Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

To get the most general form of the estimates,

some additional function space trickery is required.

As before, a simple rescaling argument extends

estimate [26] to the case of data F, u

, u

, whose

spatial Fourier trans forms are localized in the

annulus 2

j1

jj2

jþ1

; we obtain

jð1=pþn=qÞ

kuk



jn=2

þ 2

jðn=21Þ

þ 2

jð1=

þn=

2Þ

kFk

Finally, if the data are arbitrary, we may decompose

them as series of localized functions, and summing

the corresponding estimates we obtain the general

Strichartz estimates for the wave equation [25]: for

all (p, q) and (

q)asin[27],

kuk

1=pþn=q

q;2



n=2

þku

n=21

þkFk

þn=

2

½28

Here, given a decomposition f =

j2Z

, the

homogeneous Besov and Sobolev norms are defined,

respectively, by the identities (obvious modification

for r = 1):

kf k

q;r

j2Z

jsr

;

kuk

¼kjj

’kuk

2;2

It is easy to convert the estimates [28] into a form

that uses only the more traditional nor ms

kf k

kjDj

f k

; jDj



F

1

j j



since by the Besov–Sobolev embedding we have

q;2



for 2  q < 1;

q;2



for 1 < q  2

Notice that if we apply to the equation and the

data the operator jDj



= F

1

j j



F, which commutes

with

, the Strichar tz estimate [28] can be rewritten

in an apparently more general form:

kuk

1=pþn=qþ

q;2



n=2þ

þku

n=21þ

þkFk

þn=

2þ

½29

In particular, it is possible to choose the indices in

such a way that no derivatives appear on u and F:

this choice gives

kuk

ðR

nþ1



1=2

þku

1=2

þkFk

ðR

nþ1

p ¼

2ðn þ 1Þ

n  1

which is the estimate originally proved by Strichartz.

Global Large Waves

As for ord inary differential equations (ODEs), the

local solutions constructed in Theorem 1 can be

extended to a maximal time interval [0, T



], and a

natural question arises: are these maximal solutions

global, that is, is T



= 1?

For generic nonlinearities and large data, the

answer is negative; in a dramatic way, in general

the norm ku(t,  )k

is unbounded as t"T



< 1.

The reason for this is simple: using the finite speed

of propagation, we can localize the equation and

work on a cone; then if we take constant functions

as initial data, the solution inside the cone does not

depend on x, and the equation restricted to the cone

effectively reduces to an ODE:

&u ¼ f ðuÞ()y

ðtÞ¼f ðyÞ;

yðtÞuðt; xÞ

½30

By this remark it is elementary to construct solutions

of the IVP [6] that blow up in a finite time.

This construction does not apply if the equation

has some positive conserved quantity. Indeed, con-

sider a general gauge-invarian t equation

&u þ gðjuj

Þu ¼ 0;

uð0; x Þ¼u

ðxÞ; u

ð0; xÞ¼u

ðxÞ

½31

for some smooth function g(s). Writing G(s) =

g()d, multiplying the equation by u

, and

integrating over R

, it is easy to check that the

nonlinear energy

EðtÞ¼

þjr

þ Gðjuj

dx  Eð0Þ½32

is constant in time, provided the solution u is

smooth enough. When G(s) has no definite sign,

we can proceed as above and construct solutions

that blow up in finite time; this is usually called the

‘‘focusing’’ case . However, if we assume that

G(s)  0 (‘‘defocusing’’ case), the energy E(t)is

non-negative. The corresponding ODE, which is

þ g(y

)y = 0, has only global solutions, and one

may guess that also the solutions of [31] can be

extended to global ones.

This innocent-looking guess turns out to be one of

the most difficult problems of the theory of nonlinear

waves, and is actually largely unsolved at present.

The only general result for eqns [31] is Segal’s

theorem, stating that the IVP has always a global

weak solution:

Theorem 2 Let g(s) be a C

non-negative function

on [0, þ1), write G(s) =

g()d and assume that

for some constant C

sgðs

ÞCGðs

Þ; lim

s!þ1

GðsÞ ¼ þ1 ½33

522 Semilinear Wave Equations

Then for any (u

, u

) 2 H

 L

such that G(ju

)

2 L

, the IVP [31] has a global solution u(t, x) in the

sense of distributions, such that u

2 L

(R, L

))

and F(u) 2 L

(R, L

)).

The proof (see Shatah and Struwe (1998))is

delicate but element ary in spirit: by truncating the

nonlinear term, we can approximate the problem at

hand with a sequence of problems with global

solution; then the conservation law [32] yields

some extra compactness, which allows us to extract

a subsequence converging to a solution of the

original equation.

Thus we see that, despite its generality, this result

does not shed much light on the difficulties of the

problem. Indeed, the weak solution obtained might

not be unique, nor smooth, and in these questions

the real obstruction to solving [31] is hidden.

