Symmetry and separation of variables for the

Symmetry and separation of variables for the Hamilton–Jacobi equation W 2 t −W 2 x −W 2 y =0 C. P. Boyer, E. G. Kalnins, and W. Miller Jr. Citation: Journal of Mathematical Physics 19, 200 (1978); doi: 10.1063/1.523539 View online: http://dx.doi.org/10.1063/1.523539 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/19/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Variable-separation theory for the null Hamilton–Jacobi equation J. Math. Phys. 46, 042901 (2005); 10.1063/1.1862325 Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation J. Math. Phys. 38, 6578 (1997); 10.1063/1.532226 Symmetries of the Hamilton–Jacobi equation J. Math. Phys. 18, 1032 (1977); 10.1063/1.523364 Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation J. Math. Phys. 16, 2476 (1975); 10.1063/1.522489 Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation J. Math. Phys. 16, 1461 (1975); 10.1063/1.522694

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Symmetry and separation of variables for the Hamilton-Jacobi equation W~-W~-W~ = 0 c.

P. Boyer

Instituto de Investigaciones en Matemtiticas Aplicadas y en Sistemas, Universidad Nacional Autonoma de Mexico, Mexico 20, D. F., Mexico

E. G. Kalnins Mathematics Department, University of Waikato. Hamilton, New Zealand

W. Miller, Jr. School of Mathematics, University of Minnesota, Minneapolis. Minnesota (Received 3 December 1976) We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R -separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra 0 (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split ofT, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases.

INTRODUCTION

This is the first of two papers in which we study the problem of separation of variables for solutions of W;- W;- W~=O,

(0.2)

known 1,2

As is well the symmetry algebra of (0. 1) defined in terms of operators acting on t,x,y is 0(3,2). Furthermore, there is an isomorphism between the symmetry algebras of (0.1) and (0.2). In Refs. 3- 5 it was shown that every R-separable coordinate system for (0.2) is characterized by a pair of commuting second order symmetric differential operators in the enveloping algebra of 0(3,2). Furthermore, coordinates whose operators lie on the same orbit under the adjoint action of 0(3,2) can be considered as equivalent. The separated special function solutions are eigenfunctions of the commuting symmetry operators and this relationship is a powerful tool for the derivation of special function identities. In Ref. 2 it was shown that the symmetry algebra of the Hamilton-Jacobi equation for the free particle (0.3) (acting in t,x,S space) was also isomorphic to 0(3,2). The problem of additive R-separation of variables for (0.3) was studied in Ref. 6. There all solutions of the form S= R(u, v)

+ U(ll) + V(v)

(0.4)

were classified where u, v is a new coordinate system, either R == 0 or R, 0 and is not expressible as a sum of a function of 11 alone and a function of valone, and U, V are arbitrary solutions of first order ordinary differen200

J. Math. Phys. 19( 1), January 1978

(0.5)

(0.1)

the characteristic equation of the wave equation 'lttt-'ltlCX-'lt~y=O.

tial equations. It was shown that the separable coordinates agree exactly with the separable coordinates for the free-particle Schrodinger equation

as derived in Ref. 7. There coordinates are all associated with the Schrodinger subalgebra of 0(3,2). Furthermore, the other elements of 0(3,2) lead to symmetry adapted solutions which do not separate additively as in (0.4) but in some more complicated fashion. In exact analogy with the linear results in Ref. 7 it was also shown that the Hamilton-Jacobi equations for the harmonic oscillator, repulsive oscillator, and linear potential are equivalent to (0.3). Finally, it was pointed out that all these equations are equivalent to (0.1) in the sense that (0.1) is the equation of the graph of each of the considered Hamilton-Jacobi equations. Here, we look for additive R-separable solutions of (0 1) in the form 0

3

W=R(u k ) + 0 UJ(u,).

,.1

(0.6)

We show by example that the separable systems are the same as those derived in Refso 3- 5 for (0.2) and, properly interpreted the Lie algebraic characterizations of the systems are the same. The proper interpretation of the symmetry algebra of (001) is that it is an algebra of functions in the six-dimensional phase space. {t,x,y;po=W t , P1=Wx, P2=Wy}, linear in the p" where the commutator is the Poisson brackeL The second order symmetries are formed by taking linear combinations of products of these functions. For the linear case, separable solutions are eigenfunctions of commuting second order differential operators. For the Hamilton-Jacobi case, separable solutions are those for which the corresponding second order functions, commuting under the Poisson bracket operation,

0022-2488178/1901-0200$1.00

© 1978 American Institute of Physics

200


take constant values. The orbit analysis for the classification of separable systems is identical to that in the linear case.

