558
9 Variational Principles in Mechanics and Their Applications
If the radius vector and velocity are expressed in scalar, since V (r) = V (x, y, z),
the Euler equation is
−m
dv
dt
− V r = 0
but
V r = V x
dx
dr
+ V y
dy
dr
+ V z
dz
dr
= (V x i + V y j + V z k) ·
dx
dr
i +
dy
dr
j +
dz
dr
k
= ∇V ·
d(x i + y j + zk)
dr
= ∇V ·
dr r
0
dr
= r
0
· ∇V
Note that the direction of r
0 is consistent with the direction of ∇V , therefore we
have
m
dv
dt
= −r
0
· ∇V = −r
0
· r
0
|∇V | = −|∇V |
Using Eqs. (1.3.7), (1.3.10) and (1.3.19), the above result can be obtained directly.
Multiplying both sides of the above equation by r
0 , we get
m
dv
dt
r
0
= m
dv
dt
= −|∇V |r
0
= −∇V
It follows that two kinds of representations are equivalent.
Example 9.6.6 A circular cylinder of mass m, radius r makes a pure roll on the
inner surface of a hollow circular cylinder. The mass of hollow circular cylinder is
M, the radius is R, which can rotate around the center horizontal O axis, as shown in
Fig. 9.8. The two circular cylinders are homogeneous. Write the differential equations
of motion of the system.
Solution The system has two degrees of freedom. Choosing the rotation angle ϕ
of the hollow circular cylinder and the rotation angle θ of the line of centers of the
two circular cylinders as the generalized coordinates. The rotational inertia of the
circular cylinder is J =
1
2
mr
2 . Let the angular velocity of the circular cylinder be ω.
The velocity of the center of the circular cylinder is
v 1 = (R − r ) ˙
θ
The angular velocity of the circular cylinder is
ω =
R ˙
ϕ
r
−
R − r
r
˙
θ =
1
r
[R ˙
ϕ − (R − r ) ˙
θ]
The kinetic energy of the system is