3-geometries and the hamilton–jacobi equation

3-geometries and the Hamilton–Jacobi equationPatricia Garcıa-Godınez, Ezra Ted Newman, and Gilberto Silva-Ortigoza Citation: Journal of Mathematical Physics 45, 2543 (2004); doi: 10.1063/1.1753667 View online: http://dx.doi.org/10.1063/1.1753667 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/45/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Killing tensors, warped products and the orthogonal separation of the Hamilton-Jacobi equation J. Math. Phys. 55, 013505 (2014); 10.1063/1.4861707 On the Hamilton-Jacobi theory for singular lagrangian systems J. Math. Phys. 54, 032902 (2013); 10.1063/1.4796088 A geometric framework for discrete Hamilton-Jacobi equation AIP Conf. Proc. 1460, 164 (2012); 10.1063/1.4733374 Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints J. Math. Phys. 53, 072905 (2012); 10.1063/1.4736733 Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation J. Math. Phys. 38, 6578 (1997); 10.1063/1.532226

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3-geometries and the Hamilton–Jacobi equationPatricia Garcıa-GodınezFacultad de Ciencias Fı´sico Matema´ticas de la Universidad Auto´noma de Puebla,Apartado Postal 1152, Puebla, Pue., Me´xico

Ezra Ted NewmanDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania15260

Gilberto Silva-OrtigozaFacultad de Ciencias Fı´sico Matema´ticas de la Universidad Auto´noma de Puebla,Apartado Postal 1152, Puebla, Pue., Me´xico

~Received 1 March 2004; accepted 23 March 2004;published online 21 May 2004!

In the first part of this work we show that on the space of solutions of a certain classof systems of three second-order PDE’s,uaa5Y(a,b,u,ua ,ub), ubb

5C(a,b,u,ua ,ub) and uab5V(a,b,u,ua ,ub), a three-dimensional definite orindefinite metric, gab , can be constructed such that the three-dimensionalHamilton–Jacobi equation,gabu,au,b51 holds. Furthermore, we remark that thisstructure is invariant under a subset of contact transformations. In the second part,we obtain analogous results for a certain class of third-order ordinary differentialequation~ODE’s!, u-5L(s,u,u8,u9). In both cases, we apply our general resultsto the cental force problem. ©2004 American Institute of Physics.@DOI: 10.1063/1.1753667#

I. INTRODUCTION

In the early years of the 20th century, while studying the structure and transformation prop-erties of second- and third-order ODE’s, Lie, Tresse, Wu¨nschmann1–4 among others, discoveredthat there was a rich differential geometry induced on the solution spaces of the differentialequations by the equations themselves. This work was greatly developed and generalized byCartan and Chern5–9 in the 1930–1940s. Robert Bryant,10 in more recent years, studied the ge-ometry associated with fourth-order ODE’s. Paul Tod11 showed how third-order ODE’s couldgenerate three-dimensional Einstein–Weyl metrics.

With a totally different motivation and from a different point of view originating with GeneralRelativity, Frittelli, Kozameh and Newman, in a series of papers12–17 came to the same set ofissues and problems. Rather than starting with given differential equations, the point of view ofthese authors began with three- and four-dimensional conformal Lorentzian manifolds, alreadycontaining a metric. They then studied families of complete solutions to the eikonal equation onthese manifolds. From these solutions, by the elimination of the space–time coordinates, thedifferential equations of Cartan and Chern were reobtained. However, from this point of view,unwittingly, the Cartan–Chern work was generalized from ODE’s to pairs of second-order PDE’swhose solution spaces could be identified with any four-dimensional manifold with a conformalLorentzian metric. In particular, they showed that the Einstein equations could be reformulated interms of pairs of second-order PDE’s.

Later, with Kamran and Nurowski,18–22this work was connected with the Cartan–Chern workfor both the equivalence problem for differential equations under a variety of transformations andwith the theory of Cartan normal conformal connections. With this consideration one saw how thedifferential equations~both the third-order equation and the pair of second-order equations! had tolie in a restricted class defined by the vanishing of the so-called Wu¨nschmann~or generalizedWunschmann! equation.

An underlying unifying theme in many of the discussions was the existence of the eikonalequation and families of complete solutions. These solutions could be obtained either via the given

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 7 JULY 2004

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third-order ODE or pair of second-order PDE’s or alternatively from the solutions to the eikonalequation on the given conformal background space.

In a recent work,23 we turned to the geometry associated with a new class of second-orderODE’s. The connecting link with the earlier work is that now we used the time-independenttwo-dimensional Hamilton–Jacobi equation rather than the eikonal equation to obtain this newclass. We showed~via two different procedures! that, in the solution spaces of the ODE’s, eithera two-dimensional Riemannian or Lorentzian metric can be constructed in a natural way and thatthe metric structure associated with each differential equation is preserved when the equation istransformed by a subset of contact transformations~namely canonical transformations!.

