Page:Elementary Principles in Statistical Mechanics (1902).djvu/36

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12

EXTENSION-IN-PHASE

second configuration will coincide with the third system of coördinates. This will give $D_{2}'=D_{3}''$ . We have therefore $D_{1}'=D_{2}'$ .

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 $q_{1},\ldots q_{n}$ , $Q_{1},\ldots Q_{n}$ are two systems of coördinates, and $p_{1},\ldots p_{n}$ , $P_{1},\ldots P_{n}$ the corresponding momenta, we have to prove that

\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

{\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

dQ/dp

are equal to zero, the determinant reduces to a product of two, and we have to prove that

{\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

{\dot {Q}}

's are linear functions of the

{\dot {q}}

's and therefore of the

p

's, with coefficients involving the

q

's, so that a differential coefficient of the form

d{\dot {Q}}_{r}/dp_{y}

is function of the

q

's alone, we get^[1]

↑ The form of the equation

{\frac {d}{dp_{y}}}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{x}}}={\frac {d}{d{\dot {Q}}_{x}}}{\frac {d\epsilon _{p}}{dp_{y}}}

in (27) reminds us of the fundamental identity in the differential calculus relating to the order of differentiation with respect to independent variables. But it will be observed that here the variables

{\dot {Q}}_{x}

and

p_{y}

are not independent and that the proof depends on the linear relation between the

{\dot {Q}}

's and the

p

's.

[1] The form of the equation

${\frac {d}{dp_{y}}}{\frac {d\epsilon _{p}}{d{\dot {Q}}_{x}}}={\frac {d}{d{\dot {Q}}_{x}}}{\frac {d\epsilon _{p}}{dp_{y}}}$
in (27) reminds us of the fundamental identity in the differential calculus relating to the order of differentiation with respect to independent variables. But it will be observed that here the variables ${\dot {Q}}_{x}$ and $p_{y}$ are not independent and that the proof depends on the linear relation between the ${\dot {Q}}$ 's and the $p$ 's.

[1]