that the principle of conservation of extension-in-phase may be expressed as a differential relation between the coördinates and momenta and the arbitrary constants of the integral equations of motion. Now the integration of the differential equations of motion consists in the determination of these constants as functions of the coördinates and momenta with the time, and the relation afforded by the principle of conservation of extension-in-phase may assist us in this determination.

It will be convenient to have a notation which shall not distinguish between the coördinates and momenta. If we write $r_{1}\ldots r_{2n}$ for the coördinates and momenta, and $a\ldots h$ as before for the arbitrary constants, the principle of which we wish to avail ourselves, and which is expressed by equation (37), may be written

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}=\operatorname {func.} (a,\ldots h).$

(71)

Let us first consider the case in which the forces are determined by the coördinates alone. Whether the forces are 'conservative' or not is immaterial. Since the differential equations of motion do not contain the time ($t$) in the finite form, if we eliminate $dt$ from these equations, we obtain $2n-1$ equations in $r_{1},\ldots r_{2n}$ and their differentials, the integration of which will introduce $2n-1$ arbitrary constants which we shall call $b\ldots h$. If we can effect these integrations, the remaining constant ($a$) will then be introduced in the final integration, (viz., that of an equation containing $dt$,) and will be added to or subtracted from $t$ in the integral equation. Let us have it subtracted from $t$. It is evident then that

${\frac {dr_{1}}{da}}=-{\dot {r}}_{1},\quad {\frac {dr_{2}}{da}}=-{\dot {r}}_{2},\quad \mathrm {etc.}$

(72)

Moreover, since $b,\ldots h$ and $t-a$ are independent functions of $r_{1},\ldots r_{2n}$, the latter variables are functions of the former. The Jacobian in (71) is therefore function of $b,\ldots h$, and $t-a$, and since it does not vary with $t$ it cannot vary with $a$. We have therefore in the case considered, viz., where the forces are functions of the coördinates alone,

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}=\operatorname {func.} (b,\ldots h).$

(73)

Now let us suppose that of the first $2n-1$ integrations we have accomplished all but one, determining $2n-2$ arbitrary constants (say $c,\ldots h$) as functions of $r_{1},\ldots r_{2n}$, leaving $b$ as well as $a$ to be determined. Our $2n-2$ finite equations enable us to regard all the variables $r_{1},\ldots r_{2n}$, and all functions of these variables as functions of two of them, (say $r_{1}$ and $r_{2}$,) with the arbitrary constants $c,\ldots h$. To determine $b$, we have the following equations for constant values of $c,\ldots h$.

$dr_{1}={\frac {dr_{1}}{da}}\,da+{\frac {dr_{1}}{db}}\,db,$

$dr_{2}={\frac {dr_{2}}{da}}\,da+{\frac {dr_{2}}{db}}\,db,$

whence

${\frac {d(r_{1},r_{2})}{d(a,b)}}\,db=-{\frac {dr_{2}}{da}}\,dr_{1}+{\frac {dr_{1}}{da}}\,dr_{2}.$

(74)

Now, by the ordinary formula for the change of variables,

${\begin{aligned}&\int \ldots \int {\frac {d(r_{1},r_{2})}{d(a,b)}}\,da\,db\,dr_{3}\ldots dr_{2n}=\int \ldots \int dr_{1}\ldots dr_{2n}\\&\qquad \qquad {}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,da\ldots dh\\&{}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}\,da\,db\,dr_{3}\ldots dr_{2n},\end{aligned}}$

where the limits of the multiple integrals are formed by the same phases. Hence

${\frac {d(r_{1},r_{2})}{d(a,b)}}={\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}},$

(75)

With the aid of this equation, which is an identity, and (72), we may write equation (74) in the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}\,db={\dot {r}}_{2}\,dr_{1}-{\dot {r}}_{1}\,dr_{2}.$

(76)

The separation of the variables is now easy. The differential equations of motion give ${\dot {r}}_{1}$ and ${\dot {r}}_{2}$ in terms of $r_{1},\ldots r_{2n}$. The integral equations already obtained give $c,\ldots h$ and therefore the Jacobian $d(c,\ldots h)/d(r_{3},\ldots r_{2n})$, in terms of the same variables. But in virtue of these same integral equations, we may regard functions of $r_{1},\ldots r_{2n}$ as functions of $r_{1}$ and $r_{2}$ with the constants $c,\ldots h$. If therefore we write the equation in the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,db={\frac {{\dot {r}}_{2}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{2},$

(77)

the coefficients of $dr_{1}$ and $dr_{2}$ may be regarded as known functions of $r_{1}$ and $r_{2}$ with the constants $c,\ldots h$. The coefficient of $db$ is by (73) a function of $b,\ldots h$. It is not indeed a known function of these quantities, but since $c,\ldots h$ are regarded as constant in the equation, we know that the first member must represent the differential of some function of $b,\ldots h$, for which we may write $b'$. We have thus

$db'={\frac {{\dot {r}}_{2}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}}\,dr_{2},$

(78)

which may be integrated by quadratures and gives $b'$ as functions of $r_{1},r_{2},\ldots c,\ldots h$, and thus as function of $r_{1},\ldots r_{2n}$.