The aim of the present work is to generalize our previous result to the three-dimensionaltime-independent Hamilton–Jacobi equation. In Sec. II we begin with a three-dimensional mani-fold, M, with no further structure and then investigate arbitrary two-parameter families of sur-faces onM given byu5constant5Z(xa,a,b). ~Thexa are local coordinates onM anda andbparametrize the families and take values onS 2, S 13R or onR 2.) More specifically, we then askwhen do such families of surfaces define a three-dimensional metric,gab(x

a), such that

gab¹aZ~xa,a,b!¹bZ~xa,a,b!51. ~1!

We will show that theu5Z(xa,a,b) must also satisfy a system of three second-order PDE’s

]aaZ5Y~u,v,w,a,b!,

]bbZ5C~u,v,w,a,b!, ~2!

]abZ5V~u,v,w,a,b!,

with

v[]aZ, and w[]bZ,

and whereY, C, andV are restricted to satisfy certain ‘‘Wu¨nschmann-type’’ conditions.Here] denotes partial derivative. Observe that in the solutionsu5Z(xa,a,b), thexa are three

constants of integration for Eqs.~2! while the ‘‘a’’ and ‘‘ b’’ are two integration constants for Eq.~1!.

Before proceeding we make the following important remark:Remark 1: The time-independent Hamilton–Jacobi equation for a particle, with mass m and

energy E, in a three-dimensional Riemannian space under the influence of a potential and theeikonal equation describing the evolution of the light rays in a three-dimensional isotropic opticalmedium characterized by its index of refraction, i.e., either of

g* ab¹aS~xa,a,b!¹bS~xa,a,b!5E2V~xa!,

g* ab¹aS~xa,a,b!¹bS~xa,a,b!5n~xa!,

can be rewritten in the form of Eq. (1) by dividing the equations by either E2V(xa) or by n(xa)and simultaneously rescaling the metric by the same factors.

This action has the effect of changing certain properties of solutions to the Hamilton–Jacobi~H-J! equation. Normally for the three-dimensional H-J equation a complete integral containsthree constants of integration, where one of them isE. In our case, after the conformal rescaling,E is hidden as afixedconstant in the metricgab and the solution will depend now on only twoparameters,a andb. With an abuse of language, we will refer tou5Z(xa,a,b) as a ‘‘restrictedcomplete’’ integral to the H-J equation. In Sec. II we also show that any two arbitrary restrictedcomplete integrals of the same H-J equation are connected via a special contact transformation.This result allows one to establish that if two systems of three second-order PDE’s are connected

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via that special contact transformation then in their spaces of solutions the same three-dimensionalmetric can be constructed. Finally, our result is applied to the central force problem in sphericalpolar coordinates.

In Sec. III we present analogous results for a certain class of third-order ODE’s. That is, webegin with a three-dimensional manifold,M, with no further structure and then investigate arbi-trary one-parameter families of surfaces onM given byu5constant5Z(xa,s). ~Thexa are localcoordinates onM and s parametrizes the families and takes values onS 1 or on R 1.) Morespecifically, we ask when do such families of surfaces define a three-dimensional metric,gab(x

a),such that

gab¹aZ~xa,s!¹bZ~xa,s!51. ~3!

We will show that theu5Z(xa,s) must also satisfy a third-order ODE,

u-5L~s,u,u8,u9!, ~4!

whereL is restricted to satisfy a certain ‘‘Wu¨nschmann-type’’ condition.Here the prime denotes ‘‘s’’ derivatives. Observe that in the solutionsu5Z(xa,s), which we

will refer to as a one-parameter family of solutions to the Hamilton–Jacobi equation, thexa, arethree constant of integration for Eq.~4! while the ‘‘s’’ is an integration constant for Eq.~3!. In thissection it is also remarked that the three-metric is invariant when the third-order ODE is trans-formed by a subclass of contact transformations. Finally, these results are also applied to thecentral force problem in spherical polar coordinates.

II. THE SYSTEM OF PDE’s CASE

In this section we prove that in the space of solutions of a certain class of systems of threesecond-order PDE’s, a three-dimensional definite or indefinite metric,gab , can be constructedsuch that the solutions satisfy the three-dimensional H-J equation. We start with a three-dimensional manifoldM @with local coordinatesxa5(x0,x1,x2)] and assume we are given atwo-parameter set of functionsu5Z(xa,a,b), the parametersa and b can take values onS 2,S 13R or on R 2. We also assume that for fixed values of the parametersa and b the levelsurfaces

u5constant5Z~xa,a,b!, ~5!

locally foliate the manifoldM and thatu5Z(xa,a,b) satisfies the H-J equation

gab~xa!¹aZ~xa,a,b!¹bZ~xa,a,b!51 ~6!

for some unknown metricgab(xa).

Remark 2: If gab contains the E simply as a number or fixed parameter it will appear in thesolution to the H-J equation. i.e., u5Z(xa,E,a,b).