Notice that in the one-dimensional case n = 1 the

solution is always unique and smooth when the data

are smooth, since in this case E(t) controls the L

norm of u. For higher dimensions n  2, something

more can be proved if we assume that the nonlinear

term has a polynomial growth:

sgðs

Þ¼jsj

p1

s for s large; p > 1 ½34

In particul ar, the defocusing wave equation with a

power nonlinearity

&u þjuj

p1

u ¼ 0 ½35

has been studied extensively. Notice that when p is

close to 1, the term juj

p1

u becomes singular near 0;

this introduces additional difficulties in the problem;

for this reason, it is better to consider a smooth term

as in [34].

We can summarize the best-known results con-

cerning [31] under [34] as follows. Let p

(n) be the

number

ð1Þ¼p

ð2Þ¼1

ðnÞ¼1 þ

n  2

for n  3

Then



in the subcritical case 1  p < p

(n), for any data

, u

) 2 H

 L

, there exists a unique solution

u 2 C(R; H

) such that u

2 C(R; L

);



the same result holds in the critical case p = p

(n)

for n  3; and



when 3  n  7, 1  p  p

(n), the solution is

smoother if the data are smoother.

These results have been achieved in the course of

more than 30 years through the works of several

authors (it is indispensable to mention at least the

names of K Jo¨ rgens, I Segal, W Strauss, W von

Wahl, P Brenner, H Pecher, J Ginibre, G Velo,

R Glassey and the more recent contributions of

J Shatah, M Struwe, L Kapitanski, M Grillakis,

omitting many others). Actually modern proofs are

remarkably simple, and are based again on a

variation of the fixed-point argument. Roughly

speaking, the linear equation

u þ g(jvj

)v = 0

defines a mapping v 7!u; the Strichartz estimates

localized on a cone imply that this mapping is

Lipschitz continuous in suitable spaces, the Lipschitz

constant being estimated by the nonlinear energy of

the solution restricted to the cone. In order to show

that this mapping is actually a contraction, it is

sufficient to prove that the localized energy tends to

zero near the tip of the cone, that is, it cannot

concentrate at a point. Once this is known, it is easy

to continue the solution beyond any maximal time

of existence and prove the global exis tence and

uniqueness of the solution.

In the supercritical case p > p

(n), very little is

known at present; there is some indication that the

problem is much more unstable than in the

subcritical case (Kumlin, Brenner, Lebeau), and

there is some numerical evidence in the same

direction.

Global Small Waves

It was noted already in the 1960s (Segal, Strauss)

that the equation in dimension n  2

&u ¼f ðuÞ ; uð0; xÞ¼"u

ðxÞ; u

ð0; xÞ¼"u

ðxÞ

f ðuÞ¼Oðjuj



Þ for u  0

with small data can be considered as a perturbation of

the free wave equation and admits global solutions.

The phenomenon may be regarded as follows: the

wave operator tends to spread waves and reduce their

size (see [21]); the nonlinear term tends to concen-

trate the peaks and make them higher, but at the same

time it makes small waves smaller. If the rate of

dispersion is fast enough, the initial data are small

enough, and the power of the nonlinear term is high

enough, the peaks have no time to concentrate, and

the solution quickl y flattens out to 0. Notice that in

dimension 1 there is no dispersion, and this kind of

mechanism does not occur.

It was, however, F John who initiated the modern

study of this question by giving the complete picture

in dimension 3: for the IVP

&u ¼juj



; uð0; xÞ¼"u

ðxÞ

ð0; xÞ¼"u

ðxÞ; n ¼ 3

Semilinear Wave Equations 523

he proved that, for fixed u

, u

2 C



if >1 þ

ﬃﬃﬃ

and " is small enough, the solution

is global and



if 1 <<1 þ

ﬃﬃﬃ

and the data are not identically

zero, the solutions blow up in a finite time for all "

(i.e., the L

-norm is unbounded).

Later Schaeffer proved that blow-up occurs also at

the critical value  = 1 þ

ﬃﬃﬃ

W Strauss guesse d the correct critical value for all

dimensions – 

(n) is the positive root of the

algebraic equation

n  1

 

n þ 1



 ¼ 1

and conjectured that the same picture as in dimen-

sion 3 is valid for all dimensions n  2.

Soon Sideris proved that, for 1 <<

(n) and the

quite general and small data, one always has blow-up.

Also it was proved by Klainerman, Shatah, Christo-

doulou, and others that the positive part of the

conjecture was true for >

(n), with a small gap

near the critical value. The gap was closed by

Georgiev, Lindblad, Sogge, who proved global exis-

tence for all >

(n). We also mention that the

solution at the critical value  = 

(n) always seems to

blow up; this is settled for low dimension (Schaeffer,

Yordanov, Zhang and others), but the question is still

not completely clear for large dimensions.

This problem has spurred a great deal of

creativity, eventually leading to very fruitful results:

the different approaches have proved useful in a

variety of problems, sometimes quite different from

the origin al semilinear equation. We mention a few:



The weighted estimates of F John are estimates of

the solution in spacetime L

norms with weights

of the form (1 þjtjþjxj)



(1 þktjjxk)



.An

extension of this method was also used in the

final complete proof of the conjecture.



