energy is always to be treated in the differentiation as function of the ${\displaystyle p}$'s and ${\displaystyle q}$'s.

We have then

${\displaystyle {\dot {q}}_{1}={\frac {d\epsilon _{p}}{dp_{1}}},\quad {\dot {p}}_{1}=-{\frac {d\epsilon _{p}}{dq_{1}}}+F_{1},\quad \mathrm {etc.} }$

(3)

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

${\displaystyle F_{1}=-{\frac {d\epsilon _{q}}{dq_{1}}},\quad F_{2}=-{\frac {d\epsilon _{q}}{dq_{2}}},\quad \mathrm {etc.} }$

(4)

where ${\displaystyle \epsilon _{q}}$ is a function of the coördinates which we shall call the potential energy of the system. If we write ${\displaystyle \epsilon }$ for the total energy, we shall have

${\displaystyle \epsilon =\epsilon _{p}+\epsilon _{q}}$

(5)

and equations (3) may be written

${\displaystyle {\dot {q}}_{1}={\frac {d\epsilon }{dp_{1}}},\quad {\dot {p}}_{1}=-{\frac {d\epsilon }{dq_{1}}},\quad \mathrm {etc.} }$

(6)

The potential energy (${\displaystyle \epsilon _{q}}$) may depend on other variables beside the coördinates ${\displaystyle q_{1},\ldots q_{n}}$. We shall often suppose it to depend in part on coördinates of external bodies, which we shall denote by ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, etc. We shall then have for the complete value of the differential of the potential energy^[1]

${\displaystyle d\epsilon _{q}=-F_{1}\,dq_{1}\ldots -F_{n}\,dq_{n}-A_{1}\,da_{1}-A_{2}\,da_{2}-\mathrm {etc.} ,}$

(7)

where ${\displaystyle A_{1}}$, ${\displaystyle A_{2}}$, etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy (${\displaystyle \epsilon }$) we shall have

${\displaystyle {\begin{aligned}d\epsilon ={\dot {q}}_{1}\,dp_{1}\ldots &+{\dot {q}}_{n}\,dp_{n}-{\dot {p}}_{1}\,dq_{1}\ldots \\&-{\dot {p}}_{n}\,dq_{n}-A_{1}\,da_{1}-A_{2}\,da_{2}-\mathrm {etc.} \end{aligned}}}$

(8)

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

1. ↑ It will be observed, that although we call ${\displaystyle \epsilon _{q}}$ the potential energy of the system which we are considering, it is really so defined as to include that energy which might be described as mutual to that system and external bodies.