Time scaling as an infinitesimal canonical transformation

TIME SCALING AS A N I N F I N I T E S I M A L TRANSFORMATION CANONICAL JOS]~ F. C A R I N E N A Departamento de Fisica Te6rica, Universidad de Zaragoza, 50.009 Zaragoza, Spain. L U I S A. I B O R T * Department of Mathematics, University of California, Berkeley, California 94720, U.S.A. and E R N E S T O A. L A C O M B A Departamento de Matemiiticas, Universidad Autbnoma Metropolitana, Iztapalapa, P.O. Box 55-543 M~xico D. F., M~xico. (Received: 28 January, 1987; accepted: 14 July, 1987) Abstract. We show that time scaling transformations for Hamiltonian systems are infinitesimal canonical transformations in a suitable extended phase space constructed from geometrical considerations. We compute its infinitesimal generating function in some examples: regularization and blow up in celestial mechanics, classical mechanical systems with homogeneous potentials and Scheifele theory of satellite motion. 1. Introduction The purpose of this paper is to show how any change of time scale in Hamiltonian systems can be interpreted as an infinitesimal canonical transformation. In particular, we are interested in the application to these ideas to the regularization and blow up of collisions and other regularizations of problems in celestial mechanics, and to the theory developed by Bond and Janin (1981) (see also the references therein), describing canonical orbital elements in terms of arbitrary variables and its applications to satellite theory. There are some conceptual difficulties in celestial mechanics regarding the interpretation of the regularization of binary collisions and of the blow up of total collisions, McGehee (1974). These transformations always follow two steps: we first change phase space coordinates so that all the terms in the energy relation are bounded at the specific singularity, and then change the time scale through multiplication of the vector field by a convenient function k to prevent time from becoming infinite at the singularity. The transformation in phase space coordinates can always be made canonical. This was first shown for regularization of binary collisions and the three body problem by Levi-Civita (1906 and 1920). That this is also the case for blow up of total collisions has recently been shown by Lacomba and Ibort (1986). The time scaling defines a new system of differential equations in terms of a fictitious * On leave of absence from the Departamento de Matemfitica Aplicada, Universidad de Zaragoza, 50.009 Zaragoza, Spain. Research partially supported by CONACYT (M6xico), Grant PCCBBNA 022553 and CICYT (Spain). Celestial Mechanics 42 (1988) 201-213. 9 1988 by Kluwer Academic Publishers. 202 JOSl~ F. CARII~ENA ET AL. time 7, but the physical time t has to be computed on each solution by a quadrature from d t / d z = k. There is of course the well-known Poincar6 trick (Siegel and Moser, 1971 p. 35), allowing us to interpret the scaled vector field k X n of the vector field X n with Hamiltonian function H, as derived from a new Hamiltonian function K h = k ( H - h) on each energy level h. This is not completely satisfactory because there is no a canonical transformation taking X n into Xr~ (see Abraham and Marsden 1978, p. 384 for a discussion of canonical transformations for time dependent systems). The above discussion suggests that if we want to understand time scaling of Hamiltonian systems as infinitesimal canonical transformations we should add the time t as another coordinate. In this way we get an odd dimensional evolution space which is a presymplectic manifold. Also our vector field X n is made time-dependent by addition of a trivial time component. A coisotropic imbedding of the evolution space would allow us to form an extended phase space by adding one dimension, completing the presymplectic form to a symplectic form depending on the Hamiltonian. In this extended phase space a simple Hamiltonian function defines a vector field whose restriction to the original phase space gives the scaled vector field. Furthermore, the scaling can be easily understood as an infinitesimal canonical transformation generated by a special class of functions. The paper is organized as follows: In Section 2 we review the extended and the coisotropic imbedding formalism for time-dependent Hamiltonian systems. Section 3 is devoted to show how we can use the constructions explained in Section 2 to describe time scaling as an infinitesimal canonical transformation. Finally, in Section 4 we study specific examples of this theory taken from regularization and blow up of singularities in celestial mechanics, mechanical systems with homogeneous potentials and Scheifele theory of satellite motion in extended phase space. 