It is our hope that this treatment of a most interesting example will aid in the establishment of more general results for Hamilton-Jacobi equations.

We follow closely the procedure of Ref. 3 and concentrate here on those systems where one coordinate can be split off by diagonalizing a first order symmetry, leaving a reduced equation in two variables, The additively separable coordinates for the reduced equations (Hamilton-Jacobi equations for the free particle, harmonic oscillator, repulsive oscillator, linear potential, the equation of geometrical optics, etc.) correspond to proper subalgebras of 0(3,2). However, utilizing the full symmetry algebra, we find many solutions of these reduced equations which separate nonadditively, On one hand each reduced equation is a special case of (0. 1), but also (0. 1) is the equation of the graph of the reduced equation, Thus each reduced equation is equivalent to (0.1) and admits the symmetry algebra 0(3,2),

1. BASIC PRINCIPLES

We show that the passage from a Hamilton-Jacobi equation to the associated Hamiltonian system provides us with the analogy of a momentum space mOdel in the linear theory. We also indicate how the results of Ref. 4 concerning cyclidic R- separable coordinates for (0,2), in which it is impossible to split off one variable at a time, carryover directly to (0, I), In the second paper we will provide complete proofs concerning the identity of separable coordinates for these two equations, Finally, due to the fact that Lie algebra computations are much easier (though equivalent) for (0,1) than for (0,2), we have been able to find and correct some computational errors in Refs, 3 and 8. Although this paper concerns only the nonlinear equation (0.1), it should be obvious to the reader that our Lie algebraic procedure can be applied with little change to more general Hamilton-Jacobi equations. Indeed there has been a recent revival of interest in separation of variables 9,10 for general Hamilton-Jacobi equations owning to its usefulness as a solution technique for the Einstein and Einstein-Maxwell equations. 11 (For the claSSical literature see Ref. 12, and the book by Hagihara, 13 where many applications to celestial mechanics are given. ) Of the recent literature dealing with separation of variables for the Hamilton-Jacobi and related second order differential equations in general Riemannian (and pseudo-Riemannian) spaces we mention the works of Havas, 9 Dietz,10 and Woodhouse. 10 Havas 9 has given the general form of the metric tensor for coordinates which admit complete or partial separation. He also gives the general form of linear and quadratic integrals of motion, Dietz lO and Woodhouse lO consider a much more restrictive definition of separation of variables; they additively split off a single variable at a time. In this way one cannot obtain the more general type separable coordinates. 14 None of the above authors allows nontrivial R separation or considers nonadditive separation such as appears here. Furthermore, this and our subsequent paper are the only ones to associate with separable systems orbits of second order members of the enveloping algebra of the symmetry algebra as the correspondings integrals of the motions, thus allowing us to give explicit lists of separable coordinates classified in equivalence classes. 201

J. Math. Phys., Vol. 19, No.1, January 1978

We begin with the equation W~" ==

W;o - W;l- W;2 == 0,

a"" W(x)

(1.1)

for the characteristics of the wave equation. As is well known1,2 the space-time symmetry algebra of (1. 1) is 0(3,2). That is, the set of Lie derivatives 2

L=

0

".0

a"(x)a~,,

such that L W is a solution of (1, 1) whenever W is a solution forms the Lie algebra 0(3,2) under the operations of addition of Lie derivatives and commutator bracket. A basis for 0(3,2) is provided by the elements

(1.2) 2 K"=2x,,x.a,,-x a"",0'""I1,v,",,2,

where Xo=xo, Xi = - xi for j = 1, 2, x 2=x"x" and the Einstein summation convention for repeated indices is adopted, The commutation relations are [M"v,

M...l =g".M.a + g""l\f,.. -

[M".,

Pal =g.a P " - g"aP.,

g "aM •• - g •• M"a,

[p", pvl = [K", K.l = [M"., D] = 0,

(1. 3)

[D,P"]=-P,,, [D,K"]=K,,,

[M,,", ~1=g .aK" - g "aK., [K", p.] = - 2(M"v + g ,,»), where goo=- g,,= 1, gl'V=O for 11

* v.

