This form is due to Lord Kelvin. When q1, q2, ... qm have been determined, as functions of the time, the velocities corresponding to the cyclic co-ordinates can be found, if required, from the relations (7), which may be written
| χ˙ = | ∂K | − α1q˙1 − α2q˙2 − ..., |
| ∂κ |
(18)
| χ˙′ = | ∂K | − α′1q˙1 − α′2q˙2 − ..., |
| ∂κ′ |
&c., &c.
It is to be particularly noticed that
(r, r) = 0, (r, s) = −(s, r).
(19)
Hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q˙1, q˙2, ... q˙m respectively, and add, we find
| d | (⅋ + K) = Q1q˙1 + Q2q˙2 + ..., |
| dt |