2. Time Dependent Hamiltonian Systems Time dependent Hamiltonian systems are usually described in the framework of presymplectic geometry (Abraham and Marsden, 1978, p. 370) and the presymplectic 2-form (i.e. a closed, possibly degenerate, 2-form of constant rank) is defined using the Hamiltonian of the system. More explicitely, if M is a 2n-dimensional phase space for any fixed time and 09 is a symplectic form (i.e. o9 is closed and nondegenerate), a new manifold N = M • ~ is considered, and a Hamiltonian H can be used to define a closed 2-form o9n in N as follows: con = ~z~ 9co + dH A dt, where r~ denotes the projection on the first factor rc~:M x ~ , M and t: M • ~ ~ ~ is the projection on the second factor. The pair (N, o9/~) is a presymplectic manifold, sometimes called a geometrical mechanical system (Curtis and Miller, 1985), the rank of o9/~ is 2n everywhere and the dynamics is given by the uniquely defined vector field X n in N = M • ~ satisfying X~/~ Ker o9n and X n ( t ) = 1. In canonical coordinates (q~,p~) for 09 we have 09n = Z(dq ~ A dp~) + dH A dt, and the explicit form of X n is given by Xu = c3t + i= ~1 ~qi ~qi ~-Pi " (2.1) TIME SCALING 203 Its integral curves will be solutions of Hamilton's equation, the equation t ' = 1 meaning that the parameter of such curves is just the time t. However a different approach has been developed by several authors (see e.g. Kuwabara, 1984 and Asorey et al., 1983) to fit these systems into the same symplectic formalism as an autonomous Hamiltonian system. This formalism is called the extended formalism. In such an approach, an extended manifold M e = N • ~*, a symplectic form f~e in M e given by n ' c o + dt A du, and a new Hamiltonian He = H o Pl + u are considered. Here u denotes the new coordinate u:M e ~ ~, i.e. the projection on the second factor, and n and p l are the natural projections n: M e , M and p~:M e , M • [~ = N. Notice that dt A du is the constant canonical symplectic structure defined in E • E*. Then (Me, f~e, He) is a Hamiltonian system, the corresponding dynamical field X e is defined by the equation i(Xe)f2 e = dH e, and in appropriate coordinates is written (Asorey et al., 1983) c3 " (C3pH c3 X e = ~t + i= 21 Oq i 0H c3) c~qi ~-Pi c3H c3 (2.2) ~t ~U" Therefore X e is p~-projectable on N. In this approach the dynamical evolution is given by the one-parameter subgroup of canonical transformations generated by X e. This method of dealing with time-dependent systems is, no doubt, an ad hoc method and in a more recent paper (Carifiena et al., 1987) it was shown to be a particular case of a more general way of extending the presymplectic system (N, coil, 0) to a symplectic, one. Given a presymplectic manifold (S,a) it is possible to imbed it in a higher dimensional symplectic manifold (P, f2) in such a way that the pull-back of f2 to S by the imbedding coincides with a and the symplectic orthogonal complement of the tangent space of S defined by T S n = { X ~ T P I f 2 ( X , Y ) = O V Y ~ T S } is just Kera. Such imbedding exists, it is called a coisotropic imbedding, and it is essentially unique (Gotay 1982 and Marle, 1983). In the particular case we are dealing with, the coisotropic imbedding for the presymplectic system (N, con) is explicitly constructed using the characteristic bundle E~,, of con. The characteristic bundle E,o,~ is made up with the null vectors of cou, i.e. E,o~ = { X ~ TNIcou(X, Y) = 0 V Y ~ T N } . It is easily shown that E,oH is a one-dimensional vector bundle over N. In fact E,o" is trivial because X n is a global non-vanishing section. Moreover X n provides a global coordinate system E" E~,, , ~ assigning to each point ~ on E,o" its coordinate E with respect to X u, i.e. ~ = E X n. Obviously we can identify E,o,~ with p = N • ~ using the map q~(x) = (p~(x), E) where p~ denotes the projection of E~,, on N and E the coordinate of r as before. The