The basic idea now is to solve Eq.~6! for the components of the metric in terms of¹aZ(xa,a,b). To do so, we will consider a number of parameter derivatives of the condition~6!,and then by manipulation of these derivatives, obtain both the three-dimensional metric and thethree partial differential equations defining the surfaces plus the conditions they must satisfy. Wewill refer to them as the Wu¨nschmann-type conditions.

Remark 3: The notation is as follows: there will be two types of differentiation, one is withrespect to the local coordinates, xa, of the manifoldM, denoted by¹a or ‘‘ comma a, ’’ the otheris with respect to the parametersa and b, denoted by]a and ]b .

From the assumed existence ofu5Z(xa,a,b), we define three parametrized scalarsu i in thefollowing way:

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u05u[Z~xa,a,b!,

u15v[]aZ~xa,a,b!, ~7!

u25w[]bZ~xa,a,b!.

Remark 4: For each value ofa and b Eqs. (7) can be thought of as a coordinate transfor-mation between the xa’ s and(u, v, w).

We also define the following three important scalars

Y* 5]aaZ~xa,a,b!,

C* 5]bbZ~xa,a,b!, ~8!

V* 5]abZ~xa,a,b!.

In what follows we will assume that Eqs.~7! can be solved for thexa’s; that is,

xa5Xa~u,v,w,a,b!,

so that Eqs.~8! can be rewritten as

]aaZ5Y~u,v,w,a,b!,

]bbZ5C~u,v,w,a,b!, ~9!

]abZ5V~u,v,w,a,b!.

This means that the two-parameter family of level surfaces, Eq.~5!, can be obtained as solutionsto the system of three second-order PDE’s~9!. Note that~Y,C,V! satisfy the integrability condi-tions,Y,b5V,a andV,b5C,a .

The solution space of Eqs.~9! is three-dimensional. This can be seen in the following way.The system of PDE’s~9! is equivalent to the vanishing of the three one-forms,v i ,

v05du2vda2wdb,

v15dv2Yda2Vdb, ~10!

v25dw2Vda2Cdb.

A simple calculation, using the integrability conditions on~Y,C,V!, leads todv i50 ~modulov i)from which, via the Frobenius Theorem, the solution space of Eqs.~9! is three-dimensional.

From the three scalars,u i , we have their associated gradient basisu i,a given by

u i,a5¹au i5$Z,a ,]aZ,a ,]bZ,a%, ~11!

and its dual vector basisu ia, so that

u iau j

,a5d ij , u i

au i,b5db

a. ~12!

Definition 1: The totala and b derivatives of a function F5F(u,v,w,a,b) are defined by

DaF[Fa1Fuv1FvY1FwV,

DbF[Fb1Fuw1FvV1FwC, ~13!

respectively.



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It is easier to search for the components of the three-dimensional metric in the gradient basisrather than in the original coordinate basis. Furthermore, it is preferable to use the contravariantcomponents rather than the covariant components of the metric; that is, we want to determine

gi j ~xa,s!5gab~xa!u i,au ,b

j . ~14!

The metric components and the Wu¨nschmann-type conditions, for this case, are obtained byrepeatedly operating with]a and]b on Eq.~6!, that is, by definition, on

g005gabZ,aZ,b51. ~15!

Applying ]a to Eq. ~15! yields ]ag0052gab]aZ,aZ,b50, i.e.,

g1050. ~16!

In the same way we obtain that]bg0052gab]bZ,aZ,b50 and thus,

g2050. ~17!

A direct computation shows that

]aa~g00/2!5gab]aaZ,aZ,b1gab]aZ,a]aZ,b5gabY,aZ,b1g1150. ~18!

Since, by the assumed linear independence of (Z,a ,]aZ,a ,]bZ,a),

Y,a5YuZ,a1Yv]aZ,a1Yw]bZ,a , ~19!

Eq. ~18!, using Eqs.~16!, ~17!, and~19!, is equivalent to

g1152Yu . ~20!

In exactly the same way we find that

]ab~g00/2!5Vu1g2150,

]bb~g00/2!5Cu1g2250. ~21!

Therefore, the final result is

~gi j !5S 1 0 0

0 2Yu 2Vu

0 2Vu 2Cu

D . ~22!

Remark 5: We require thatdet(gij)5D be different from zero, with

D[~YuCu2Vu2!. ~23!

The metricity or Wu¨nschmann-type conditions are obtained from the conditions]aaag00

50, ]baag0050, ]bbag0050, and]bbbg0050. A direct computation shows that they are equiva-lent to

Yua1Yuuv1YuvY1YuwV52~YvYu1YwVu!,

Yub1Yuuw1YuvV1YuwC52~VvYu1VwVu!,~24!

Vub1Vuuw1VuvV1VuwC5VvVu1VwCu1CvYu1CwVu ,



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Cub1Cuuw1CuvV1CuwC52~CvVu1CwCu!.

