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390 Dynamics of Mechanical Systems

where as before, n is the number of degrees of freedom of the system. (The minus sign is
chosen so that P is positive in the usual physical applications.)
To illustrate the consistency of this definition with the intuitive concept, consider a
particle Q having a mass m in a gravitational field. Let Q be at an elevation h above a
fixed level surface S as in Figure 11.11.1. Then, if Q is released from rest in this position,
the work w done by gravity as Q falls to S is:

w = mgh (11.11.2)

From a different perspective, if h is viewed as a generalized coordinate, the velocity v
of Q and, consequently, the partial velocity v of Q (relative to h) are:
h
˙
v = h k and v = k (11.11.3)
h
where k is the vertical unit vector as in Figure 11.11.1. The generalized force due to gravity
is then:

F =− mg ⋅kv = − mg (11.11.4)
h h

Let P be a potential energy defined as:

P = w = mgh (11.11.5)

Then, from Eq. (11.11.1), F is:
h

F =−∂P h ∂ =− mg (11.11.6)
h
which is consistent with the results of Eq. (11.11.4).
As a second illustration, consider a linear spring as in Figure 11.11.2. Let the spring have
modulus k and natural length . Let the spring be supported at one end, O, and let its
other end, Q, be subjected to a force with magnitude F producing a displacement x of Q,
as depicted in Figure 11.11.2. The movement of Q has one degree of freedom represented
by the parameter x. The velocity v and partial velocity v ˙ x of Q are then:

v = ˙ x n and v = n (11.11.7)
˙ x
Q(m)

k

k
h Q
F
O
S
n
x
FIGURE 11.11.1 FIGURE 11.11.2
A particle Q above a level surface S. A force applied to a linear spring.