Nonlinear Mechanics - Physics at Oregon State University

1.3. VIRTUAL WORK AND GENERALIZED FORCE 9
acceleration is that we can separate out the forces of constraint, which are
always perpendicular to the direction of motion and hence do no work. The
trick is to write this in terms of generalized coordinates and velocities. This
is rather technical, but the underlying idea is simple, and the result looks
much like (1.8).
The qk’s are all independent, so we can vary one by a small amount δqk
while holding all others constant.
δri = ∑ ∂ri
δqk
∂qk
k
(1.9)
This is sometimes called a virtual displacement. The corresponding virtual
work is
δWk = ∑
(
∂ri
Fi ·
∂qk
)
δqk
(1.10)
We define a generalized force
i
ℑk = ∑
i
Fi · ∂ri
∂qk
(1.11)
The forces of constraint can be excluded from the sum for the reason explained
above. We are left with
ℑk = δWk
δqk
The kinetic energy is calculated using ordinary velocities.
T = 1
2
i
∑
mi ˙ri · ˙ri
i
∂T
=
∂qk
∑ ∂ ˙ri
mi ˙ri · =
∂qk
∑ ∂ ˙ri
pi ·
∂qk
∂T
∂ ˙qk
= ∑ ∂ ˙ri
mi ˙ri ·
∂ ˙qk
i
i
= ∑
i
pi · ∂ri
∂qk
(1.12)
(1.13)
(1.14)
(1.15)
Equation (1.7) was used to obtain the last term. A straightforward calculation
now leads to
??ℑk = d
( )
∂T
−
dt ∂ ˙qk
∂T
∂qk
(1.16)
which is the generalized form of (1.8).