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
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