symplectic structure f2 on E,o" is given by p~*coH + dt A dE, which in local coordinates is written as E(dq ~ A dp i) + dt A d(E - H). The symplectic structure f~ in E,o,~, identified in what follows with N • ~, induces a new Poisson bracket on it, denoted by {, }n- As it is easily checked, the basic commutation relations for the set (q~,p~, t, E) on N • ~ are {qi, pj}t a = 6ji {qi, E}n = OH/c3p, { p , , E } , = - c~H/c~q~ {t,E}n = 1. (2.3) Obviously the map ~,:M e , N x ~ given by ~(q,p,t,u) = (q,p,t,u + H(q,p,t)) (2.4) 204 JOS~ F. CARII~IENA ET AL. is a symplectomorphism between (Me, De) and (N x ~,D). The original manifold N can be thought as being the zero section N • {0} of N x ~, defined by the constraint function E - 0 . The submanifold N x {0} is coisotropic, because its codimension is one. It is obvious from the commutation relations (2.3) that the dynamics in the space N • ~ is given by the constraint function E = 0. In other words, Hamilton's equations for the Hamiltonian E with respect to {, }n are just the dynamics defined by the vector field X u when restricted to N. Consequently the Hamiltonian dynamical system (P,f~,E) provides an extension of the time-dependent Hamiltonian system (N, coH). More details can be found in Carifiena et al. (1987). The previous discussion can be summarized in the following theorem. T H E O R E M . Let H be a time-dependent Hamiltonian defined for each time t in the phase space (M, co) with Poisson bracket {, },o. Then there exists a canonical extension (P, D, E) of the time dependent system (M x ~, cou), which is equivalent to the extended phase space (Me, De,He) via the translation by H of the energy variable u in M e (see 2.4). In addition the equations of motion for E are written in canonical form as dF dt = {F,E}n where F is any function on P, { , )~ is the Poisson bracket associated to the symplecticform D whose commutation relations are given in (2.3). The space P is identified with Mx~x [~*, and E is the projection on the last factor. The extension given in the theorem above will be called the canonical extended phase space. The expression in canonical coordinates for the vector field corresponding to a constraint function gE with g being any differentiable function in N • E is given by XoE - ~ ( OH c3g + E ~ -t g Opi c~Pi Ec3Pi c3E 63qi i=1 OH i= + c~g OH + ~qi OEJOpi 1 o+E Ot E 0 OE -+ (2.5) In particular, when g = 1 we obtain XE=~tFi:x-~pi~-qi Oqi i The restriction of X oE to N reduces to i=1 c3q i where k is the function k = giN. The integral curves of Xg~I N are determined by the 205 TIME SCALING following equations d q i = k aH dPi _ dz dr - dt dz c3Pi k(q, p, t) dE 9(q, P, t, 0) dr OH k--aqi 0. (2.6) The point we want to stress here is that equations (2.6) look like Hamilton's equations, in which the time variable has been scaled by a function k(q, p, t) = 9(q, P, t, 0). Hence, the scaled Hamiltonian vector field can be regarded as the restriction to N of the Hamiltonian vector field X : in the extended symplectic structure where k is the scaling function. This generalizes and provides a geometrical meaning for the Poincar6 trick mentioned in the introduction. 3. Time Scaling as a Canonical Transformation In the preceding section a description of the time scaling of Hamilton's equations was given in the framework of the coisotropic imbedding formalism for time-dependent systems. We have seen how to derive the scaled vector field as the restriction of a Hamiltonian vector field in the extended space. We want to go one step further showing that time;scaling is an infinitesimal canonical transformation. We will prove first the following result: T H E O R E M . Usin9 the notation introduced in Section 2, the infinitesimal canonical transformation 9enerated by the Hamiltonian f E defined in (N x ~,f~) for any function f on N x ~, transforms the vector field XHon N into the new vector field { f E}nl{E=o}X n on N. Proof. The action of the infinitesimal canonical transformation generated by f E on the vector field X E, which is the generator of the dynamics on N x E, is obtained computing the commutator [XyE, XE]; namely, [X fE, XE] = --X{fE, E}" = --X{y,E}, E. From the relation Xel/E=o/= XH, we get [XfE, XE-II{E=0} = {E,f}oI{E=o}XEI{E=O} = {E,f}.I{E=o}XH. Another way of stating this theorem is that the vector field