These operators can be exponentiated to yield a local Lie transformation group2 of symmetries of (1.1). indeed the operator M"., P" generate the Poincare symmetry group A EO SO(I, 2),

a = (a o, al> a2),

W(x) - W(A -1 (x - a)),

(1, 4)

the dilatation operator generates the symmetry (expill) W(x) =X (e~x) and the K" generate the special conformal transformations

"

( x.2 a"x2 + 22 )

exp(a K,,)W(x.)=W 1

-

a·x

a x

•

(1.6)

We shall also consider the inverSion, space reflection, and time reflection symmetries of (1.1), IW(x) = W(- x/x 2 ),

RW(x) = W(x o, xl>

- X2),

TW(x) = W(-XO,Xl>X2),

which are not generated by the Lie derivatives (1. 2). In Ref. 6 the full infinite-dimensional symmetry algebra of an arbitrary first order partial differential equation was computed and shown that this algebra splits into symmetries which are contact transformaBoyer, Kalnins, and Miller

201


tions and an ideal of characteristics. Now there is a well-known Lie algebra isomorphism between canonical vector fields on phase space with the usual Lie brackets and functions on phase space (modulo constants) with the Poisson brackets. j Explicitly, given phase space with coordinates (x'", p,J we have the canonical 2-form w =dP,,/\ rb:", Then with each vector field X on phase space which leaves w invariant, we can associate a function F(x",pv) such that X Jw=dF,

(10 7)

subject to the requirements

oW ox" at;=p"at;, j=1, ... ,n, H(x"(t, ••. ,tn),P" (tl> .•• , tn» = 0, Then, provided

Hpo'" HPn oXo

det

where J denotes the inner product between vector fields and forms, To tl;le Lie brackets for vector fieldS there correspond the Poisson brackets

toG of of OG) {F(x, P), G(x,P)}= 2.: ,ax" ap" - ox" 'iJP ..

(108)

for functions, Explicitly for the Lie derivatives (1. 2) we have, using (1. 7),

M"v=x"Pv-x"p",

oX

at; ... at;n

atoXo

n

aXn

' ..

3t;

on the surface, the solutions of (1.12) with initial data (1. 13) generate a local solution of (L 11). The function W can be obtained either from the equation dW

dT- p - "H p ..

(1.14)

or the defining relations Pv = WxV.

P"=P,,, D=x"P",

(1. 9)

2

K" = 2x" (XVpv) - x p IJ"

Conversely, let (1. 15)

W=f(x", al,'" ,an) +a o One can easily check that the basis functions (L 9) satisfy relations (L 3) under the Poisson bracket operation, From the point of view of separation of variables of (10 1), the Lie algebraic characterization (L 9) in terms of functions on phase space is superior to that of Lie derivatives. By taking all possible products of operators (10 2) we can generate an enveloping algebra 15 of so(3, 2). Furthermore, we can identify the subspace S~ of homogeneous symmetric kth-order elements in the enveloping algebra with the space 5 k of kth- order polynomials in the basis functions (L 9), That is, the two subspaces are isomorphic as vector spaces and the adjoint action of so(3, 2) on 5~ induced by the commutator [, , .J agrees with the adjoint action on S k induced by the Poisson bracket, In particular, 5f is spanned by elements of the form [LI> L2J.= L1L2 + L2LI> where the L j are Lie derivatives belonging to the symmetry algebra, Let L 1, L2 be the corresponding functions in the Lie algebra (1. 9), Then the correspondence (1.10)

extended by linearity provides the stated isomorphism between 52 and S2. As is well known there is an intimate relationship between a first- order partial differential equation

oW

H(x",P,,)=O, p,,=ox'"

O~J-L~n,

(1.11)

and the Hamiltonian system of ordinary differential equations 16

op" a:r=-Hx '"

dx" --;;:r=Hp",

O~J-L~n,

(L 12)

Indeed, consider the n-dimensional surface x" =x"(t1 , •• ,, t n ) and prescribe initial data on this surface: (1. 13) 202


be a complete integral of (1 11), 1. e" W is a solution of (1.11) for each choice of the n + 1 real constants {l" and the n x (n + 1) matrix 0

(oa,o""f)

= (fa},e")

has rank n. Then (1. 15) and relations

faj(x",a,)=II."

j=1,oo.,n

Pv=f"v(x",a,),

v=O" .. ,n

(1. 16)

with a o, ' , , ,am 11.1>"" An fixed, define a solution of the characteristic system (L 12), It is also well known that the canonical transformation generated by (1. 12) preserves Poisson brackets, 1 Thus, if

Fj(x", Pv) = F,(x"(T),Pv(T», j

= 1,2,

where

X"(T) =X"(T, x"',Pv'), Pv(T) = Pv(T,X"',Pv')'

(1.17)

are solutions of (1,12) such thatx"(O)=x", Pv(O)=Pv, then (L 18)

