G.2 Lagrangian Formulation
663
G.2.2 The Lagrange Equations
Let us determine the time rate of change of the generalized momentum p k as
dp k
dt
=
d
dt
∂T
∂ ˙
q k
.
(G.2.7)
Newton’s equations of motion in Cartesian form are
m i ¨
r i = F i ,
( i= 1, · · · , N) .
(G.2.8)
We may utilize the form
T =
N
k=1
1
2 m i ˙
r 2
i
for the kinetic energy T to obtain the expression
∂T
∂ ˙
q k
=
N
i=1
m i
˙
r i ·
∂ ˙
r i
∂ ˙
q k
(G.2.9)
for the derivative of the kinetic energy with respect to the generalized velocity ˙
q k
that enters into the defining relation Eq. (G.2.7) for the time rate of change of the
generalized momentum.
Let us examine briefly now an arbitrary component of the derivative
∂ ˙
r i
∂ ˙
q k
, say
∂ ˙
x i
∂ ˙
q k
: from Eq. (G.1.3), we may write
∂ ˙
x i
∂ ˙
q k
=
3N
j =1
∂
∂ ˙
q k
∂x i
∂q j
˙
q j
+
∂
∂ ˙
q k
∂x i
∂t
=
3N
j =1
∂x i
∂q j
δ jk =
∂x i
∂q k
.
We obtain this result because the Cartesian coordinates x i and their derivatives with
respect to the generalized coordinates q i are functions only of the q i and t, while the
time derivatives ˙
x i [through Eqs. (G.1.3)] are also functions of the time derivatives
˙
q j of the generalized coordinates. This result allows us to replace
∂ ˙
r i
∂ ˙
q k
in Eq. (G.2.9)
with
∂r i
∂q k
. Once we have carried out this step, we may differentiate Eq. (G.2.9) with
respect to t to obtain