Figure 1.12. Finsler billiard reflection
Let w be the intersection point of the tangent lin
points u and v. Then w = u + a£)=v + bri where a, b
It follows that Dw(f) = l,Dw(g) = - 1 and Dw(f +
is tangent to the mirror Z, then X is a critical point o
/ -f g, Finsler length of the broken line AXB. This
Finsler reflection law.
Of course, if the indicatrix is a circle, one obtains t
of equal angles. For more information on propagatio
Finsler geometry, in particular, Finsler billiards, see [2
1.4. Digression. Brachistochrone. One of the mos
lems in mathematical analysis concerns the trajectory
going from one point to another in least time, subject
tional force. This curve is called brachistochrone (in G
time"). The problem was posed by Johann Bernoulli
the 17th century and solved by him, his brother Ja
L'Hospital and Newton. In this digression we describ
of Johann Bernoulli who approached the problem fro
view of geometrical optics; see, e.g., [44] for a historic
Let A and B be the starting and terminal points
curve, and let x be the horizontal and y the vertic
convenient to direct the y axis downward and assum
coordinate of A is zero. Suppose that a point-mass dro
distance y. Then its potential energy reduces by mgy
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