
90 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Now, all of the indices, i, j, and k are dummies. If we change i to j in the rst term on the right-hand
side and i to k in the second term, then both terms become identical, and we have
q
T
q
aq
i
i
i
n
jk jk
jk
n
∂
∂
∑∑
,
.
(4.6)
This is a special case of Euler’s theorem, which states that if f(x
i
) is a homogeneous function of the
x
i
of order n, that is,
f(λx
i
) = λ
n
f(x
i
)
then
x
f
x
nf
i
i
i
∂
∂
=
∑
.
As an illustration of Euler’s theorem, consider the motion of a particle of mass m under the inuence
of a force derived from a potential dependent on position only. The kinetic energy T in Cartesian ...