Furthermore,

:TF'=F""Hp" -Fp"H",,={nr, FT} so that

r

= F if F commutes with H,

Applying this theory to Eqo (L 1), we find

H=P5- pi -

pL

so that the associated Hamiltonian system is

-dp" -dx" = dT = 0'dT

l •.IV -g pv, 2

0~

/I r-,

V ~

2

'

(1,20)

Thus we can obtain a solution of (L 1) by prescribing initial data for W, x", and P" on a two-dimensional Boyer, Kalnins, and Miller

202


surface in x-space and solving Eqs. (1. 20). For some of our computations we shall choose this surface and data in the special form xO=O, Po=(Pi+pW /2 , xl=tf, X2=t2' (1.21) PI =Pl(tf, t2 ), P2 =P2(t l , t 2).

al. In particular, on 32 we have the relations

Note that the basis functions (1. 8) restricted to this surface become

We shall now classify all separable solutions of (1.1) which are analogous to the separable solutions of the wave equation (1.23), studied in Refs. 3-5. In these references each (R -) separable solution of (1. 23) was characterized by the fact that the corresponding separated solutions were common eigenfunctions of a commuting pair 51,52 of second-order differential symmetry operators from the enveloping algebra of so(3, 2). Analogously we will characterize separated solutions W of (1. 1) by the conditions Fl (X" ,P~) = A, F2 (x", Pv) = !J., where F t , F2 E32 and {Ff, F 2}= O. Here P.,= Wx" and A, !J. are the separation constants. For all cases treated in this paper we shall see that coordinates which yield separation for (1. 23), corresponding to commuting operators 51> 52> also yield separation for (1.1) corresponding to commuting functions Ff, F2 E 3 2. Here,S, and F J are related by (1. 10). We also mention, although we will not make use of it, that corresponding to each commuting pair (Ff, F2) there is via (1. 7) a commuting pair of vector fields (X j ,X2 ) on phase space and hence a two-parameter local Abelian group of symmetries.

Po= (pi +p~)1/2=PO' P,=Ph 1YiI,=xtPrXjP/> lYioJ=x'Po, Ko=[(xd+(X2)2]pO' D=XiPI,

(1.22)

1

K, = 2x,(x p;) + [(Xl)2 + (X2)2]P J, i,j = 1, 2.

Model (1. 22) and its relationship to (1. 1) via integration of the Hamiltonian system (1. 20) is an analogy of the Fourier transform model for the solution space of the wave equation (0.2) which was treated in Ref. 3. We now introduce another basis for the symmetry algebra which makes explicit the isomorphism with the usual matrix realization of 0(3,2). The matrix algebra 0(3,2) is usually defined as the ten-dimensional Lie algebra of 5 x 5 matrices A such that AG + GA t = 0, where 0 is the zero matrix and

(1.23)

I,

Let C be the 5 x 5 matrix with a 1 in row i, column j and zeros elsewhere. Then the matrices If'.b=Cab-Cba=-Ir b., If'.B={.B+{Ba=-If'B.,

a*b, l~a,b~3,

(1. 24)

If'45=C 54 -(:45=-If'54, B=4,5 form a basis for 0(3,2) with commutation relations [If' "'B, If'yc] = GByIf' ",c + G"CIrBY+ Gy",il'CB + GCBil'y",. (1.25) This basis is related to our basis (1. 2) by the identifications Po = r

14

+ r 45 , PI = r 25 + r t2 , P 2= r35 + r 13 ,

~=r45-rt4'

Kl =r 25 - r12 ,

K2=r35-rt3,

(1.26)

Mt2 =r 32 , Mot=r 24 , Mo2=r 34 , D=r 51 . Here and hereafter we denote by r(r) the vector field (func tion) which corresponds to the matrix If'. Returning to our Poisson bracket model (1. 9), we note that if we impose the relation

K~ - Ki - K~ = 0,

ri2 + ri3 + r~3 =r1 5,

Mi2 - M%1 - M%2 = - D2,

r1 5 - r~t - r;1 = r~3'

(1.28)

[Of course, these relations also hold in the model (1. 22).