The vector field approach of S Klainerman. If we

regard energy estimates as norms generated by the

plain derivatives, it is natural to extend them to

more general norms generated by vector fields

commuting, or quasicommuting, with the wave

operator. The conservation of energy expressed in

these generalized norms has a built-in decay that

allows us to prove global existence of small waves.

This circle of ideas led very far, and we might even

regard Christodoulou and Klainerman’s proof of

the stability of Minkowski space for the Einstein

equation as an extreme consequence of this

approach.



The normal forms of J Shatah. The idea is to

apply a nonlinear (and nonlocal) transformation

to the equation in order to increase the power .

This method is effective for a variety of equations,

including the semilinear wave, Klein–Gordon, and

Schro¨ dinger equations.



The conformal trans form method of D Christo-

doulou. The Penrose transform takes the wave

operator on R

1þn

to the wave operator on a

bounded subset of R  S

, the so-called Einstein

diamond (here S

is the n-dimensional sphere).

Thanks to the fact that a problem of global

existence is converted into a problem of local

existence, the proof reduces to showing that the

lifespan of the local solution becomes large

enough to cover the whole diamond when "

decreases.

A similar theory has been developed for the more

general semilinear equation

&u ¼ Fðu; u

Þ; Fðu; u

Þ¼Oðju; u



Þ for u  0

but the results are less complete. The general picture

is similar: for   2 when n  4, and for   3 when

n = 3, one has global small solutions, while for 

close to 1 one in general has blow-up.

A very interesting phenomenon in this context

was discovered by S Klainerman: some nonlinea-

rities with a special structure, called ‘‘null struc-

ture,’’ behave better than the others. This structure

is clearly related to the wave operator, and in the

end it can be precisely explained in terms of

interaction of waves in phase space. We illustrate

these ideas in the most interesting special case.

Consider the equation in three dimensions

&u ¼ FðD

u; D

uÞ; F ¼ Oðju



Þ; n ¼ 3

In the ‘‘cubic’’ case  = 3, one has global existence

for all data small enough. On the other hand, in the

‘‘quadratic’’ case  = 2, it is possible to construct

examples where the solution blows up in a finite

time no matter how small the data. Now, assume

that the nonlinear term has the following structure:

Fðu

Þ¼aQ

ðu

Þþ

0j<k3

ðu

ÞþOðju

Þ½36

which is called a ‘‘null structure’’. Here a,c

are

constants, and the quadratic forms Q are the

following:

ðu

Þ¼jD

jD

½37

ðu

Þ¼D

u  D

u  D

u  D

u; j ¼ 1; 2; 3 ½38

ðu

Þ¼D

u  D

u  D

u  D

j; k ¼ 1; 2; 3; j < k

½39

524 Semilinear Wave Equations

Then the problem has a global solution for all small

enough data. The extensions and applications of this

idea are v ery wid e (see the ‘‘Furth er readi ng’’ section

for further information). Another situation where

the null structure plays an important role is

discussed in the next section.

Low Regularity

Theorem 1, although optimal in the classical frame-

work, is not satisfactory for a few reasons. From a

physicist’s point of view, requiring n=2 þ 1 deriva-

tives of the data is not meaningful, since the

measurable quantities involve only low-order deri-

vatives, the most important one being the energy,

that is, the H

-norm of the solution. Moreover, the

wave equation has a rich set of conserved quantities,

symmetries and decay properties which may be

useful to prove stronger results, and in particular the

global existence. However, many of these structures

appear only at a low-regularity level ( H

or even

); in order to exploit them it is essential to work

with low-regularity solutions .

As an example, if we were able to prove Theorem 1

for k = 1, then we could deduce that the local

solutions can be extended to global ones in all cases

when the H

-norm is conserved. For instance, this

would allow us to solve globally the equations of

the form

&u þ G

ðjuj

Þu ¼ 0; GðsÞ0

The problem of the lowest value of s such that

a unique local solution exists in H

is quite

difficult, and still not completely solved. In order

to state the results we precise the definition of

solution as follows: the IVP is said to be locally

well posed in H

, if, for all (u

, u

) in a bounded

set B of H

 H

s1

,thereexistaT > 0, a B anach

space X

(depending on B) continuously

embedde d in C([0, T]; H

), and a unique solution

u 2 X

, such that the map (u

, u

) 7!u is contin-

uous from B to X

For the wave equation with a power nonlinearity

&u ¼juj

½40

or more generally

&u ¼ FðuÞ; FðuÞ¼0

jFðuÞF ðvÞj  Cju  vjðjuj

p1

þjvj

p1

½41

the picture is almost complete. Indeed, by using the

scaling

t 7!t; x 7!x ½42

and the Lorentz transformation

t 7!

t  x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



 1

; x 7!

t  x

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



 1

½43

it is possible to show by explicit constructions that



the equation is not locally well posed for p(n=2  s) 

(n=2 þ 2  s) (scaling) and



the equati on is not locally well posed for p(n=4 þ

1=4  s)  n=4 þ 5=4  s (Lorentz).

