A Lie connection between Hamiltonian and Lagrangian optics

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

A Lie connection between Hamiltonian and Lagrangian optics

Introduction In Hamiltonian optics rays are described using the Hamiltonian H # ##n 2 # p 2 # 1#2 (1) The use of a Hamiltonian formulation is advantageous because Hamiltonian flows produce symplectic maps, and there is a well developed calculus, using both characteristic functions and Lie algebraic methods, for handling symplectic maps in an efficient and economical way [1, 2]. However, the use of a Hamiltonian approach has the consequence, perhaps at first surprising, that the map describing simple transit in a uniform medium (free flight in optical parlance, and a drift in accelerator parlance), is nonlinear. Therefore, in a Hamiltonian approach to optics, aberrations (nonlinearities) arise not only from transfer maps associated with lens interfaces, but also from simple transit within and between lenses. This circumstance is perhaps of les

A Lie connection between Hamiltonian and Lagrangian optics

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