12.2 Calculus of Variations 671Thus
δA=∫x 2x 1[
∂F
∂u−
d
dx(
∂F
∂u′)
+
d^2
dx^2(
∂F
∂u′′)]
δu dx+
[
∂F
∂u′−
d
dx(
∂F
∂u′′)]
δu∣
∣
∣
∣
x 2x 1+
[(
∂F
∂u′′)
δu′]∣
∣
∣
∣
x 2x 1= 0 (12.9)
Sinceδuis arbitrary, each term must vanish individually:
∂F
∂u−
d
dx(
∂F
∂u′)
+
d^2
dx^2(
∂F
∂u′′)
= 0 (12.10)
[
∂F
∂u′−
d
dx(
∂F
∂u′′)]
δu∣
∣
∣
∣
x 2x 1= 0 (12.11)
∂F
∂u′′δu′∣
∣
∣
∣
x 2x 1= 0 (12.12)
Equation (12.10) will be the governing differential equation for the given problem and
is called Euler equation or Euler–Lagrange equation. Equations (12.11) and (12.12)
give the boundary conditions.
The conditions
[
∂F
∂u′
−
d
dx(
∂F
∂u′′)]∣∣
∣
∣
x 2x 1= 0 (12.13)
∂F
∂u′′∣
∣
∣
∣
x 2x 1= 0 (12.14)
are callednaturalboundary conditions (if they are satisfied, they are calledfreebound-
ary conditions). If the natural boundary conditions are not satisfied, we should have
δu(x 1 ) = 0 , δu(x 2 )= 0 (12.15)δu′(x 1 ) = 0 , δu′(x 2 )= 0 (12.16)in order to satisfy Eqs. (12.11) and (12.12). These are calledgeometricorforced
boundary conditions.
Example 12.1 Brachistochrone Problem In June 1696, Johann Bernoulli set the fol-
lowing problem before the scholars of his time. “Given two pointsA andB in a
vertical plane, find the path fromAtoBalong which a particle of massmwill slide
under the force of gravity, without friction, in the shortest time” (Fig. 12.2). The term
brachistochronederives from the Greekbrachistos(shortest) andchronos(time).
Ifsis the distance along the path andνthe velocity, we have
ν=ds
dt=
(dx^2 + dy^2 )^1 /^2
dt=
[1+(y′)^2 ]^1 /^2
dtdxdt=1
ν[1+(y′)^2 ]^1 /^2 dx