Summarizing:~a! If we start from a restricted complete integral,u5Z(xa,a,b) to the H-J equa-tion, ~6!, then it satisfies the system of three second-order PDE’s~9!, with Y, C andV satisfyingthe Wunschmann-type conditions~24!; in other words, in the solution space of Eqs.~9! there is thenaturally defined metric,

ds25Fu,au,b21

D~Cuv ,av ,b2Vu~v ,aw,b1w,av ,b!1Yuw,aw,b!Gdxadxb, ~25!

whereD is defined by Eq.~23!. ~b! If we start with a system of three second-order PDE’s~9!,whereY, C, andV satisfy Eqs.~24! and the integrability conditions, then in its solution spacethere exist a natural three-dimensional metric given by Eq.~25!. Though it might appear as if themetric components depend on the parameters~a,b!, the Wunschmann-type conditions guaranteesthat they do not. Furthermore, the solutionsu5Z(xa,a,b) satisfy the H-J equation

gab¹aZ~xa,a,b!¹bZ~xa,a,b!51

with the just determined metric, Eq.~25!.Remark 6: From the results presented above we conclude that solving the three-dimensional

H-J equation is equivalent to solving a system of three second-order PDE’s.In some of the earlier work on the eikonal equation in three and four dimensional Lorentzian

spaces, it was proved that the conformal Lorentzian metrics associated with third-order ODE’s andpairs of second order PDE’s satisfying the Wu¨nschmann condition and generalized Wu¨nschmanncondition, is preserved when the differential equation is transformed by a contact transformation.For our present case, there is an analogous result given by the following:

Theorem 1: Let Eqs. (9) be a system of three second-order PDE’s, withY, C, and Vsatisfying the conditions~24! and let

]aaZ5Y~ u,v,w,a,b !,

]bbZ5C~ u,v,w,a,b !, ~26!

]abZ5V~ u,v,w,a,b !,

be a system of three second-order PDE’s locally equivalent to Eqs. (9) under the subset of contacttransformations generated by the generating function

H~a,b,u,a,b,u!5u2u2G~a,b,a,b !. ~27!

Then under this subset of contact transformations the metric given by Eq. (25) is preserved.The proof of this theorem is exactly as that presented in Ref. 18 for a system of two second-

order PDE’s such that in its space of solutions is living a four-dimensional conformal Lorentzianmetric,gab, such thatgabu,au,b50 holds. Here we only justify the form of the generating function~27!. We first review the definition of a general contact transformation.

Theorem 2: Every contact transformation which is not a prolonged point transformation is

determined in terms of a generating function H(a,b,u,a,b,u) by solving the following five

implicit equations fora, b, u, v5]au,w5]bu:

H~a,b,u,a,b,u!50,

Ha1vHu50, H a1 vHu50, ~28!

Hb1wHu50, H b1wHu50.



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The generating function H(a,b,u,a,b,u) is an arbitrary smooth function, subject only to the

solubility of Eqs. (28) fora, b, u, v, w.For a proof of this theorem see, for example, Olver.24

Without loss of generality one can take

H5u2V~u,a,b,a,b !, ~29!

so that the contact transformation has the form

u5V~u,a,b,A~a,b,u,v,w!,B~a,b,u,v,w!!,

a5A~a,b,u,v,w!,

b5B~a,b,u,v,w!, ~30!

v5Va~u,a,b,A~a,b,u,v,w!,B~a,b,u,v,w!!,

w5Vb~u,a,b,A~a,b,u,v,w!,B~a,b,u,v,w!!,

whereA(a,b,u,v,w) andB(a,b,u,v,w) are obtained by solving

Va1vVu50,

Vb1wVu50, ~31!

for a and b in terms ofa, b u, v, andw.As was pointed out earlier, for each value ofa andb, the three-parameter family of solutions

u5Z~xa,a,b!, ~32!

of ~9! is also a two-parameter family of solutions of Eq.~6!, i.e., are ‘‘restricted complete’’integrals of Eq.~6!. We now invoke the envelope construction to take one restricted completeintegral of Eq.~6! into another such solution. Consider the functionu5Z(xa,a,b) defined by

u5V~u,a,b,a,b !, ~33!

whereu is defined by Eq.~32! anda andb are defined implicitly as functions ofxa, a andb bythe envelope condition18,25

Vuv1Va50,

Vuw1Vb50. ~34!

Note that although Eqs.~34! have the same form as Eqs.~31!, they involve the variablesxa, a, andb. Using both Eqs.~33! and ~34!, we have that

u,a5Vuu,a . ~35!

By direct substitution ofu,a into the H-J equation, Eq.~6!, we see that it is a new restrictedcomplete integral if and only ifVu

251. That is, u5V(u,a,b,a,b) has the form u56u

1G(a,b,a,b). For simplicity, taking the positive sign, we have that ifu(xa,a,b) is a restrictedcomplete integral of Eq.~6! then



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u5u1G~a,b,a,b !, ~36!

wherea andb are defined implicitly as a function ofxa, a, andb by the envelope conditions

v1Ga50,

w1Gb50, ~37!

is a new restricted complete integral of Eq.~6!. Equations~36! and~37! define a particular subsetof the contact transformations given by contact transformations

u5u1G~a,b,a,b !, ~38!

v52Ga , ~39!

w52Gb , ~40!

v5Ga , ~41!

w5Gb . ~42!

