A Lie connection between Hamiltonian and Lagrangian optics

Abstract : It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.
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Discrete Mathematics and Theoretical Computer Science, DMTCS, 1997, 1, pp.149-157
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Alex J. Dragt. A Lie connection between Hamiltonian and Lagrangian optics. Discrete Mathematics and Theoretical Computer Science, DMTCS, 1997, 1, pp.149-157. 〈hal-00955697〉

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