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5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

(p',q') = ~(p,q). 222 We

(p',q') = ~(p,q). 222 We claim that • is area-preserving. That is, • preserves the symplectic form dp^ dq, hence is a symplectomorphism on the symplectic manifold P = (-1,1) x C ~ ~ x S 1. In two dimensions, a symplectomorphism is the same as an area-preserving difeomorphism. In higher dimensions, symplectomorphisms are volume-preserving, but the converse is false - the symplectic structure implies additional constraints. normal Fig.l.1 Reflection of a ray or particle at a curve. p= cos 0 Fig.l.2 Coordinates for a single bounce. tangent To prove that • is area-preserving we show that det D~ = 1. We begin with a simpler case, in which C consists of two straight lines inclined at an external angle ~x as in Fig. 1.3. Here the reference point on C is distance d from the corner. We have Now C e ~ q' OS 0' q'= d+x = d+(d-q) sin0 sin0' (1.1) 0'= o~-0 (1.2) ~P--~' = 0 (1.3) ~q

~p'_-sin0' ~0' sin0' 3p -sin0 30 sin0 Bq_._~' = sin0 3q sin0'" ,4 q 223 Fig. 1.3 A single bounce on a polygonal curve. Thus D~ has the form sin0' 0 1 sin0 sin0 , - sin0----- 7 where * = -~ 3q' need not be computed. Clearly det D~ -- 1 as required. 0' q' v (1.4) By composing finitely many such polygonal arcs we obtain the analogous result when C is a convex polygon. Then, passing to the limit, we see that when C is a piecewise smooth curve, the map • is a symplectomorphism. This simple example provides informal motivation for the use of symplectic geometry in optics. In the next section we describe the underlying concepts in a more formal setting. Another proof that • is symplectic is given at the end of §4. See Rychlik [1989] for a more detailed analysis and further references. 2 Symplectic Manifolds, Phase Space and Lagrangian Varieties In this section we give precise definitions of some of the basic concepts in symplectic geometry, and standardise notation. Symplecfic Manifolds A symplectic manifold is a manifold P of even dimension dim P -- 2n, equipped with a nondegenerate closed 2-form co, its symplectic form. In local coordinates ~ is (1.5)

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