Figure 1.13. Brachistochrone

a(t) the angle between 7 and the respective level curv

A generalization of equation (1.11) is given by the follo

Theorem 1.15. Along a trajectory 7, one has:

cosa(£)

= const.

t

Exercise 1.16. a) Let the speed of light be given b

v(x,y) = y. Prove that the trajectories of light are

centered on the line y = 0.

b) Let the speed of light be given by the function v(x,y)

Prove that the trajectories of light are arcs of parabola

c) Let the speed of light be v(x,y) — y/l ~ x2 — y2.

trajectories of light are arcs of circles perpendicular to

centered at the origin. £

To conclude this chapter, let us mention numerou

the billiard set-up. For example, one may consider bill

tial fields. Another interesting modification, popular i

literature, is the billiard in a magnetic field; see [1

strength of a magnetic field, perpendicular to the plan

a function on the plane B. A charge at point x is acte

Lorentz force, proportional to B(x) and to its speed

force acts in the direction perpendicular to the moti

path of such a point-charge is a curve whose curvature

is prescribed by the function B. For example, if the

is constant, then the trajectories are circles of the