On the positive side, local well-posedness has been

almost fully proved in the complem entary region of

indices, with the exception of a tiny spot near the

endpoint s = 0, p = (n þ 5)=(n þ 1) where the pro-

blem is still open (and the conjecture is that the

equation is ill posed for indices in that region).

These results are due to several authors, among the

others we cite C Kenig, G Ponce, L Vega,

H Lindblad, C Sogge, L Kapitanski, and T Tao.

When the nonlinearity depends also on the first-

order derivatives of u, the situation b ecomes more

complex. In the general case, the best result

available is still the local existence theorem

(Theorem 1); the only possible r efinement is the

use of fractional Sobolev spaces H

, but in general

local solvability only holds for s > n=2 þ 1. If we

assume that F = F(u

) is a quadratic form in the

first-order derivatives, a clever use of Strichartz

estimates allows us to prove l ocal solvability down

to s > n=2 þ 1=2forn  3ands > 7=4forn = 2

(Ponce and Sideris).

However, exactly as in the case of the small

nonlinear waves examined in the previous section, if

the nonlinear term has a null structure the result can

be improved. Indeed, when F(u

) is a combination of

the forms [37]–[39], then local solvability and

uniqueness can be proved for all s > n=2, as in the

case of a nonli near term of the type F(u). This result

is due to Klainerman, Machedon, and Selberg.

Again, the proof is based on a variation of the

contraction method; the additional ingredient here

is the use of suitable function spaces, which are

the counterpart for the wave equation of the spaces

used by Bourgai n in the study of the nonlinear

Schro¨ dinger equation. The norm of these spaces is

defined as follows:

kuk

s;

khi

hjtjj ji



uð; Þk

ðR

nþ1

where hi= (1 þjj

)

1=2

and

u is the spacetime

Fourier transform of u(t, x). The wave operator can

be regarded as a spacetime Fourier multiplier of the

form 

jj

= (jtjjj)(jtjþj j), and we see that

‘‘inverting’’ the operat or

has a regularizing effect

in the scale of H

s, 

spaces, since it decreases both

Semilinear Wave Equations 525

s and  by one unit. Substantiating this formal

argument and complementing it with suitable esti-

mates for the nonlinear term requires some hard work,

which is contained in the theory of bilinear estimates

developed by Klainerman and his school.

See also: Evolution Equations: Linear and Nonlinear;

Symmetric Hyperbolic Systems and Shock Waves; Wave

Equations and Diffraction.

Further Reading

Choquet-Bruhat Y ( 1988) Global existence for nonline ar

-models, Rend. Sem. Mat. Univ. Pol. Torino, Fascicol o

speciale ‘‘Hyperbolic equations,’’ 65–86.

D’Ancona P, Georgiev V, and Kubo H (2001) Weighted decay

estimates for the wave equation. Journal of Differential

Equations 177: 146–208.

Georgiev V, Lindblad H, and Sogge CD (1997) Weighted

Strichartz estimates and global existence for semilinear wave

equations. American Journal of Mathematics 119: 1291–1319.

Ginibre J and Velo G (1982) The Cauchy problem for the

o(N), cp(N  1), and G

(N, p) models. Annals of Physics 142:

393–415.

Ginibre J and Velo G (1995) Generalized Strichartz inequalities

for the wave equation. Journal of Functional Analysis 133:

50–68.

Ho¨ rmander L (1997) Lectures on Nonlinear Hyperbolic Equa-

tions. Berlin: Springer.

John F (1979) Blow-up of solutions of nonlinear wave equations

in three space dimensions. Manuscripta Mathematica 28(1–3):

235–268.

Keel M and Tao T (1998) Endpoint Strichartz estimates.

American Journal of Mathematics 120: 955–980.

Klainerman S (1986) The null condition and global existence to

nonlinear wave equations. Lecture in Applied Mathematics

23: 293–326.

Klainerman S and Selberg S (2002) Bilinear estimates and

applications to nonlinear wave equations. Communications

in Contemporary Mathematics 4: 223–295.

Lindblad H and Sogge C (1995) Existence and scattering with

minimal regularity for semilinear wave equations. Journal of

Functional Analysis 130: 357–426.

Schiff LI (1951) Nonlinear meson theory of nuclear forces I.

Physical Review 84: 1–9.

Segal IE (1963) The global Cauchy problem for a relativistic

scalar field with power interaction. Bulletin de la Socie´te´

Mathe´matique de France 91: 129–135.

Shatah J (1988) Weak solutions and the development of

singularities in the SU(2) -model. Communications on Pure

and Applied Mathematics 41: 459–469.

Shatah J and Struwe M (1993) Regularity results for nonlinear

wave equations. Annals of Mathematics 138: 503–518.

Shatah J and Struwe M (1998) Geometric Wave Equations.

Courant Lecture Notes in Mathematics, vol. 2. New York:

Courant Institute, New York University.

Strauss W (1989) Nonlinear Wave Equations. CBMS Lecture

Notes, vol. 75. Providence, RI: American Mathematical Society.