The generating function for this set of contact transformations is given by

H~a,b,u,a,b,u!5u2u2G~a,b,a,b !50, ~43!

thus justifying our choice of the generating function, Eq.~27!.As an example we apply our results to the central force problem in spherical polar coordi-

nates,xa5(r ,u,f). For this problem the time-independent Hamilton–Jacobi equation is given by

S 1

2m~E2V! D S u,r2 1

u,u2

r 2 1u,f

2

r 2 sin2 u D 51, ~44!

wherem is the mass,V(r ) the potential energy, andE is the total energy of the particle, respec-tively. By the method of separation of variables one finds that a complete solution to Eq.~44! canbe written in the following form

u5Z~xa,E,a,b!5EA2m~E2V!2b2

r 2 dr1EAb22a2

sin2 udu1af, ~45!

wherea andb are two constants of separation. For this problem,a, is the magnitude of the totalangular momentum andb is the value of the angular momentum about the polar axis. In this case,our restricted complete integral is obtained from Eq.~45! by fixing E. Now we obtain the systemof three second-order PDE’s associated with this restricted complete integral. A direct computationshows that

v5]aZ5E 2adu

sin2 uAb22a2

sin2 u

1f,

w5]bZ5E 2bdr

r 2A2m~E2V!2b2

r 2

1E bdu

Ab22a2

sin2 u

. ~46!



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By using Eqs.~45! and ~46!, one sees that the Jacobian of the coordinate transformation,u i

5u i(xa,a,b), for this case, is given by

J5]~u,v,w!

]~r ,u,f!5

22m~E2V!b

Ab22a2

sin2 uA2m~E2V!2

b2

r 2

. ~47!

Therefore,b cannot be zero.Using Eqs.~46!, we see that for the central force problem in spherical polar coordinates

]aaZ52b2F~u,a,b!,

]bbZ52G~r ,b!2a2F~u,a,b!, ~48!

]abZ5abF~u,a,b!,

where

F~u,a,b!5E du

sin2 uS b22a2

sin2 u D 3/2,

G~r ,b!5E 2m~E2V!dr

r 2S 2m~E2V!2b2

r 2 D 3/2. ~49!

On the other hand, Eqs.~45! and ~46! imply that

r 5R~u,v,w,a,b!,

u5Q~u,v,w,a,b!. ~50!

Therefore, the system of three second-order PDE’s for the central force problem in sphericalpolar coordinates is given by

]aaZ5Y52b2F~Q~u,v,w,a,b!,a,b!,

]bbZ5C52G~R~u,v,w,a,b!,b!2a2F~Q~u,v,w,a,b!,a,b!, ~51!

]abZ5V5abF~Q~u,v,w,a,b!,a,b!.

Sinceu(xa,a,b) given by Eq.~45! with E fixed is a restricted complete integral to the H-Jequation~44!, then Y, C, and V given in Eqs.~51! satisfy Eqs.~24! and, therefore, a three-dimensional metric can be defined in the space of solutions of Eqs.~51!. By comparison of Eqs.~6! and~44! it is clear what that metric should be. However, here we show the steps to explicitlyobtain this metric by the procedure developed in earlier. What is remarkable is that this metric canbe obtained without the evaluation of the integrals that arose in this problem. As we can see fromEq. ~25!, to obtain the three-dimensional metric,gab , associated with this problem, we need tocompute:u i

,a5(u,a ,v ,a ,w,a), Yu , Cu , Vu , andD. By using Eqs.~45! and ~46!, we have that



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u,a5SA2m~E2V!2b2

r 2 , Ab22a2

sin2 u,a D ,

v ,a5S 0,2a

sin2 uAb22a2

sin2 u

, 1D , ~52!

w,a5S 2b

r 2A2m~E2V!2b2

r 2

,b

Ab22a2

sin2 u

, 0D .

From Eq.~47! we have that these three vectors are linearly independent whenb is different fromzero.

Since, for example,

Cu5~]C/]r !~]r /]u!1~]C!/]u)~]u/]u!,

then to computeYu , Cu , Vu , andD we need to obtain (]u/]u) and (]r /]u). From Eqs.~45! and~46!, via implicit derivations, we obtain that

]r

]u5

A2m~E2V!2b2

r 2

2m~E2V!,

]u

]u5

Ab22a2

sin2 u

2mr2~E2V!. ~53!

By using the definition ofD given by Eqs.~23!, ~51!, and~53!, a direct computation shows that

Yu

D522mr2~E2V!1b2,

Cu

D522mr2~E2V!sin2 u1a2, ~54!

Vu

D5

a@2mr2~E2V!2b2#

b.