1

In analogy with Ref. 3 we shall begin by studying separable systems which can be characterized by commuting functions Ff, F2 such that F t =A2 for some A E so(3, 2) with {A, F2}= O-and where A has a nontrivial centralizer. In Ref. 3 the corresponding systems were called "semi-subgroup" coordinates. However, as has been pOinted out by Winternitz, this name is not appropriate because many coordinate systems which correspond to the restriction of so(3, 2) to a subalgebra are not "semisubgroup" systems. Thus, we Shall now call these systems "semi- split" coordinates. In the following seven sections we shall study systems of the form A 2 , F2 by first splitting off the coordinate associated with A to obtain a reduced equation (t) such that F2 belongs to the space of second- order elements in the enveloping algebra of the symmetry algebra (centralizer of A) of (t). We shall then investigate the coordinate systems in which (t) separates. In the last section we shall give an example of a completely nonsplitting type coordinate. In the study of separable coordinates it is common to associate with each coordinate system x"=x"(y,,) a metric tensor g"v and a quadratic differential form

(1. 27) to obtain solutions of (1. 1), we introduce linear dependencies among the elements of 52. Although there are formally 35 independent terms LjL j where the L/ run over a basis for 0(3,2), among the explicit functions (1.9) subject to (1.27) there are 20 independent relations obeyed by the LIL j • Hence, if.!h is the subspace of 52 which is mapped to zero under this identification, thenj2 is actually an ideal under the adjoint action of 0(3,2) and the factor space 32 == 52/.!h is 15-dimension203


(1.29) then the Hamilton-Jacobi equation (1.1) takes the form (1.30) in the new coordinate system, where g"" is the inverse tensor to g "V given by g"" =

(- 1)" ..... cofg !!~

(1.31)

detg" .. Boyer, Kalnins, and Miller

203


We also mention for convenience that for orthogonal coordinates (1. 29) takes the form ds 2 =

'0 h~(d~i)2

= Ina + il n In(A + cosa)_

and the relation between the canonical moment in p" in Cartesian coordinates and arbitrary curvalinear coordinates ~i is

1hj23£1 P(i. ax"

(1. 33)

ri2+ri3+r~3=A2,

(2.4c)

(2,5a)

Y3=ksn(o,k)sn(a,k),

_ A2k 2 (sn2 Q1 _ sn 2a)

(2,1)

Xl-

Y2

- YI - cos1/! '

x2 _

Y3

leads to the separated solution 5=J (-A 2a 2 sn 2 a+/J.)I/zdQl+ J (-A zaZ sn 2a+/J.)1/2da, (2,5c)

There is a close relationship between our own study of r 45 and the Hamilton-Jacobi equation for the Kepler problem with closed orbits in two-dimensional space, Indeed, on the surface (1,21) the condition r 45 = A for a solution of (1," 1) becomes

- YI - cos1/! ' (2.2)

yi +y~ +y~= L

Thus, choosing any parametrization Y i (a, a) of the unit sphere 52, we obtain a new set of coordinates for space time, In these coordinates we have r 23 = Y2al'3 - Y3a:.>.!, r l3 = YI aY3 - Y3aYI'

The equation r 45 =A or, what is the same thing, r 45 w = A implies

W=- AJ1I2 2{3L (]I2+{32 , (5,9a)

L - v1) + 1: + ~ r((v- 0)2 + (32)«(1)- 0)2 + (32>Jl/2 YO- 2 ((]I2+{32) 2 2{3L 02+{32

with 0 ~ 1) < 00, solutions are

ex;

< v < 0, and

0,

The solutions are fl)1/2

{3 reaL The separated

cd + (32]1 /2

(5,13) Again implementing the canonical transformation Pj j - x , xi - - Pj, we obtain (5.14)

(5,9b)

with separation constant oM61 - 2{3M12 :lI02

== 1)V(V - 1)rl[ ((1) - U)2 + (32)p; -

((v - 0)2 + (32)p~1 = flo

(5,9c) (20) Elliptic-parabolic: For simplicity we consider the nondegenerate point a = 1; the coordinates are Zp + cos 2E!) 1,0:= 1. (COSh . 2 coshp cosE! '

(5, lOa)

Vi = - tanhp tanE!

with -

,Ycos 2E!)1 /2

J

00

0

(50 lOb)

1. e., the positive energy Kepler problem o It is clear that under the above canonical transformation the reduced symmetry algebra 0(2,1) is preserved,

Again, we emphasize that Eqo (1. 1) can also be interpreted as the equation for the graph of a solution 5(1),0/) of (5.1), here parametrized by the spherical coordinates (5,5a)o Thus 0(3,2) is the full symmetry group of (5.1) and every separated solution of (L 1) gives rise via the graph to a symmetry adapted solution of (5. I),

6, THE NONRELATIVISTIC FREE PARTICLE We now look at the only partial separation of (L 1) which involves nonorthogonal coordinates, Since this case waS already treated in detail in Ret 6, we will be brief here o ConSidering the reduced equation corresponding to the operator Po + Pj, we set Po + PI = A; then (1, 1) reduces to

with constant of the motion

where

== (coshZp - cos 2Btl (coshZpp~ + cos 2BP~) = flo

(5,10c)