Separation of Variables for Differential Equations

S Rauch-Wojciechowski, Linko

ping University,

Linko

ping, Sweden

K Marciniak, Linko

ping University, Norrko

ping,

Sweden

Introduction

The method of separation of variables (SoV) is a

way of finding particular and general solutions of

certain types of partial differential equations (PDEs).

Its main idea is to consider the additive ansatz

u(x) =

, ) or the multiplicative ansatz

u(x) =

, ) for a solution of a PDE that

allows for reducing this PDE to a set of (uncoupled)

ordinary differe ntial equations (ODEs) for the

unknown functions w

, )oru

, )ofone

variable x

, where x = (x

, ..., x

). Locally, the

additive ansatz is, through the change of variables

u(x) = exp(

, )), equivalent to the multi-

plicative ansatz.

Many well-known equations of mathematical

physics such as the heat equation, the wave

equation, the Schro¨ dinger equation, and the

Hamilton–Jacobi equation are solved by separating

variables in suitably chosen systems of coordinates.

Fourier Method

The SoV method can be attributed to Fourier

(1945), who solved the heat equation

u ¼ @

u ½1

for distri bution of temperature u(x, t ) in a one-

dimensional metal rod (of length L) by looking

first for special solutions of the product type

u(x, t) = X(x)T(t). This ansatz, substituted to [1],

reduces it to two O DEs: @

T = k

T and @

X =

k

X that can be solved by quadratures:

ðtÞ¼Ae

k

; X

ðxÞ¼B cosðkxÞþC sinðkxÞ

Due to linearity of [1], any formal linear combina-

tion u(x, t) =

(x)T

(t) is again a solution of

the heat equation and can be used for solving an

initial boundary-value problem (IBVP). For instance,

526 Separation of Variables for Differential Equations

in the case of the IBVP on the interval 0  x  L

and with zero boundary conditions

u ¼ @

uð0; tÞ¼uðL; tÞ¼0;

uðx; 0Þ¼f ðxÞ;

0 < t; 0 < x < L

0 < t

0 < x < L

only a countable set of values for the separation

constant k is admissible: k

= (n=L), n = 1, 2, ....

Then the general solution has the form of the

Fourier series

uðx; tÞ¼

n¼1

exp k



sinðk

xÞ

where the coefficients c

are given by the integrals

f ðxÞsinðk

xÞdx

The sequence of functions sin(k

x) is complete on

the interval [0, L]. That means that any regular

(continuous and differentiable) initial data function

f (x) such that f (0) = f ( L) = 0 can be uniquely

expressed as an infinite convergent sum of the

orthogonal set of functions sin(k

x). The study of

mathematical properties of the Fourier expansion

gave rise to the classical theory of Fourier series and

Fourier integrals.

Separability of PDEs in General Setting

A general setting for an additive separability of a

single, usually nonlinear, PDE has been developed

by Levi-Civita (1904) and by Kalnins and Miller

(1980) (see also Miller (1983)). Let

Hðx

; ...; x

; u; u

; u

ijk

; ...Þ¼E

1  i; j; k  n ½2

be a finite -order PDE for an unknown function u(x),

where u

(x) = @

i u, u

= @

j @

i u, etc., and E is a

constant. A separable solution u(x) =

)

satisfies the simpler equation

E ¼ Hðx; u; u

; u

; ...ÞH½x; u½3

where all mixed derivatives u

, etc., disappear. If a

separable solution is admissible by eqn [2], then the

function H(x; u, u

, u

, ...) has to satisfy a set of

integrability conditions following from the total

derivatives of [3]. Let

¼ @

i þ u

i;1

þ u

i; 2

i;1

þþu

i; m

þ1

i; m



þ u

i; m

þ1

i; m

(where u

i,1

= u

, u

i, jþ1

= @

i u

i, j

, etc., and m

is the

largest number l such that @

i, l

H 6¼ 0) denote the

operator of total derivative with respect to (w.r.t.) x

;

then, D

H[x, u] = 0or

i;m

þ1

¼

where H

, m

= @

, m

H. The integrability conditions

i, m

þ1

= 0, j 6¼ i, give rise to a large set of

differential conditions to be satisfied by H[x, u]:



þ H



¼ H



þ H



½4

In general, the conditions [4] are restrictions for

both H and the form of a particular separable

solution u(x). If [4] is satisfied identically w.r.t. all

u, u

k, l

, we say that the corresponding coordinate

system x

is a regular separable coordinate system;

then the PDE [3] admits a (

þ 1)-parameter

family of separable solutions. Most cases considered

in literature are regular; since then the separable

solution is usually sufficiently general for solving

various IBVPs.

A given PDE, however, usually does not satisfy

[4]; since these equations are not of tensorial type,

the natural question arises if there exists a suitable

change of coordinates y(x) such that the transformed

PDE satisfies [4]. Such separation coordinates may

or may not exist; it is usually very difficult to decide.

Here and in what follows, we speak about

separability of a single (scalar) PDE. The theory of

separability of systems of PDEs is still not devel oped

fully, although it is of relevance in the theory of

Maxwell equations and of the Dirac equation.