Finally, substituting Eqs.~52! and~54! into Eq.~25! we obtain the three-dimensional metric livingin the solution space of the PDE’s~51!;

ds25gabdxadxb52m~E2V!~dr21r 2du21r 2 sin2 udf2!, ~55!

the desired result.In the special case of the Kepler problem with energy equal to zero, i.e., forV52k/r and

E50, we have for the three PDE’s.



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]aaZ5b cot~]bZ1arccose!

a22b2 ,

]bbZ5a2 cot~]bZ1arccose!

b~a22b2!1

4

Z2a]aZ2b]bZ, ~56!

]abZ5a cot~]bZ1arccose!

b22a2 ,

where

e5S 4b22~Z2a]aZ2b]bZ!2

4b21~Z2a]aZ2b]bZ!2D . ~57!

III. THE THIRD-ORDER ODE’s CASE

In this section we prove that in the solution space of a certain class of third-order ODE’s athree-dimensional definite or indefinite metric,gab , can be constructed directly from the solutionssuch that the three-dimensional time-independent H-J equation holds. We begin with a three-dimensional manifoldM ~with local coordinatesxa5(x0,x1,x2)) and assume we are given aone-parameter set of functionsu5Z(xa,s); the parameters can take values onS 1 or on R. Wealso assume that for a fixed value of the parameters, the level surfaces

u5constant5Z~xa,s!, ~58!

locally foliate the manifoldM and thatu5Z(xa,s) satisfies the H-J equation

gab¹aZ~xa,s!¹bZ~xa,s!51, ~59!

for some unknown metricgab(xa). That is,u5Z(xa,s), is a one-parameter family of solutions to

the three-dimensional time-independent H-J equation.The basic idea now is to solve Eq.~59! for the components of the metric in terms of

¹aZ(xa,s). To do so, we will consider a number of parameter derivatives of the condition~59!,and then by manipulation of these derivatives, obtain both the three-dimensional metric and theODE defining the surfaces and the Wu¨nschmann-type condition it must satisfy.

Remark 7: The notation is as follows: as in the previous section, there will be two types ofdifferentiation, one is with respect to the local coordinates, xa, of the manifoldM, denoted by¹a

or ‘‘ comma a, ’’ the other is with respect to the parameter s, denoted by a prime or by]s .We first note that the one-parameter family of ‘‘level’’ surfaces, Eq.~58!, can be obtained as

solutions to the third-order ODE

u-5L~u,u8,u9,s! ~60!

by first calculating

u-~xa,s![L* ~xa,s! ~61!

and then by inverting the relations

u5Z~xa,s!,

u85Z8~xa,s!,

u95Z9~xa,s!,



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obtaining

xa5Xa~u,u8,u9,s!. ~62!

The threexa can be eliminated inL* yielding Eq.~60!.Remark 8: Eq. (62) can be thought of as an‘‘ s’’ dependent coordinate transformation between

xa and (u,u8,u9).We define three parametrized scalars:

u i5$u0,u1,u2%[$u,u8,u9%5$Z~xa,s!,Z8~xa,s!,Z9~xa,s!%, ~63!

which for each value ofs form a coordinate system intrinsically adapted to the surfaces.From the three scalars,u i , we have their associated gradient basisu i

,a given by

u i,a5¹au i5$Z,a ,]Z,a ,]2Z,a%, ~64!

and its dual vector basisu ia, so that

u iau j

,a5d ij , u i

au i,b5db

a. ~65!

Definition 2: The total s derivative of a function F5F(s,u,u8,u9) is defined by

DF[Fs1Fuu81Fu8u91Fu9L. ~66!

As in the previous section, it is easier to search for the components of the three-dimensionalmetric in the gradient basis rather than in the original coordinate basis. Furthermore, it is prefer-able to use the contravariant components rather than the covariant components of the metric; thatis, we are interested in

gi j ~xa,s!5gab~xa!u i,au j

,b . ~67!

The metric components and the Wu¨nschmann-type condition, for this case, are obtained by re-peatedly operating with]s on Eq.~59!, that is, on

g005gabZ,aZ,b51. ~68!

Applying ]s on Eq.~68! yields

]s~gabZ,aZ,b!52gabZ,a8 Z,b52g1050, ~69!

where we have used that]sgab50. Applying ]s on Eq.~69! we obtain

]s2~g00/2!5gabZ,a9 Z,b1gabZ,a8 Z,b8 50, ~70!

which is equivalent tog1152g20. In the same manner we find that

]s3~g00/2!5gabZ,a-Z,b12gabZ,a9 Z,b8 1gabZ,a8 Z,b9 5gabL ,aZ,b13gabZ,a9 Z,b8 50. ~71!

Since

L ,a5LuZ,a1Lu8Z,a8 1Lu9Z,a9 , ~72!

then Eq.~71! is equivalent to

g2152 13 @Lu1Lu9g

20#. ~73!