(21) Hyperbolic-parabolic: Again for a = 1 the coordinates are

=

}

1. (COSh2p + cos 2 E!) 2

sinhp sinB

'

.v l = cothp cotE! with 0 "-. p < 00 and 0 ~ E! < 7T S=

0

The separated solutions are

J ([P(fl- A2/sinh2p)1I2 + J dB(- fl- ,\,z/sin2B)1I2 (5,lIb)

with separation constant -

(.HOI

+ ,\112)2 + M~2

== (sinh2p + sin2B)-1 (sinh2pp; - sin2BP~) =flo 208

(6, la)

AS t - S;= 0,

.H~z + (;1101 + NliZ)2

,II

(50 12c)

As with the sphere in Sec. 2 there is a close relationship between our model on the hyperboloid and the Hamilton-Jacobi equation for the Kepler problem with unbounded orbits (positive energies), Indeed the operator D = r 15 is conformally equivalent to r 14 = t(P o - Ko)o Explicitly, Adexp(trrr45)r15=r14o Thus r 14 ='\', and on the surface (1,21) we have

5=J dV(fl + iJ., 2v)1/2v- l / 2[(v_ 0)2 + {321-1I2 + J d1)(fl + tA21))1/21)-1/2[ (1) -

(5,12b)

with the constant of the motion ~,\I02(A101 - ;1[12) = (1;2 + 1)2rl(1)2p~ - ep~) = flo

. 1 - (]It + (32 , 12 - 2 ((]I2+{32)

0

s= J d1)(fl + ,\.2/1)2)11 2 + J d~(,\.2/~2 -

(19) 5emilz:l'per/JOlic- The coordinates are

v2 _

2 _ (~2 + 1)2)2 - 4 Y 81;1)


(5. lIc)

f=(x O _x l ),

y=x 2

and (6,lc)

Clearly (6. 1) is equivalent to the equation studied in Ref, 6 (take t - - A-It and y - x), Its reduced symmetry algebra is the Schrodinger algebra sl generated by {po - Pj, Ko + Kj, D + MOb P 2 , M02 - Mi2' Po + Pi}' Notice that in this case we no longer have a Lie algebra direct sum of the operator corresponding to the partial separation (here E = Po + PI) and its centralizer. However, since E is in the center of sj, we can consider the factor algebra S/Eo Because the partial separation in this case involves nonorthogonal coordinates, the R-separable coordinates of the reduced equation (6. 1a) are nonorthogonal and are characterized by orbits in the factor Boyer, Kalnins, and Miller

208


algebra

StiE.

(27) r~2' r~s + r 14 r 4S +arrs, a> -

The list of representatives is:

(3") Po + Pi> P 2 ,

(28) r~2' arts + r 14 r 4S ,

free particle,

(23) Po + Pi> Po - PI - t(Ko + K l ), attractive oscillator,

(29) r~2' rts + a2r~s, O- I t 10 ) In analogy with the previous reduced equations it is easy to show that (1.1) is the equation of the graph of (701). Thus, 0(3,2) is the full symmetry algebra of (70 It

8. THE SYMMETRY

1=-[(-1)s+21/4.

As usual we try to associate separable coordinates for (8.4) with the nine orbits of second order symmetries in the enveloping algebra for 0(2,1). It is guaranteed that there are separable coordinates corresponding to the three orbits which correspond to squares of first order symmetries:

(32) L2, C 2,

In particular, (1/) is equivalent to (1)0 To obtain the remaining systems, we note that for K = 0 the operators and coordinates (8,3) agree with system (15) on the hyperboloid, i. eo, coordinates (5.5a). Since the separable coordinates for (8.4) must be independent of K it follows that a separable system for (8 04) must be one of the systems (7/), (15)-(22). However, one of the latter systems need not necessarily yield separation for (8 4L 0

We are guaranteed success for systems (32) and (33). For (32) we set coshz = cosh~ cosh?), tan!/> = tanh~/ sinh7J to obtain __1 __ ~

r 23 - r 45

i-

cosh 2 ~"

We next separate a variable from (10 1) by requiring r 4s)=K In terms of the coordinates (20 2 ), (204) with (3= a + 1jJ, rp = a -1jJ we have L =P8' so

L=~(r23-

52 {

+ ___-=1,.--_~-

(1 - cosh 2 ~ cosh21])

x (2K cosh~ sinh7J cosh7JS~ - 2K sinh~ cosh27JS" + K2) = 0,

(8.6a) The condition

W=:K{3+S(rp,a),

C = - 5" + K sinh~/ (cosh2 ~ cosh27J - 1) = iJ.

where cot2aS~ + 2K(csc 2a + I)S~

+ S; + K2 cot2a = O.

