Symplectic and Lie algebraic techniques in Geometric Optics (original) (raw)

We will show the usefulness of the tools of Symplectic and Presymplectic Geometry and the corresponding Lie algebraic methods in different problems in Geometric Optics. 1 Introduction: Symplectic and Presymplectic geometry Geometric techniques have been applied to physics for more than 50 years in many different ways and they have provided powerful methods of dealing with classical problems from a new geometric perspective. Linear representations of groups, vector fields, forms, exterior differential calculus, Lie groups, fibre bundles, connections and Riemannian Geometry, symmetry and reduction of differential equations, etc..., are now well established tools in modern physics. Now, after more than twenty years of using Lie algebraic mehods in Optics by Dragt, Forest, Sternberg, Wolf and their coworkers, we aim here to establish the appropriate geometric setting for Geometric Optics. Applications in computation of aberrations for different orders will also be pointed out. The basic geometric structure for the description of classical (and even quantum) systems is that of symplectic manifold. A symplectic manifold is a pair (M, ω) where ω is a nondegenerated closed 2-form in M. If ω is exact we will say that (M, ω) is an exact symplectic manifold. Let ω : X(M) → 1 (M) be given byω(X) = i(X) ω,ω(X)Y = ω(X, Y). The two-form ω is said to be nondegenerate whenω is a bijective map. Then M is evendimensional and it may be used to identify vector fields on M with 1-forms on M. Vector fields X H corresponding to exact 1-forms dH are called Hamiltonian vector fields. The 2-form ω is said to be closed if dω = 0. The simplest example is R 2n with coordinates (q 1 ,. .. , q n , p 1 ,. .. , p n) endowed with the constant 2-form ω = n i=1 dq i ∧dp i. Closedness of ω is very important because Darboux theorem establishes that for any point u ∈ M there exists a local chart (U, φ) such that if φ = (q 1 ,. .. , q n ; p 1 ,. .. , p n), then ω| U = n i=1 dq i ∧ dp i. Consequently, the example above is the local prototype of a symplectic manifold.It is also well known that if Q is the configuration space of a system, its cotangent bundle, T * Q = q∈Q T * q Q, called phase space, is endowed with a canonical 1-form θ on T * Q such that (T * Q, −dθ) is an exact symplectic manifold. More especifically, if (q 1 ,. .. , q n) are coordinates in Q then (q 1 ,. .. , q n , p 1 ,. .. , p n) are coordinates in T * Q and θ = n i=1 p i dq i , ω = n i=1 dq i ∧ dp i. A Hamiltonian dynamical systems is a triplet (M, ω, H) where M is a differentiable manifold, ω ∈ Z 2 (M) is a symplectic form in M and H ∈ C ∞ (M) is a function called Hamiltonian. The