In the same way we find that



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]s4~g00/2!5DLu2 1

3 LuLu91@DLu9213 ~Lu9!

223Lu8#g2013g2250, ~74!

]s5~g00/2!52W(1)@L#2g20W(2)@L#50, ~75!

where

W(1)@L#[2D2Lu2 13 LuDLu91

73 Lu9DLu14LuLu82

49 Lu~Lu9!

2, ~76!

W(2)@L#[2 49 ~Lu9!

312Lu9DLu922Lu8Lu92D2Lu913DLu826Lu . ~77!

Remark 9: Observe that W(2)@L# is the standard Wu¨nschmann invariant for third-orderODE’s, which is an invariant under a general contact transformation.

From Eq.~75! one sees that there are, at this point, four different possible cases to look at:

~a! W(1)@L#Þ0, W(2)@L#Þ0; in this case the three-dimensional metric can be completely re-constructed;

~b! W(1)@L#50, W(2)@L#Þ0; in this case we obtain a degenerate metric, which must be ex-cluded by the assumption of the H-J equation. See Remark 10 below, for the proof;

~c! W(1)@L#50, W(2)@L#50; in this case the metric is not completely determined. See theconjecture below;

~d! The case ofW(2)@L#50 with nothing said aboutW(1)@L# arises not with the H-J equationbut with the time dependent three-dimensional eikonal equation. It leads to a conformalmetric on the solution space.12–17As this case has been extensively studied, nothing furtherwill be said here about it.

We thus consider only the first case which leads to the final result

gi j ~xa,s!5hi j ~xa,s!1gci j ~xa,s!, ~78!

where

~hi j !5S 1 0 0

0 0 2 13 Lu

0 2 13 Lu 2 1

3 DLu1 19 LuLu9

D , ~79!

~gci j !5g20S 0 0 1

0 21 2 13 Lu9

1 2 13 Lu9 2 1

3 DLu9119 Lu9

21Lu8

D , ~80!

with

g2052S W(1)@L#

W(2)@L# D . ~81!

The Wunschmann-type condition, which is obtained from]s6(g00/2)50, is

DS W(1)@L#

W(2)@L# D5S 2

3D S Lu9W(1)@L#

W(2)@L#2LuD . ~82!

Remark 10: When W(1)@L#50, using Eqs. (81) and (82), one sees from



132.174.255.116 On: Tue, 02 Dec 2014 21:14:25

det~gi j !52Lu

2

91

g20

3@DLu2LuLu9#1

~g20!2

3 FDLu922

3~Lu9!

223Lu8G1~g20!3 ~83!

that det(gij)50. This implies a degenerate metric thus excluding this case.Summarizing, we conclude the following: IfL is such thatW(1)@L#Þ0, W(2)@L#Þ0 and

satisfies the Wu¨nschmann-type condition~82!, then in the solution space of the ODE,u-5L(u,u8,u9,s), there is defined either a three-dimensional Riemannian or Lorentzian metricgiven by

ds25gabdxadxb5gi j ui,au j

,bdxadxb, ~84!

with (gi j ) given by Eqs.~78!–~81!. Furthermore the solution satisfies the H-J equation with thismetric. One can think of the ODE and the H-J equation as dual to each other; each being satisfiedby the same function but with respect to different variables.

In earlier work on the eikonal equation in three-dimensional Lorentzian spaces, it was provedthat the conformal Lorentzian metric associated with a third-order ODE satisfying the Wu¨n-schmann condition, is preserved when the differential equation is transformed by a contact trans-formation. For our present case, there is an analogous result given by the following:

Theorem 3: Let Eq. (60) be a third-order ODE, withL satisfying the condition~82! withW(1)@L#Þ0, W(2)@L#Þ0 and let

u-5L~ u,u8,u9,s!, ~85!

be a third-order ODE locally equivalent to Eq. (60) under the subset of contact transformationsgenerated by the generating function

H~s,u,s,u!5u2u2G~s,s!. ~86!

Then under this subset of contact transformations the metric given by Eqs. (78)–(81) is preserved.The proof of this theorem is exactly as that presented in Ref. 18 for the third-order ODE case

such that a conformal Lorentzian metric,gab, lives on its space of solutions is living, such that theeikonal equationgabuaub50 is satisfied. The justification of this choice of generating function~86! is the same as that given in the previous section.

We return to the example from the previous section of the central force problem but now withonly one parameter,s, in the solution. We see that ODE’s can be constructed from the two-parameter solution, Eq.~45!, i.e., from

u5Z~xa,E,a,b!5EA2m~E2V!2b2

r 2 dr1EAb22a2

sin2 udu1af, ~87!

by choosinga andb as functions ofs. To illustrate what can occur we take three different cases:

~a! (a5s,b5b0)⇒W(1)@L#50 andW(2)@L#50;~b! (a5a0 ,b5s)⇒W(1)@L#50 andW(2)@L#50;~c! (a5as,b5s),⇒W(1)@L#50, the three vectorsu i

,a are not linearly independent;~d! (a5s2,b5s),⇒W(1)@L#Þ0, W(2)@L#Þ0.

