Mathematics of Physics and Engineering

Lagrange's Equations^89

function: F = — VV; the negative sign is used by convention. The func-
tion V is called the potential of the force field F. The work done by a
conservative force along a path r(t), a < t < b, is

rb rb

w

b w,„uw (2.3.24)

pO no

= / F-rdt = / -W-rdt

J a J a

= _ [b (^te + ^dy + dV_dz\dt=_ fb dV(r(t)) M

Ja \ dx dt dy dt dz dt J Ja dt

= -{V(r(b))-V(r(a))),

That is, dW = —dV, and the work done by a conservative force equals the
change in potential.
Let us assume that all the forces Fi,...,Fn are conservative, and
denote by V\,...,Vn the corresponding potentials. By construction, we
have Fi = Fi(xuyi,Zi), i = l,...,n and so V* = Vi(xi,yi,Zi). De-
fine V = Vi + • • • + Vn. According to (2.3.21), dW = X)"=1 Ft • dr{ =
— Y17=i dVi = —dV. We now write V as a function of the generalized coordi-
nates: V = V(qi,...,qk), and get dW = -(dV/dqi)dqi (dV/dqk)dqk.
Comparison of the last equality with (2.3.21) results in the conclusion
Qj = —dV/dqj, j — l,...,fc. Thus, for conservative forces, equations
(2.3.23) become

±(0£K\_ dtK=_dV^ j = 1 (2325)

dt \ dq\j J dqj dqj'

We will transform (2.3.25) even further by noticing that V does not depend
on the velocities, and therefore is independent of qy. dV/dqj = 0. We then
define the Lagrangian

L = SK-V, (2.3.26)

and re-write equation (2.3.25) as

d fdL\ dL

dtvdqj-wr

0

(2

3

27)

Equation (2.3.27) has at least three advantages over (2.3.23): (a) a more

compact form; (b) a higher level of abstraction; (c) a possibility to include

constraints.

To conclude this section on Lagrange's equations, let us look briefly

at some NON-CONSERVATIVE FORCES. AS before, q = (q\,...,qk) is the

vector of generalized coordinates and q = (gi, • • • ,<7fc), the vector of the