Lagrange's equations.
We may proceed to the continuous motion of a system. The equations of
motion of any particle of the system are of the form
mx" = X, my" = Y, mz" = Z (1)
Now let x + [delta]x, y + [delta]y, z + [delta]z be the co-ordinates
of m in any arbitrary motion of the system differing infinitely little
from the actual motion, and let us form the equation
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) =
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) (2)
Lagrange's investigation consists in the transformation of (2) into an
equation involving the independent variations [delta]q1, [delta]q2,
... [delta]q_n.
It is important to notice that the symbols [delta] and d/dt are
commutative, since
d dx d
[delta]x` = --(x + [delta]x) - -- = --[delta]x, &c. (3)
dt dt dt