Page:Elementary Principles in Statistical Mechanics (1902).djvu/39

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

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)