X yE is an infinitesimal symmetry of X H that scales the integral curves of X n in such a way that t' = {E,f}nl/E_o/. Thus the important point for applications is whether or not there exists a function f ~ C~(N x E) satisfying k = {f, (3.1) for a given scaling function k E C~ The difference between two functions fl and f2 satisfying (3.1) is a constraint function, that is, a multiple of the function E, because N has codimension 1 in N • R. Then we must just find a particular solution for k = {f, I The following comments show that (3.1) is locally equivalent to solving a linear 206 JOSl~ F. CARII~IENA ET AL. partial differential equation by the method of characteristics. We observe that (3.1) is x [~) such that k = gl{e=o}. In local equivalent to g + Eh = { E , f } n with g , h ~ C~ coordinates this is expressed by E1 g + Eh = c3--i- i= ~qi Oqi 9 Furthermore, we can choose f as being functionally independent of E (up to a redefinition of h) and finally, when restricted to {E = 0} we must look for a solution of the partial differential equation (a linear Hamilton-Jacobi equation) Of Of (~f OH k = Ot + { f , H } ~o = c~t ~ ~qi Op i Of OH t~pi Oqi (3.2) with f and k being C ~ functions in N. A set of 2n + 1 functionally independent local first integrals {~i} of the characteristic system dq i dt - dpi dz Hpi 1 Hqi k of (3.2) always exists locally because X n has no equilibrium points. It will be enough to write the general solution of (3.2) in an implicit form r p, t, z) = ~(~1,...,~2,+ 1) where tI) is an arbitrary function such that OO/c~z :~ O. Then, (3.1) has at least a local solution that can be global or not depending on the topological properties of the manifold N. The result of this discussion is summarized in the following COROLLARY. A n y scaling dr~dr = k of a time dependent Hamiltonian can always be considered as an infinitesimal canonical transformation in the canonical extended phase space of the system ( M x N, oM). The computation of f in (3.2) will be illustrated with various examples from celestial mechanics. 4. Applications to Celestial Mechanics In this Section we describe the regularization and blow-up techniques, used in celestial mechanics to deal with problems with singularities, using the above formalism. The behaviour of trajectories of mechanical systems, described by potentials with a singularity at one or more points of the configuration space, when close to the singular points are better studied after a process of regularization about the singular point has been carried out. Most of these regularization procedures consist on first blowing the singularity up, then performing a transformation which scales the time in such a way that the singularity disappears from the vector field. This process adds new points to the energy levels. But the time rescaling step is a formal manipulation of the dynamical equations in which the time derivative is replaced by making use of a relation dt/dz - k, where k is a function that usually only depends on the norm TIME SCALING 207 r = Ixl of the position vector x. The function k is suitably chosen to remove the singularity of the vector field. In spite of its computational usefulness, this time rescaling process is not fully satisfactory. To improve numerical algorithms and to use perturbation methods it would be desirable to understand reparametrizations along the integral curves of the equations of motion as infinitesimal transformations performed in the phase space variables, (qi, pi, t). The corollary of the theorem stated in Section 3 implies that any rescaling can be considered as an infinitesimal canonical transformation scaling the dynamical vector field X u. As mentioned in the introduction, in all examples to follow, the blow-up part of the process can be considered as a preliminary transformation in phase space that makes all the terms in the energy relation H = h bounded. Then, we compute (when possible) the function f in (3.2) which gives the infinitesimal canonical transformation XIE for the required scaling k. Example 1. Let us first consider the Levi-Civita regularization of the Kepler problem in one dimension. In normalized coordinates, we denote position and velocity (momentum) respectively by x > 0 and p~[~, with total energy (Hamiltonian) H ( x , p ) = p 2 / 2 l 1 / x . The energy relation