​

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 ${\displaystyle r_{1}\ldots r_{2n}}$ for the coördinates and momenta, and ${\displaystyle a\ldots h}$ 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

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

(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 (${\displaystyle t}$) in the finite form, if we eliminate ${\displaystyle dt}$ from these equations, we obtain ${\displaystyle 2n-1}$ equations in ${\displaystyle r_{1},\ldots r_{2n}}$ and their differentials, the integration of which will introduce ${\displaystyle 2n-1}$ arbitrary constants which we shall call ${\displaystyle b\ldots h}$. If we can effect these integrations, the ​remaining constant (${\displaystyle a}$) will then be introduced in the final integration, (viz., that of an equation containing ${\displaystyle dt}$,) and will be added to or subtracted from ${\displaystyle t}$ in the integral equation. Let us have it subtracted from ${\displaystyle t}$. It is evident then that

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

(72)

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

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

(73)

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

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

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

whence

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

(74)

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

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

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

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

(75)

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

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

(76)

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

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

(77)

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

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

(78)

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

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 (${\displaystyle a}$), is also a quadrature, since the equation to be integrated may be expressed in the form

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

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

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

and ${\displaystyle {\dot {r}}_{1}}$ and ${\displaystyle {\dot {r}}_{2}}$ are given in terms of ${\displaystyle r_{1},\ldots r_{2n}}$ by the differential equations of motion. When we have obtained ${\displaystyle 2n-1}$ integral equations, we may regard ${\displaystyle {\dot {r}}_{2}}$ and ${\displaystyle {\dot {r}}_{1}}$ as known functions of ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$. 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

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

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

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

(80)

Therefore the multiple integral

${\displaystyle \int \ldots \int da\,db'\,dc\ldots dh}$

(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 ${\displaystyle r_{1}\ldots r_{2n}}$, and ${\displaystyle t}$. We cannot use the principle of conservation of extension-in-phase until we have made ${\displaystyle 2n-1}$ integrations. Let us suppose that the constants ${\displaystyle b,\ldots h}$ have been determined by integration in terms of ${\displaystyle r_{1}\ldots r_{2n}}$, and ${\displaystyle t}$, leaving a single constant (${\displaystyle a}$) to be thus determined. Our ${\displaystyle 2n-1}$ finite equations enable us to regard all the variables ${\displaystyle r_{1}\ldots r_{2n}}$ as functions of a single one, say ${\displaystyle r_{1}}$.

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

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

(82)

Now

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

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

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

(83)

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

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

(84)

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

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

(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'

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

(86)

for the integration of the equation

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

(87)

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