We present here the most classical part of SoV theory:

orthogonal separability of the Hamilton–Jacobi

equation for geodesic motions on Riemannian

manifolds.

Configurational Separation

of Hamilton–Jacobi Equation

on Riemannian Manifolds

Around 1842, C G J Jacobi invented the method of

generating function for solving the canonical

Hamilton equations

x ¼

@Hðx; yÞ

;

y ¼

@Hðx; yÞ

x ¼ðx

; ...; x

Þ y ¼ðy

; ...; y

½5

where H(x, y) is a Hamiltonian and dot denotes the

time derivative (Landau and Lifshitz 1976). In this

Separation of Variables for Differential Equations 527

method, one looks for a generating function W(x, )

of a canonical transformation

y ¼

@Wðx;Þ

;¼

@Wðx;Þ

@

that transforms Hamiltonian equations [5] into simple

equations for the new variables  2 R

,  2 R

.Since

the transformation is canonical, the transformed

equations are again Hamiltonian with the new

Hamiltonian

H(, ) = H(x(, ), y(, )). If we

choose this transformation so that

H(, ) = 

,then

the transformed Hamilton equations become

 ¼

Hð; Þ

@

¼ð1; 0; ...; 0Þ

 ¼

Hð; Þ

@

¼ 0

so that (t) = (t þ 

, 

, ..., 

), (t) =

(

, ..., 

) = const. and the solution x(t), y(t)of

the Hamilton equations [5] is then given implicitly

by the equations

ðtÞ¼

@WðxðtÞ;Þ

@

; yðtÞ¼

@WðxðtÞ;Þ

Since

y ¼

@Wðx;Þ

the generating function W(x, ) has to satisfy (identi-

cally w.r.t. (x, )) the first-order nonlinear PDE

Hx;

@Wðx;Þ



¼ 

½6

This equation is called the Hamilton–Jacobi

equation for the generating function W(x, ). It is

solved when its complete integral W(x, ), complete

means that

det

Wðx;Þ

@



6¼ 0

depending on n independent constants  is known.

In general, it is very difficult to find solutions of [6].

The most important method is the method of

separation of variables when one looks for a

solution in the form W(x, ) =

k = 1

, )

which is a sum of n functions W

, ), each

depending on a single variable x

and, possibly, all

constants a. If the Hamilton–Jacobi equation [6]

admits such a solution, then integrating this

equation is reduced to integrating n (uncoupled)

first-order ODEs for functions W

, ). The

constants 

acquire then the meaning of integration

constants.

A separable solution W(x, )of[6] exists when-

ever the Hamiltonian H(x, y) satisfies (identically)

the integrability conditions [4] which in this case

acquire the (nonlinear) form

ðHÞ@

H þ @

 @

H  @

¼ 0 for all i; j ¼ 1; ...; n ½7

= @=@x

, @

= @=@y

) found by Levi-Civita (1904).

In classical mechanics the most important

Hamiltonians are natural ones:

Hðx; yÞ¼

i;j

ðxÞy

þ VðxÞG þ V ½8

They are defined on the cotangent bundle T



Q of a

configurational Riemannian manifold Q with the

metric tensor g. The function G is the geodesic

Hamiltonian associated with the metric tensor g.For

such natural Hamiltonians, the Levi-Civita condition

(G þ V) = 0 splits into the condition L

(G) = 0

and a condition for the potential V(x). The condition

(G) = 0, depending solely on the kinetic energy

term, is thus a necessary condition for coordinates x

on Q to be separation coordinates for [8].

In the fundamental case of orthogonal separation

(i.e., when g

= 0 for i 6¼ j), the Levi-Civita condi-

tions L

(G þ V) = 0 read

 @

ln g



 @

ln g



¼ 0; i 6¼ j ½9

V  @

ln g



 @

ln g



V ¼ 0; i 6¼ j ½10

The main questions arising here are

1. What is the algebraic form of orthogonally

separable Riemannian metri cs?

2. What is the form of separable coordinates on

Riemannian manifolds?

The first question is answered by the Sta¨ckel

theorem (Sta¨ ckel 1891) that provides an algebraic

characterization of orthogonal separability of a

natural Hamiltonian H = G þ V.

Theorem 1 The Hamilton –Jacobi equation for the

natural Hamiltonian

H ¼ G þ V ¼

ðxÞy

þ VðxÞ

528 Separation of Variables for Differential Equations

is separable in the (orthogonal) coordinates x if and

only if

(i) There exists a matrix =[’

)], det () 6¼ 0

(so that the row i depends only on x

) such that

, ..., g

] is the first row of the inverse

matrix =

1

(ii) The potential V has the form V(x) =

where each f

) is a function of one variable x

only.

Such matrix  is called a Sta¨ckel matrix.

Proof If

; ...; g



’

ðx

Þ’

ðx

’

ðx

Þ’

ðx

¼ 1; 0; ...; 0½ ½11

then the Hamilton–Jacobi equation for H can be

written as



ðx

Þ¼

¼ 

’

ðx

Þþ

’

ðx

þþ

’

ðx

Þ½12

This equation admits an additively separable

solution W =

), where the functions W

satisfy n ODEs (separation equations):



þf

ðx

¼ 

’

ðx

Þþ

’

ðx

Þþþ

’

ðx

i ¼ 1; ...; n ½13

By differentiating [13] w.r.t. 

