Page:Elementary Principles in Statistical Mechanics (1902).djvu/27

ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS

CHAPTER I.

GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE.

We shall use Hamilton's form of the equations of motion for a system of $n$ degrees of freedom, writing $q_{1},\ldots q_{n}$ for the (generalized) coördinates, ${\dot {q}}_{1},\ldots {\dot {q}}_{n}$ for the (generalized) velocities, and

$F_{1}\,dq_{1}+F_{2}\,dq_{2}\ldots +F_{n}\,dq_{n}$

(1)

for the moment of the forces. We shall call the quantities $F_{1},\ldots F_{n}$, the (generalized) forces, and the quantities $p_{1}\ldots p_{n}$, defined by the equations

$p_{1}={\frac {d\epsilon _{p}}{d{\dot {q}}_{1}}},\quad p_{2}={\frac {d\epsilon _{p}}{d{\dot {q}}_{2}}},\quad \mathrm {etc.,}$

(2)

where $\epsilon _{p}$ denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates,^[1] and on this account we denote it by $\epsilon _{p}$. This will not prevent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the ${\dot {q}}$'s and $q$'s. But in expressions like $d\epsilon _{p}/dq_{1}$, where the denominator does not determine the question, the kinetic

1. The use of the momenta instead of the velocities as independent variables is the characteristic of Hamilton's method which gives his equations of motion their remarkable degree of simplicity. We shall find that the fundamental notions of statistical mechanics are most easily defined, and are expressed in the most simple form, when the momenta with the coördinates are used to describe the state of a system.