This integration gives us the last of the arbitrary constants which are functions of the coördinates and momenta without the time. The final integration, which introduces the remaining constant ($a$), is also a quadrature, since the equation to be integrated may be expressed in the form

$dt=F(r_{1})\,dr_{1}.$

Now, apart from any such considerations as have been adduced, if we limit ourselves to the changes which take place in time, we have identically

${\dot {r}}_{2}\,dr_{1}-{\dot {r}}_{1}\,dr_{2}=0,$

and ${\dot {r}}_{1}$ and ${\dot {r}}_{2}$ are given in terms of $r_{1},\ldots r_{2n}$ by the differential equations of motion. When we have obtained $2n-1$ integral equations, we may regard ${\dot {r}}_{2}$ and ${\dot {r}}_{1}$ as known functions of $r_{1}$ and $r_{2}$. The only remaining difficulty is in integrating this equation. If the case is so simple as to present no difficulty, or if we have the skill or the good fortune to perceive that the multiplier

${\frac {1}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}},$

(79)

or any other, will make the first member of the equation an exact differential, we have no need of the rather lengthy considerations which have been adduced. The utility of the principle of conservation of extension-in-phase is that it supplies a 'multiplier' which renders the equation integrable, and which it might be difficult or impossible to find otherwise.

It will be observed that the function represented by $b'$ is a particular case of that represented by $b$. The system of arbitrary constants $a,b',c\ldots h$ has certain properties notable for simplicity. If we write $b'$ for $b$ in (77), and compare the result with (78), we get

${\frac {d(r_{1},\ldots r_{2n})}{d(a,b',c\ldots h)}}=1.$

(80)

Therefore the multiple integral

$\int \ldots \int da\,db'\,dc\ldots dh$

(81)

taken within limits formed by phases regarded as contemporaneous represents the extension-in-phase within those limits.

The case is somewhat different when the forces are not determined by the coördinates alone, but are functions of the coördinates with the time. All the arbitrary constants of the integral equations must then be regarded in the general case as functions of $r_{1}\ldots r_{2n}$, and $t$. We cannot use the principle of conservation of extension-in-phase until we have made $2n-1$ integrations. Let us suppose that the constants $b,\ldots h$ have been determined by integration in terms of $r_{1}\ldots r_{2n}$, and $t$, leaving a single constant ($a$) to be thus determined. Our $2n-1$ finite equations enable us to regard all the variables $r_{1}\ldots r_{2n}$ as functions of a single one, say $r_{1}$.

For constant values of $b,\ldots h$, we have

$dr_{1}={\frac {dr_{1}}{da}}\,da+{\dot {r}}_{1}\,dt.$

(82)

Now

${\begin{aligned}&\int \ldots \int {\frac {dr_{1}}{da}}\,da\,dr_{2}\ldots dr_{2n}=\int \ldots \int dr_{1}\ldots dr_{2n}\\&\qquad \qquad {}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,da\ldots dh\\&{}=\int \ldots \int {\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}\,da\,dr_{2}\ldots dr_{2n},\end{aligned}}$

where the limits of the integrals are formed by the same phases. We have therefore

${\frac {dr_{1}}{da}}={\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}},$

(83)

by which equation (82) may be reduced to the form

${\frac {d(r_{1},\ldots r_{2n})}{d(a,\ldots h)}}\,da={\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dt.$

(84)

Now we know by (71) that the coefficient of $da$ is a function of $a,\ldots h$. Therefore, as $b,\ldots h$ are regarded as constant in the equation, the first number represents the differential of a function of $a,\ldots h$, which we may denote by $a'$. We have then

$da'={\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}{}'}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dt,$

(85)

which may be integrated by quadratures. In this case we may say that the principle of conservation of extension-in-phase has supplied the 'multiplier'

${\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}$

(86)

for the integration of the equation

$dr_{1}-{\dot {r}}_{1}\,dt=0.$

(87)

The system of arbitrary constants $a',b,\ldots h$ has evidently the same properties which were noticed in regard to the system $a,b',\ldots h$.

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1. See Boltzmann: "Zusammenhang zwischen den Sätzen über das Verhalten mehratomiger Gasmolecüle mit Jacobi's Princip des letzten Multiplicators. Sitzb. der Wiener Akad., Bd. LXIII, Abth. II., S. 679, (1871).