Calculus of variations
Example
Solve the problem
Z 1
0
tx+
x
2
dt!min,x( 0 )= 1.
The EulerñLagrange is
0 =Fx^0 = d
dt
Fx^0 = d
dt
t+ 2 x
= 1 + 2 x.
C=t+ 2 x, x( 0 )= 1
x=C
2
t
2 , x(^0 )=^1
x(t) 1 =
C
2 t
t^2
4.
x(t)=
t^2
4 +
C
2 t+^1
Solve the problem
Z 1
0
tx+
x
dt!min,x( 0 )= 1.
The EulerñLagrange is
0 =Fx^0 = d
dt
Fx^0 = d
dt
t+ 2 x
= 1 + 2 x.
C=t+ 2 x, x( 0 )= 1
x=C
2
t
2 , x(^0 )=^1
x(t) 1 =
2 t
t^2
4.
x(t)=
t^2
4 +
2 t+^1