H = h can be written a s x p 2 / 2 - 1 = hx. The Levi-Civita transformation gives new coordinates (~,r/) where x = ~2, p = r//(2~). This transformation is canonical, and the Hamiltonian in the new coordinates becomes H(~,r/) = (r/2/8 - 1 ) / ~ 2 which is a homogeneous function of degree - 2 in ~. The new energy relation (4.1) I"/2/8 - - 1 - - h ~ 2 is already regular for ~ = 0. Writing Hamilton's equations in these new coordinates, we find that a change of time scale dt/dr = ~2 is necessary to make them regular at ~ = 0. To compute the time rescaling generator f of X u we find H~, H,, and substituting then into (3.2) we obtain the following first order linear partial differential equation for f ~2 = c3f/c3t + (r//4~z)(c3f /c3~) + (2/~3)(r/2/8 - 1)(0f/~r/) (4.2) whose characteristic system of ordinary differential equations is dt/l ~ d ~ / ( 4 ~ 2 ) - - d r l / ( 2 - - ~ ~ ) ~ d z / ~ 2. (4.3) From (4.3) we get the energy relation as expected. To solve (4.2) we have to consider three cases, according to the sign of the energy. We will illustrate two of the three cases: (1) If h = 0 , the energy relation (4.1) gives r/z= 8. From (4.3) we obtain d z = (4~#/r/)d~, that is, dz = x / ~ 4 d ~ whose solution is z = ~sx/~/5. As is easily checked f(~, r/, t) = ~ s x/~/5. Notice that a free use of the energy relation has been made in this reasoning, and this will be important in the simplification of other computations to follow. Similarly any other first integral of the system can be used. (2) If h > 0, we can write dz = (~3/2h)dr/. Using the energy relation one gets 2dz = h-5/2(r/2/8 - 1) 3/2 dr/ 208 JOSl~F. CARII~ENAET AL. which is easily integrated. Again using the energy relation to simplify the calculations we find that f (~, q, t) = (1/h2)[(1/64)~q(q 2 - 20) + (3/8)x/~/h In [rl/x~ + satisfies (4.2). (3) When h <0, the difference is that we get 2dz = [h1-5/2(1 --/~2/8)3/2 dr/, which can be integrated as before with a different substitution. Example 2. Consider now the Levi-Civita regularization of the Kepler problem in 2 dimensions. Identifying Nz = C, the Hamiltonian is H(x, p) = Ipl 2 - 1/Ixl 2 with p e C, x e C, x -r 0. The Levi-Civita transformation defines the new coordinates by x = ~2, p = r//2~*, where * denotes complex conjugation and ~ = ~1 + i~2, r/= r/1 + ir/2. In the new coordinates, we have Equation (4.2) is ( 1 c3f 8f 41~1z = c~t c3f t 41gl 2 g]l C~gl ~ /72 ~ 2 )2( 2 )( q 4 8 1 ~1 ~1 and the natural scaling is dt/dz =41~12. The characteristic system of the partial differential equation above is now hr at/1 = d~ 1/ ( 2 ~ ) 12. = d~2 / ( ~4~ 1 )2 = dr/l/\/ ( 2i~[-i = dz/4[~l 2. = dq2/\/f2h~2" i~[-~,]~ = (4.4) The order of (4.4) is larger than in our first example. So the computation of a solution for f is more complicated. We do have 3 first integrals h, h 1 and C given by r!2/8 = h~ 2 + hl, ri2/8 = h~ 2 + (1 - hi) , and ~r/2 - ~2g]l -- C. Example 3. We will now treat the blow up transformation of the binary collisions in the Kepler problem in one and two dimensions. For the one dimensional case we can transform to Levi-Civita coordinates, getting H ( ~ , J T ) - - ( 1 / ~ z ) ( r 1 2 / 8 - 1). Hamilton's equations read dq 2(r/8 d t = ~3 1 ) d~ r/ d t = 4~----5" (4.5) The difference between this situation and example 1 is that the time scaling is now taken to be dt/dz = ~3 instead of dt/dz = ~2. Thus we obtain the blow up equations dr/ ?/2 dz = 4 d~ 2; dr = r/ 4r TIME SCALING 209 Changing the left-hand side of (4.2) we get ~3 Of t- r/ 0f t=c~t 4~ 2 c~ ~-g 1 (4.6) c~r/ for the scaling generating function f Hence, there are only slight changes in the computations of example 1. If h = 0 , we get d z = x / ~ S d ~ , obtaining f ( ~ , q , t ) = % / ~ 6 / 6 and if h > 0, we get dz = (r/2/8 - 1)2dr//2h a, which is easily integrated to give 1( r]3 r]5 ) f(~, r/, t) = 2~t3 1 12 (4.7) 3--2-0" In contrast to example 1, (4.7) works when h < 0. We now analyze the 2-dimensional case. The transformation used here is no longer related to the corresponding Levi-Civita regularization. The reason is that we need special coordinates in configuration space. In Cartesian coordinates we write 1 H(xl, x2, pl,p2) = 2 2 (4.8) x//x2 + x 2" We take as blow up transformation the canonical transformation sending (xl,xz,pl,P2) into (2,0, o, po), where 2 = 2(x 2 + x 2 ) 1/4, v = 2 ( p l