
550 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Again, conditions (δq)
1
= (δq)
2
= 0 have been used. In Equation 16.56, we have written a total
time derivative with respect to x
i
in order to emphasize the fact that the derivative involves not only
the explicit dependence of £ on x
i
but also the implicit dependence on x
i
through q. Substituting
Equations 16.55 and 16.56 into Equation 16.54, we obtain
δq
L
qt
L
qx
L
qx
dd
i
d
i
i
∂
∂
−
∂
∂
−
∂
∂∂ ∂
=
∑
d
d
d
d/
()
1
3
dddddxxxt
123
=
∫∫∫∫
. (16.57)
Now, the variations q(x
1
, x
2
, x
3
, t) are completely arbitrary, so Equation 16.57 is satised if and
only if the integrand vanishes identically:
d
d
d
d/t
L
qx
L
qx
L
q
d
i