By the Lagrange multipliers principle, X is a critica
only if Vf(X) is orthogonal to I. The sum of the uni
A to X and from B to X is perpendicular to I if and o
BX make equal angles with /. We have again obtain
reflection law. Of course, the same argument works if
smooth hypersurface in multi-dimensional space, and
geometries other than Euclidean.
Figur e 1.9. Reflection in a curved mirror
The above argument could be rephrased using a dif
ical model. Let I be wire, X a small ring that can m
wire without friction, and AXB an elastic string fixe
and B. The string assumes minimal length, and the eq
dition for the ring X is that the sum of the two equal
along the segments XA and XB is orthogonal to I. T
equal angles condition.
1.3. Digression. Huygens principle, Finsler me
billiards. The speed of light in a non-homogeneous ani
depends on the point and the direction. Then the traje
are not necessarily straight lines. A familiar example i
going from air to water; see figure 1.10. Let c\ and CQ
of light in water and in air. Then c\ Co, and the traj
is a broken line satisfying Snell's law
cos a _ Co
cos ft c\
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