x 1 + p z X z ) / ) t , x I = 22cos 0/4, x 2 = ,~2sin0/4 and Po = x l P 2 - x z P l . The Hamiltonian (4.8) becomes 2(v2 H ( 2 , 0 , v , Po) = ,~2 4P~ ) -+ 22 2 . Indeed, Hamilton's equations are d2 4v d t - 22 d o _ 4 (v2 _ 2) + d t - 2a dO Pg' dt ~ = Po dpo dt ~ = 0. (4.10) Of course we have two first integrals, H = h and Po = c on fixed solutions. To analyze the apparent singularities, we consider two cases: (1) If c 4=0, then the energy relation 2(v2 + 4 c 2 / 2 2 - 2 ) = h 2 2 implies that 2 is bounded away from zero, as is classically known. Thus there are no binary collisions. Then no change of time scale is needed since (4.10) is regular. (2) If c = 0, we see that the change of time scale dr~dr = 23/8 transforms (4.10) into the regular system dr 2 ~ 2 - 1 d00 dr dpo dr = O. This change of scale does no make any harm if c ~ 0, and accordingly it will be considered in both cases. In fact, it is the same previously described in the one-dimensional case up to a multiplicative constant. We carry out a change of time 210 JOS~ F. CARIlqENA ET AL. scale by solving 23 ~"~" -F= t ( ] ) 2 0Of -~- [ 49 ( U- i _ _ 2 ) +( ~ ) 5 C2 ]0f (2)46~f . (4.11) F r o m the corresponding characteristic system of differential equations we can recover the two first integrals h and c. Also we obtain dz = (2/2)5v - 1 d2. Solving the energy relation for v, we obtain a solution f of (4.11) as given by the indefinite integral f(2) = _+ ~-~ h24 + 222 - 4c226 d2. (4.12) There are two branches o f f according to the sign of v, moreover f is undefined when v = 0 (turning points of the motion). When h = c = 0, (4.11) is easily integrated and gives f(2) = _ 26/192x~. To complete case (2) we can findffrom (4.6) when c = 0 and h 4: 0, but it is simpler to start from dz = h-1(2/2)4dv = (v2 - 2)2dv/4h a, using the energy relation. Therefore f(v) = h-3(v5/20 - va/3 - v). There are no branches, since we have used a variable different from that in (4.11). Notice that f is globally defined. Example 4. Consider the more general case of a celestial mechanical system with Hamiltonian H(q, p) = (pA- lpt)/2 - U(q), where A is a constant positive definite n x n matrix and U is homogeneous of degree - 1. The phase space is C x ~", where C is an open cone in ~" excluding the origin. We take the blow up canonical transformation (Lacomba and Ibort, 1986) changing to coordinates (2, Q, v,P) defined by 4 2 22 = 4x/qAq* Q = f2 q t 22 v = -~pq P = -~(p - (pQt)QA). The Hamiltonian is now 2( 2 + -fi4 p A - x p t _ H(2, Q, v,P) = 22 2U(Q) ) and using a Lagrange multiplier to take account of the constraints QAQ t = 1, PQ' = 0, Hamilton's equations for H can be written in the new coordinates as follows d--7= 5[~3 d--7=22 dt - PA- 1 -- dt = + ; PA-1U VU - U(Q) ) U + The required change of time scale is dt/d'c = 23/8, giving thepartial differential equation for the infinitesimal scaling generator 23 c~f t 22 8=ot + +- 8( 2 4 PA -~ P ' - PA -1 5- + ~- VU + U(Q) ~v + U PA x pt QA Oft (4.13) Op" TIME SCALING 211 Formula (4.11) is a particular case of(4.13). In general there is no global solution for f in terms of the variables, and multivaluated solutions may appear, as with any Hamilton-Jacobi equation. Local solutions for f a r e not easily computable, but always exist, as we mentioned at the end of Section 3. 5. The use of orbital elements in the extended phase space in terms of an arbitary independent variable (e.g. the eccentric anomaly) has been shown to be very successful in satellite theory (Bond and Janin, 1981, and references therein). The transformation from Delaunay elements to Scheifele variables (see below) follow the same scheme as the regularization and blow up examples previously described: a canonical transformation in the extended phase space, followed by scaling of the time variable relating the new time with some of the four standard anomalies. The discussion below sketches the computation of the infinitesimal generating function for time scaling in this context. Let us consider the classical phase space for the unperturbed two body problem and the canonical transformation in the direct elliptic domain 931 giving the Delaunay elements (see for example, Abraham and Marsden 1978) Example lD -- tt -- e sin u L D = x/#a go=~ h D =~ H o = cos I ~//#p where a,e,p,co,fLI and # have the standard two-body