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A Lie Connection Between Hamiltonian and Lagrangian Optics Alex J A Lie connection between Hamiltonian and Lagrangian optics Alex J. Dragt To cite this version: 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 HAL Id: hal-00955697 https://hal.inria.fr/hal-00955697 Submitted on 5 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 1, 1997, 149–157 A Lie connection between Hamiltonian and Lagrangian optics Alex J. Dragt Physics Department, University of Maryland, College Park, MD 20742, USA 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. Keywords: Lie algebra, Hamiltonian and Lagrangian optics 1 Introduction In Hamiltonian optics rays are described using the Hamiltonian 2 2 1=2 = (n p ) H (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 meth- ods, 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 acceler- ator 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 less consequence in graded index optics, where one ex- pects to have aberration effects associated with transit (at least within lenses), but it might be viewed as a drawback for the use of Hamiltonian methods for optics involving only uniform media. ` n Let D be the map for transit (drift) over a distance in a uniform medium having refractive index . Then, in Lie algebraic notation and using canonical coordinates, this map may be written in the form 2 2 1=2 = exp : `(n p ) : D (2) p for the action of this map on q and we find the result 2 2 1=2 q = q + `p =(n p ) D (3) 1365–8050 c 1997 Chapman & Hall 150 Alex J. Dragt p = p D (4) We observe, as advertised, that relation (3) is nonlinear. p Instead of using the canonical coordinates q , to specify a point in phase (ray) space, as is done in the _ q Hamiltonian formulation, one might instead use the Lagrangian coordinates q , . From (1) we have the result 2 2 1=2 _ = H=@p = p=(n p ) q (5) _ q q Suppose we let the map D act on the pair , . From (3)–(5) we find the result _ q = q + `q D (6) _ _ q = q D (7) We see that in Langrangian coordinates, unlike the case of canonical Hamiltonian coordinates, the map for simple transit is completely linear. Consequently, in a Lagrangian formulation of optics and in the case of uniform media, all aberrations arise only from transfer maps associated with lens interfaces. The purpose of this note is to study the relation between Hamiltonian and Lagrangian coordinates and dynamics in more detail. 2 Transformation between Hamiltonian and Lagrangian Coordi- nates To describe the relation between Hamiltonian and Lagrangian optics, introduce the transformation T with the property q = q T (8) 2 2 1=2 _ p = q = p=(n p ) T (9) _ T q p q Thus, maps the Hamiltonian variables , into the Langrangian variables , q . Note from (8) and (9) q p that T is not symplectic, since it does not preserve the fundamental Poisson brackets between and . Let us employ T to rewrite relations (3), (4), (6) and (7). From (6), (7), (8) and (9), we deduce the relations _ DT q = Dq = q + `q = T q + `T p = T (q + `p) 2 T exp`=) :p :]q = (10) _ _ DT p = Dq = q = T p 2 T exp [(`=) :p :]p = (11) Upon comparing (2), (10) and (11), we obtain the operator result 1 2 DT = exp [(`=) : p :] T (12) or, more explicitly, 1 2 2 1=2 2 exp `(n p ) :]T = exp [(`=) :p :] T (13) A Lie connection between Hamiltonian and Lagrangian optics 151 3 Explicit Form for Transformation T It would be useful to have T itself in explicit operator form. Although we know that is not symplectic, it may still be possible to write T in Lie form with the aid of some non-Hamiltonian vector field. Consider the vector field V defined by the equation 2 = p p (@=@p) V (14) Evidently, V has the properties q = V (15) 2 p = p p V (16) Let us use V to generate an autonomous flow parameterized by an independent variable that we will call t. Doing so gives the differential equations 0 = V q = q (17) 0 2 = V p = p p p (18) ddt) Here we have used a prime to denote the differentiation ( . Let us integrate (17) and (18). In doing so, we will find that we are well on our way to discovering an explicit representation for T . From (18) we find the result 0 2 2 2 p =(ddt)(=)(p )=(p ) p (19) 2 p Let denote the quantity , 2 = p (20) With this notation, (19) can be written in the forms 0 2 = (21) or 2 =dt d (22) 0 () = Equation (22), with the initial condition , can be integrated by quadrature to give the result 0 0 1 (t)= ( t ) (23) p p Next substitute (23) into (18). Then, for the components j of we find the differential equations 0 0 0 1 p = ( t ) p j (24) j These equations can be rewritten in the form 0 0 1 dp p = ( t ) dt j j (25) 0 p () = p and thus can also be integrated by quadrature. The results, with the initial conditions j ,arethe j relations 0 0 )=(=) log( t ) log(p p j (26) j 152 Alex J. Dragt or 0 0 1=2 p = p =( t ) j (27) j Finally, upon combining (20) and (27), we find the result 0 0 2 1=2 (t)=p =[ t(p ) ] p (28) In terms of vector fields, (28) is equivalent to the result 2 1=2 tV )p = p=[ tp ] exp ( (29) which has as a special case the result 2 1=2 =)V ]p = p=[ p ] exp (30) Note that the left-hand side of (30) resembles the left-hand side of (9). In addition, we immediately have from (15) the result =)V ]q = q exp [( (31) Next, let W be the vector field defined by the equation = p (@=@p) W (32) For this field we have the result q = W (33) p = p W (34) The vector field W can also be used to generate a flow. For this flow we get the result log n)W ]q = q exp (35) log n)W ]p = exp log n)p =(n)p exp [ (36) log n)W ] exp=)V ] Now consider the joint effect of exp [ and . From (31) and (35) we find the relation log n)W ] exp=)V ]q = q exp [ (37) From (30) and (36) we find exp [log n)W ] exp=)V ]p 2 1=2 = exp [log n)W ]fp=[ p ] g 2 2 1=2 p =[n p ] = (38) Upon comparing (8) and (9) with (37) and (38), we conclude that T is given by the operator relation = explog n)W ] exp=)V ] T (39) A Lie connection between Hamiltonian and Lagrangian optics 153 4 Variation Consider the map U defined by writing U = T explog n)W ] exp [log n)W ] exp=)V ] explog n)W ] = (40) From (35) and (36) we immediately have the relations n)W ]q = q explog (41) n)W ]p = np explog (42) Combining (8) with (41) gives the result q = q U (43) Combining (9) with (42) gives the result _ p = T np = nT p = nq U (44) T Note that U , like , is also not symplectic. Suppose we use U to rewrite the relations (3), (4), (6) and (7). From (6), (7), (43) and (44), we find the results _ DU q = Dq = q + `q = U q +(n)U p = U [q +(n)p] 2 U exp f[`=(n)] :p :gq = (45) _ _ DU p = Dnq = nDq _ = nq = U p 2 U exp f[`=(n)] :p :gp = (46) Consequently, we have the operator relation 1 2 DU = expf[`=(n)] :p :g U (47) or, more explicitly, 1 2 2 1=2 2 exp [: `(n p ) :]U = exp : [`=(n)]p : U (48) The relations (47) and (48) are particularly appealing because they relate the exact map for transit to the map for transit in the Gaussian approximation.
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