386 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
(0, 0, a)Figure 4.1. An optimal steering problem.point (0, 0, a) after time T > 0 and that, amongst all such controls, minimizes
- 11T (u^2 2
1 +u 2 )dt.
2 0Solution of the optimal steering problem. An equivalent formulation is the
following: minimize the integral
T
~ r (x^2 + i/) dt2 lo
amongst all curves x(t) joining x(O) = (0, 0, 0) to x(T) = (0, 0, a) that satisfy the
constraint
i = yi: - xy.
The calculus of variations analogue of the Lagrange multiplier theorem states that
any solution must satisfy the Euler-Lagrange equations for the Lagrangian with a
Lagrange multiplier inserted:
L ( x, x,. y, y, z, z,.. .A , >.) = 1 (. 2. 2) .A (... )
2
x + y + z - yx + xy.
The corresponding Euler-Lagrange equations are given by
x -2.Ay = 0
y + 2.Ai: = 0
,\ = 0
i - yi: +xi; = 0.From the third equation, .A is a constant, and t he first two equations state that the
particle ( x( t) , y( t)) moves in the plane in a constant magnetic field (pointing in the
z direction, with charge proportional to th e constant .A.
- Note that this optimal steering problem gives rise to an interesting mechan-
ical system, a particle in a magnetic field.