interpretation. The Hamiltonian in the Delaunay variables is given by FD = -(#2/2LD 2) + V where V represents the perturbation. The symplectic form on 931 is written in the new variables as follows: co = dl D/x dL D + dgD/x dG D + dh o ^ drip, and the extended phase space is gJl x R x R; the canonical momentum conjugate to 20 = t will be denoted A D and corresponds to minus the energy. The extended symplectic form is co + d2 D ^ dA o and the extended Hamiltonian is Fe= --(#2/2L2) + V+ A D. The canonical orbital elements will be obtained when using the canonical transformation defined by the generating function S(g D, l D, h D, 2D; G , L , H , A ) = 2DL + (#/(2L)l/Z)(lD -- 0(2D)) + + A0(2D) + Gg D + H h D with 0 a function such that c30/c32o -r 0. The coordinate transformation corresponding to this generating function is given by # I-- /~D t't (lD (2L)3/2 Ao = L + ( A -- O(~D)) LD -- ~//2L 212 JOS]~ F. CAREgIENA ET AL. and the image of the submanifold AD = 0 is given by the equation L+(A x~)t?O=o.o2D (4.14) The extended Hamiltonian F D + A D turns out to be ( A - la/(2L)l/2)(c30/C32D)+ V, or - - L + Vconsidering (4.14). The convenient scaling here is given by functions k such that kc~O/c~2D = 1. Following the theory developed in Section 1 and Section 2, the generating function for such scaling will be obtained solving the equation (3.1), or in terms of the Hamiltonian function, the equation (3.2). Because of the extremely simple form of the Hamiltonian in Schiefele variables when restricted to AD = 0, we get the following equation (in the unperturbed case V= 0) k= Ot t~l For instance, in the case k = A/2L (corresponding to 2 = v the elliptic anomaly), we get f = ( 9 / 4 ) A 3 / # 2 I, using the fact that k = (9/4)A3/# 2 on A D - 0. 5. Conclusions The main goal of this paper has been to present a geometrical framework that permits the description of time scaling in terms of infinitesimal canonical transformations in a suitable extended space. This approach gives geometrical meaning to the usual tricks and manipulations used when dealing with such transformations. The construction we have presented makes use of the coisotropic imbedding instead of the usual extended phase space. We have shown that both spaces are symplectically equivalent but not naturally equivalent because the symplectomorphism depends on the choice of a particular set of coordinates. Consequently they will provide equivalent dynamical descriptions, but from the structural viewpoint our description is easier and more natural. Furthermore, time transformations, either with a previous transformation or not, are used to improve the accuracy, stability and efficiency of numerical algorithms. Therefore, the introduction of the generating function for such transformations will permit to improve even more the analytical and numerical tools currently used. We have explicitly computed this generating function solving linear Hamilton-Jacobi equations for different problems arising in celestial mechanics as a small sample of the problems in which the applications of these ideas might be useful. Others, like the use of canonical orbital elements in terms of an arbitrary variable and its applications to satellite theory has briefly been considered and its study will be continued in the future. Acknowledgements We would like to thank the referee for his many comments and suggestions that helped us to substantially improve this work, as well as J. M. Ferrfindiz for letting us to know TIME SCALING 213 about the work of Bond and Janin discussed in example 5. Two of the authors (J. F. C. and L. A. I.) are grateful for the warm hospitality of the Departamento de Matem/ttica, U A M M6xico. References Abraham, R. and Marsden, J. E.: 1978, Foundations of Mechanics, 2nd ed., Benjamin, Reading Mass. Asorey, M., Carifiena, J. F., and Ibort, L. A.: 1983, J. Math. Phys., 24, 2745. Bond, V. and Janin, G.: 1981, Celest. Mech., 23, 159. Carifiena, J. F., Gomis, J., Ibort, L. A., and Romfin, N.: 1987, II Nouvo Cimento. 98B, 172. Carifiena, J. F., Gomis, J., Ibort, L. A., and Rom/m, N.: 1985, J. Math. Phys., 26, 1961. Curtis, W. D., and Miller, F. R.: 1985, Differential Manifolds and Theoretical Physics, Academic Press, New York. Gotay, M. J.: 1982, Proc. Amer. Math. Soc., 84, 111. 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