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Say I want to trace rays in 3 dimensions through a stack of flat plates of various refractive indices. My rays have canonical coordinates {Q,P}. The plates are normal to the z axis, all the rays start together at z=0. When a ray moves through a spacing from one surface to the next, Qout = Qin + t Pin / Pin,z, where that Pin,z is the z component of Pin and t is some constant. I can take all the first derivatives of {Qout,Pout} versus {Qin,Pin} and check that this transformation is, in fact, symplectic.

At the interface between two materials, the rays follow Snell's Law, which is only a function of Pin. Q doesn't change at the interface. But a transformation of {Q,P} that changes P and only depends on P isn't symplectic. What am I miss missing?

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  • $\begingroup$ Please use MathJax to write formulas or formula-like objects. It really improves your question. $\endgroup$ – Victor Pira Nov 7 '15 at 0:13
  • $\begingroup$ I understand nothing, surely due to your notations (and poor definitions). Please use latex math. $\endgroup$ – Fabrice NEYRET Nov 7 '15 at 10:29
  • $\begingroup$ And what do you mean by "sympleptic transform" ? $\endgroup$ – Fabrice NEYRET Nov 7 '15 at 10:31
  • $\begingroup$ A symplectic transform has a matrix of first derivatives that form a symplectic matrix link. The transform from inputs to outputs for a Hamiltonian system is always symplectic. $\endgroup$ – stackexchangesucksimleaving Nov 9 '15 at 23:33

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