, we get

’

ðx

Þ¼

@

and thus

det ’

ðx



...

det

@



6¼ 0

so that W =

) is indeed a complete integral of

the Hamilton–Jacobi equation [12]. Conversely, if

W =

) is a complete integral of the Hamilton–

Jacobi equation [12], then by differentiating it w.r.t. 

we get for j = 1

@

¼ 1

and

@

¼ 0

(for j = 2, ..., n), that is, the condition [11] for the

Sta¨ ckel matrix

 ¼

@



Further, we see that

V ¼ 



ðÞ



’

ðx

Þ

i W

ðÞ



ðx

Þ &

Remark 2 The Sta¨ ckel characterization of orthogo-

nal separability is equivalent to Levi-Civita conditions

[9] and [10]. It is in fact a solution of these conditions.

Remark 3 With every Sta¨ckel matrix, one can

relate a family of n quadratic in momenta Hamilto-

nians defined by n rows of the inverse Sta¨ckel matrix

=

1

= [

r¼1

; k ¼ 1; ...; n ½14

(so that H

= G). These Hamiltonians are linearly

and functionally independent; they Poisson-

commute (so that they form a Liouville integrable

system) and are all diagonal so that they have

common eigenvectors.

These properties are the main ingredients of an

intrinsic (coordinate-independent) characterization

of separable geodesic Hamiltonians G in terms of

involutive Killing tensors that is due to works of

Eisenhart (1934), Kalnins and Miller (1980), and

Benenti (1997).

Theorem 4 A necessary and sufficient condition

for the existence of an orthogonal additive separable

coordinate system x for the Hamilton–Jacobi

equation of the geodesic Hamiltonian H

= G

on an n-dimensional (pseudo)-Riemannian manifold

is that there exist n qua dratic forms

i, j

(x)y

such that

(i) They all Poisson-commute:{H

, H

} = 0, 1  r,

s  n.

(ii) The set {H

}

r = 1

is linearly independent.

(iii) There is a basis {!

(j)

}

j = 1

of n simultaneous

eigenforms for all H

Separation of Variables for Differential Equations 529

If conditions(i)–( iii) are satisfied then there exist

functions g

(x) such that !

(j)

= g

, j = 1, ..., n.

This theorem has been further simplified

by Benenti (1997), who has shown that for separ-

ability it is sufficient that g

admits a single Killing

2-tensor with simple eigenvalues an d normal eigen-

vectors. He has also explained the role of ignorable

coordinates.

These results are key ingredients of an answer to the

question (2). Eisenhart (1934), starting from the fact

that every separable geodesic Hamiltonian H = G

admits n quadratic (w.r.t. momenta y

)integralsof

motion, derived a set of nonlinear PDEs characterizing

separable Riemannian metrics. He has solved these

equations for spaces of constant curvature. This

solution is the basis of the Kalnins and Miller’s

(1986) diagrammatic classification of all orthogonal

separation coordinates on R

and the sphere S

Separable coordinates on the Minkowski space M

have not been classified yet.

Since the work of Robertson (1927) and Eisenhart

(1934), it is known that in R

, S

and, in general, in

the space with diagonal Ricci tensor, the (additive)

separability of Hamilton–Jacobi equation for the

natural Hamiltonian H = G þ V is equivalent

to multiplicative separab ility of the stationary

Schro¨ dinger equation with the same potential V:

ð þ VðxÞÞðxÞ¼EðxÞ½15

where

 ¼

i;j¼1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðgÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðgÞ



is the Laplace–Beltrami operator. Usually, multi-

plicative separated solutions (x) =

i = 1



(x)is

considered but the change of the dependent variable

u = ln  transforms it into an additive separable

solution. If we restrict our considerations to ortho-

gonal separation coordinates (g

= 0 for i 6¼ j), eqn

[15] becomes

i¼1



þ u



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðgÞ





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðgÞ



þ VðxÞ¼E

where u

= @

u, u

= @

u. The integrability condi-

tions [4] for regular separation lead to the Levi-Civita

condition [9] on the components g

of the metric

tensor, upon comparison of the coefficients at u

The coefficients at u

yield the Robertson condition

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðgÞ



¼ 0; i 6¼ j

and the constant terms in [4] give the Levi-Civita

equation [10] meaning that V(x) =

i = 1

Eisenhart has shown that the Robertson condition is

equivalent to the requirement that the Ricci tensor is

diagonal: R

= 0, i 6¼ j in variables x so that the

Robertson condition is satisfied automatically in the

Euclidean space, in spaces of constant curvature and in

Einstein spaces. Thus every orthogonal coordinate

system permitting multiplicative separation of the

Schro¨ dinger equation corresponds to the Sta¨ ckel form.

