Elementary Principles in Statistical Mechanics/Chapter III

1543949Elementary Principles in Statistical MechanicsChapter III. Application of the principle of conservation of extension-in-phase to the integration of the differential equations of motion.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 for the coördinates and momenta, and 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

(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 () in the finite form, if we eliminate from these equations, we obtain equations in and their differentials, the integration of which will introduce arbitrary constants which we shall call . If we can effect these integrations, the remaining constant () will then be introduced in the final integration, (viz., that of an equation containing ,) and will be added to or subtracted from in the integral equation. Let us have it subtracted from . It is evident then that

(72)

Moreover, since and are independent functions of , the latter variables are functions of the former. The Jacobian in (71) is therefore function of , and , and since it does not vary with it cannot vary with . We have therefore in the case considered, viz., where the forces are functions of the coördinates alone,

(73)

Now let us suppose that of the first integrations we have accomplished all but one, determining arbitrary constants (say ) as functions of , leaving as well as to be determined. Our finite equations enable us to regard all the variables , and all functions of these variables as functions of two of them, (say and ,) with the arbitrary constants . To determine , we have the following equations for constant values of .

whence
(74)
Now, by the ordinary formula for the change of variables,
where the limits of the multiple integrals are formed by the same phases. Hence
(75)
With the aid of this equation, which is an identity, and (72), we may write equation (74) in the form
(76)

The separation of the variables is now easy. The differential equations of motion give and in terms of . The integral equations already obtained give and therefore the Jacobian , in terms of the same variables. But in virtue of these same integral equations, we may regard functions of as functions of and with the constants . If therefore we write the equation in the form

(77)
the coefficients of and may be regarded as known functions of and with the constants . The coefficient of is by (73) a function of . It is not indeed a known function of these quantities, but since are regarded as constant in the equation, we know that the first member must represent the differential of some function of , for which we may write . We have thus
(78)
which may be integrated by quadratures and gives as functions of , and thus as function of .

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 (), is also a quadrature, since the equation to be integrated may be expressed in the form

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

and and are given in terms of by the differential equations of motion. When we have obtained integral equations, we may regard and as known functions of and . 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
(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 is a particular case of that represented by . The system of arbitrary constants has certain properties notable for simplicity. If we write for in (77), and compare the result with (78), we get

(80)
Therefore the multiple integral
(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 , and . We cannot use the principle of conservation of extension-in-phase until we have made integrations. Let us suppose that the constants have been determined by integration in terms of , and , leaving a single constant () to be thus determined. Our finite equations enable us to regard all the variables as functions of a single one, say .

For constant values of , we have

(82)
Now
where the limits of the integrals are formed by the same phases. We have therefore
(83)
by which equation (82) may be reduced to the form
(84)
Now we know by (71) that the coefficient of is a function of . Therefore, as are regarded as constant in the equation, the first number represents the differential of a function of , which we may denote by . We have then
(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'
(86)
for the integration of the equation
(87)

The system of arbitrary constants has evidently the same properties which were noticed in regard to the system .


  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).