Conjecture: Whenever one has a third-order ODE, u-5L(u,u8,u9,s), so that W(1)@L#50and W(2)@L#50 then the equation came from a solution where the one-parameter entered thesolution as a Killing trajectory; i.e., the change in the solution as‘‘ s’’ evolves can be undone bydragging the metric along the Killing trajectory.

For case~d! the metric can be found~after a lengthy calculation! from the solution



132.174.255.116 On: Tue, 02 Dec 2014 21:14:25

u~xa,s!5EA2m~E2V~r !!2s2

r 2 dr1E sA12s2csc2 u du1s2f ~88!

by the following steps:

u85E 2sdr

r 2A2m~E2V~r !!2s2

r 2

1E ~122s2csc2 u!du

A12s2 csc2 u12sf,

u95E 22m~E2V!dr

r 2F2m~E2V~r !!2s2

r 2G3/21E s csc2 u~2s2 csc2 u23!du

~12s2 csc2 u!3/2 12f. ~89!

The ODE becomes

u-52@H~r ,s!1J~u,s!#, ~90!

where

H~r ,s!5E 6sm~E2V!dr

r 4F2m~E2V~r !!2s2

r 2G5/2,

J~u,s!5E 3 csc2 udu

~12s2 csc2 u!5/2. ~91!

By inverting, we, in principle, find

r 5R~u,u8,u9,s!,

u5Q~u,u8,u9,s!,

f5F~u,u8,u9,s!, ~92!

from which

u-5L52@H~R~u,u8,u9,s!,s!1J~Q~u,u8,u9,s!,s!#. ~93!

This leads after much work to

W(1)@L#

W(2)@L#5

22mr2~E2V!1s4 csc2 u

2mr2@2mr2~E2V!2s2#~E2V!~s2 csc2 u21!~94!

and finally to

u,a5SA2mr2~E2V!2s2

r,sA12s2 csc2 u,s2D ,

u,a8 5S 2s

rA2mr2~E2V!2s2,122s2 csc2 u

A12s2 csc2 u,2sD , ~95!

u,a9 5S 22mr~E2V!

~2mr2~E2V!2s2!3/2,s csc2 u~2s2 csc2 u23!

~12s2 csc2 u!3/2 ,2D ,



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and the metric~first in tetrad components!

g0051,

g2052mr2~E2V!2s4 csc2 u

2mr2@2mr2~E2V!2s2#~E2V!~s2 csc2 u21!,

~96!

g2152s sin2 u

mr2~E2V!@2s21cos~2u!21#2 1s

@2mr2~E2V!2s2#2 ,

g2252mr2~E2V!

@2mr2~E2V!2s2#3 13s2 sin2 u24 sin4 u

2mr2~E2V!@s22sin2 u#3 ,

and then the final form,

ds25gabdxadxb52m~E2V!@dr21r 2du21r 2 sin2 udf2#. ~97!

IV. CONCLUSIONS

In this work, we have shown that the ideas and procedures developed in our recent paper23 onthe two-dimensional time-independent H-J equation can be generalized to the three-dimensionaltime-independent H-J equation. The results presented in this work show that solving the three-dimensional time-independent H-J equation is equivalent to solving a system of three second-orderPDE’s or a third-order ODE.

We point out that, though we have used, in the present work, only the three-dimensionaltime-independent H-J equation, this can be generalized. In a future paper we will present theresults for the four-dimensional H-J equation.

ACKNOWLEDGMENTS

The authors thank Carlos Kozameh for valuable conversations. P.G.-G and G.S.-O. acknowl-edge the financial support from VIEP-BUAP through Grant No. II-161G04 and from CONACyTthrough Grant No. 44515-F. Furthermore, P.G.-G acknowledges the financial support from CONA-CyT through a scholarship and G.S.-O. acknowledges the financial support from Sistema Nacionalde Investigadores~Mexico!. E.T.N acknowledges the financial support from the NSF under GrantNo. PHY-0244513.

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19S. Frittelli, N. Kamran, and E. T. Newman, Class. Quantum Grav.20, 1 ~2003!.20S. Frittelli, C. Kozameh, E. T. Newman, and P. Nurowski, Class. Quantum Grav.19, 5235~2002!.21P. Nurowski and M. Godlinski~in preparation!.22E. T. Newman and P. Nurowski, Class. Quantum Grav.20, 5235~2003!.23P. Garcı´a-Godı´nez, E. T. Newman, and G. Silva-Ortigoza, J. Math. Phys.45, 725 ~2004!.24Peter J. Olver,Equivalence, Invariants and Symmetry~Cambridge University Press, Cambridge, 1995!, p. 127.25L. Landau and E. Lifschitz,Classical Mechanics~Addison-Wesley, Cambridge, MA, 1951!.



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3-geometries and the hamilton–jacobi equation

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