(801)

The centralizer of L in 0(3,2) is {L}EB 0(2, 1), where 0(2,1) is the subalgebra with basis A,B, C such that A=i(r23 +r 45 ),

B=t(r24 +r 35 ), c=i(r25 -r 34 ), (802)

[C,A]=B,

== Sib'

- B = sinrp5e + cothz cos¢S~

C = - cos¢Se + cothz sinrpSq,

(806b)

yields the R- separated solution S=-Ktan-l(sinh~coth7J)- fl7J

+ 1 (K2-1J. 2 - 2iJ.Ksinh~)1/2/coSh~ld~0 For (33) we set coshz = He-~ + (1]2 + l)e~], tanrp = - 21]e~ / [e-~ + (7)2 - 1 )ell and use the condition A - B=5"- 2K[e- 1 + (1J2-1)e~1I{[e-~ + (r? +l)e~f- 4}

[C,Bl==Ao

Thus 0(2,1) is a symmetry algebra for the reduced equation (801). Here

+ K cos¢/ sinhz

+ K sin¢/sinhz,

(80 3 )

to obtain the R-separated solution 5= - K

(8.4) Note: The simple computation leading to this identity shows that the corresponding identity for the wave equation as given in Refs. 3 and 5 is in error. The correct

_

tan-{e~ (1 ;~2) e-~] + 1111)

f [- (2Ke-~m

+

and in terms of these symmetries equation (801) reads


(8.7a)

=IJ.

sina == tanh(z/2),

210

Indeed, the eigenvalues of - iL are As= ~(s + i), s = 0, 1, 2,···. The eigenspace V. corresponding to eigenvalue ~s is irreducible under 0(2,1) and transforms according to the unitary representation Dj where

(33) L2, (A-B)20

- (r 14 + r 45 )r15 == i(e2-

[A,Bl=C,

(8.5)

(1/) L2, A 2 ,

in which case (701) becomes

A

result for the wave equation is A2_ B2_ C 2==-&.

+ m 2 e-2/) 11/2 d ~~

(8.7b)

We have carefully studied the coordinates corresponding to system (17) and have found that they do not lead to R separation of variables for (804), It appears that only the subgroups systems (15/), (32) and (33) yield additive variable separation for this equation, although we have not explicitly checked this for all systems (15)- (22). Boyer, Kalnins, and Miller

210


Just as for the other reduced equations it is easy to show that (1. 1) is the equation of the graph of (8.4). Thus 0(3,2) is the full symmetry algebra of (8.4) and all of the additively separated solutions of (1.1) lead to symmetry adapted solutions of (8.4).

9. NONSPLIT COORDINATES In analogy with Ret 3 for the wave equation, we have listed, with the exception of some degenerate nonorthogonal systems, 5 all separable systems for (1. 1) in which it is possible to additively split off one variable. In Ret 4 a classification of all orthogonal R- separable coordinates systems for the wave equation was given for which the coordinate surfaces were families of confocal cyclides. 53 such systems were found, and, except for degenerate cases, it was shown that the variables intertwine in such a complicated fashion that it is necessary to separate them simultaneously, L e. , it is not possible to split off a single variable. Each such system was shown to be characterized by a commuting pair of second order symmetric operators in the enveloping algebra of 0(3,2). The results of Ref. 4 can be applied directly to obtain orthogonal separable coordinates for (L 1) simply by interpreting the Lie algebra of differential operators 0(3,2) as a Lie algebra of functions under the Poisson bracket.

leads to the separated solution w= (Il sin2a + lI)t /2 dO' + (Il sin2{'i + lJ)t /2

J

J

d{3

+ J (M sin2 y + lJ)t /2 dy,

In a similar fashion each of the orthogonal R-separable coordinate systems for the wave equation is additively separable for (1, I). In Paper II we shall examine the relationship between the wave equation and (1. 1) more closely and provide proofs of own assertions concerning variable separation.

ACKNOWLEDGMENT One of us (W. M. ,Jr.) would like to thank the Institute de Investigaciones en Matematicas Aplicadas y en Sistemas for financial support and their hospitality during the period of this work. lJ. E. Campbell, Introductory TYeatise on Lie's Theory of Finite Continuous Transformation Groups (Chelsea, New York, 19G6), reprint of 1903 edition, p. :12. 2C. P. Boyer and M. Penafiel N., Nuovo Cimento B 31, 195 (1976) .

