Lagrange's formula.
We may employ Hamilton's principal function to prove a very remarkable
formula connecting any _two_ slightly disturbed natural motions of the
system. If we use the symbols [delta] and [Delta] to denote the
corresponding variations, the theorem is
d
--[Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r) = 0; (1)
dt
or integrating from t to t',
[Sigma]([delta]p'_r.[Delta]q'_r - [Delta]q'_r.[delta]q'_r) = [Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r). (2)