Page:Elementary Principles in Statistical Mechanics (1902).djvu/28

4 energy is always to be treated in the differentiation as function of the $$p$$'s and $$q$$'s.

We have then

These equations will hold for any forces whatever. If the forces are conservative, in other words, if the expression (1) is an exact differential, we may set where $$\epsilon_q$$ is a function of the coördinates which we shall call the potential energy of the system. If we write $$\epsilon$$ for the total energy, we shall have and equations (3) may be written

The potential energy ($$\epsilon_q$$) may depend on other variables beside the coördinates $$q_1, \ldots q_n$$. We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by $$a_1$$, $$a_2$$, etc. We shall then have for the complete value of the differential of the potential energy where $$A_1$$, $$A_2$$, etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy ($$\epsilon$$) we shall have

It will be observed that the kinetic energy ($$\epsilon_p$$) in the most general case is a quadratic function of the $$p$$'s (or $$\dot q$$'s)