3E.G. Kalnins and W. Miller, Jr., J. Math. Phys. 16, 2507 (1975; •

4E.G. Kalnins and W. Miller, Jr., J. Math. Phys. 17, :331 (1976).

5E.G. Kalnins and W. Miller, Jr., J. Math. Phys. 17,356 (1976).

"C.p. Boyer and E.G. Kalnins, J. Math. Phys. 18, 1032 (1977).

lE.G. Kalnins and W. Miller, Jr., J. Math. Phys. 15,1728

(1974). BE.G. Kalnins and W. Miller, Jr., J. Math. Phys. 17, :369 (1976). Dp. Havas, J. Math. Phys. 16, 1461, 2476 (1975). toW. Dietz, J. Phys. A 9, 519 (1976); N.M.J. Woodhouse, Commun. Math. Phys. 44, 9 (1975). 2 x = cOSO' cos/3 cosy, HB. Carter, Commun. Math. Phys. 10, 280 (1968). 12J. Liouville, J. de Math. 11, 345 (1846); P. Stackel, In these coordinates (L 1) reduces to Habilitationschrift Halle, (1891); 1\Iath. Ann. 42, 548 (189::); 35, 91 (1890); Ann. Mat. 25, 55 (1897); T. Levi-Civita, (sin 2 /3 - sin2y)p~ + (sin2y - sin2a)p~ + (sin2Q - sin2!3)p; = O. Math. Ann. 59, :18:1 (1904); F. Dall'Acqua, Mnth. Ann. 66, 398 (1908); Rend. Circ. Matem. Palermo 33, :141 (1912); P. Burgatti, Rend. Acad. Lincei 20 (1), 108 (lUll); II. P. Robertson, Math. Ann. 98, 749 (1927); L. p. Eisenhnrt, Ann. It is not possible to additively separate one of these Math. 35, 284 (1934); M. S. Iarov-Iarovoi, J. Appl. Mnth. variables from the other two, However, use of the Mech. 27, 1499 (1964). defining symmetry elements 13y. Hagihara, Celestial Alechanics, Vol. 1 (MIT Press, Cambridge, Mnss., 1(70). 2 (Po - Pt)M02 + + 14N, Levison, B. Bogart, and R.1\1. Redhcffer, J. Appl, J\1uth. 2 sin ap ~ 7, 241 (1949); p. Moon and D. E. Spencer, J. Prnnklin Inst. 255, :302 (1953). = (sin 2a - sin 2/3)(sin 2(J1 - sin 2y) 15N. Jacobson, Lie Algebras (Intersciencc, New York, 1962).

For example, the system [311] (i) of Ret 4 leads to coordinates 2 2 2 Xo == - t(cos a + cos f:l + COs y), xl = sinO' sinj3 siny,

Po Ii

+

+ 2P2M o2

-

sin 2/3:p2B 2 (sin /3 - sin 2(J1) (sin 2/3 - sin 2y) • 2

IGR. Courant and D. Hilbert, Methods of Malllcilia/ieal Physics, Vol. II (Interscience, New York, 1%2). 17J. Patera and p. Winternitz, J. Math. Phys. 14, 11:l0

p2

SIn Y r _ (sin 2y- sin 2Q)(sin 2y- sin2!3) - Il 2HI2 +

(9.3a)

(McGraw-Hill, New York, 1951). 1~. Pock, Z. Phys. 98, 145 (1935). 2oH. Bacry, Nuovo Cimento A 41, 222 (1966); G. Gyorg-yi, Nuovo Cimento A 53, 717 (1968).

Pz

= (sin 2 Q - sin2/3)(sin 2O' - sin 2y)

+

+

211

21p. Winternitz and 1. Fris, Sov. J. Kucl. Phys. 1, 6:lG (1965).

sin Q sin2yp~ (sin 2/3- sin 2 O')(sin 2/3- sin2 y) 2

sin 2 Q sin2/3P~

(sin 2y- sin2a)(sin2y- sin 2/3) =

(1973).

lBA. Erdelyi et al., Hiffher Transcendcntal FUllctions, Vol. 2

lJ


(9.3b)

22E.G. Kalnins, SIAMJ. Math. Anal. 6, ::40 (1975). 23P. Winternitz, I. Lukac, and Y.A. Smorodinski~, Sov. J. Nucl. Phys. 7, 139 (1968). 24E.G. Kalnins and W. Miller, Jr., J. Math. Phys. 18, 1 (1977).

Boyer, Kalnins, and Miller

211