Jacobi Problem of Separability

In order to apply the separability theory to physical

Hamiltonians H = (1=2)p

þ V(q), p = (p

, ..., p

q = (q

, ..., q

), it is essential to solve the following

problem: ‘‘given a potential V(q), decide if there

exists a point transformation x(q) to some curvi-

linear coordinates x such that the Hamilton–Jacobi

equation associated with H is separable in coordi-

nates x, and if su ch transform ation exists, determine

it and solve the obtained Hamilton–Jacobi

equation.’’

This problem has been raised by Jacobi (1884) in

connection with the problem of finding geodesic

motions on a 3-axial ellipsoid. For solving this

problem Jacobi introduced his ‘‘remarkable change

of coordinates’’ to the generalized elliptic coordi-

nates x(q) defined through zeros of the rational

function

1 þ

i¼1

ðq

ðz  



ðz  x

ðz  

½16

where the constants 

> 0 are all different. From

the graph of the left-hand side of [16 ], it is easy to

see that there are exactly n simple, real zeros. For

given values of elliptic coordinates x

, the values of

)

are uniquely determined as residues at 

while

Cartesian coordinates q

are determined uniquely

only in each n-tant of R

The Jacobi elliptic coordinates play a pivotal role

in orthogonal separability on R

and S

since they

are the mother of all other separation coordinates

that can be obtained through proper and improper

degenerations of 

’s. By using these co ordinates

Jacobi solved not only the geodesic motions on the

ellipsoid but also the motion on the ellipsoid under

the action of harmonic potential V(q) = (1=2)q

.He

has also found separation coordinates for a system

of three interacting particles on the line known

today as the Calogero system. In general, however,

Jacobi considered the problem of finding separation

coordinates for a given potential V(q) to be very

difficult. In Vo¨ rlesungen u¨ ber Dynamik, ch. 26, he

writes: ‘‘The main difficulty in integrating a given

530 Separation of Variables for Differential Equations

differential equation lies in introducing convenient

variables, which there is no rule for finding. There-

fore, we must travel the reverse path and after

finding some notable substitution, look for problems

to which it can be successfully applied’’. This

statement had a profound influence on further

development of SoV theory that concentrated on

characterizing separable Hamiltonians (as expressed

in terms of separation coordinates) and on describ-

ing and classifying separation coordinates.

The original problem of Jacobi of finding separa-

tion variables for a given natural Hamiltonian has

been taken up by Rauch-Wojciechowski (1986),

who found a characterization of separable potentials

V(q) in terms of Cartesian coordinates q

. Its

invariant geometric form has been given by Benenti.

A complete criterion of separability that allows for

an effective test ing and calculation of separation

coordinates (if they exist) for V(q) has been solved

by Waksjo¨ and Rauch-Wojciechowski (2003). This

criterion is directly applicable to the problem of

finding SoV for the Schro¨ dinger equation.

Criterion of Separability for n = 2

The criter ion of separability for n = 2 can be read

from the Bertrand–Darboux theorem.

Theorem 5 (Bertrand–Darboux). For the

Hamiltonian:

H =

þ p



þ Vðq

; q

the following statements are eq uivalent:

(i) H has a functionally independent integral of

motion {H, K} = 0 of the form

K ¼ aq

þ bq

þ c









 2aq

 bq



þ d



þ kðq

; q

(ii) The potential V(q

, q

) satisfies the following linear

second-order PDE with quadratic coefficients

0 ¼2



 aq

þ bq



þ c 



þð2aq

 bq



þ dÞ @

V  @



þð6aq

þ 3bÞ@

V ð6aq

þ 3

bÞ@

V ½17

where a, b,

b, c,

c, d are some constants,

= @

, @

= @

(iii) The Hamilton – Jacobi equation for H is separ-

able in one of the four orthogonal coordinate

systems in the plane: elliptic, parabolic, polar,

or Cartesian.

Remark 6 If the potential V(q

, q

) is separable,

then it admits an integral of motion K that is

quadratic w.r.t. momenta and V satisfies (identically

w.r.t. q

, q

) eqn [17] for certain values of the

undetermined constants a, b,

b, c,

c, d. Since coeffi-

cients at linearly independent expressions of q

, q

have to be equal to zero, the parameters

a, b,

b, c,

c, d have to satisfy a set of linear, algebraic,

homogeneous equations. If there is a nonzero

solution for a, b,

b, c,

c, d, then there exists an

integral of motion K and separation coordinates

can be determined as characteristic variables for

equation [17].

Example 7 Separable cases of the Henon–Heiles

potential

V ¼

þ !



þ q



q

By substituting this form of V into [17], we get two

sets of ad missible solutions for parameters , ,

, !

: (i)  = , !

= !

with V separable in

rotated (by =4) Cartesian coordinates; (ii)

 = 6, !

, !

-arbitrary with V separable in the

shifted parabolic coordinates. In case (ii) eqn [17]

becomes

2 q



4

ð4!

 !



þ q

V  @



þ 3@

V ¼ 0

and in its characteristic coordinates defined as

ﬃﬃﬃﬃﬃ



, q

= (1=2)(  ) þ (1=4)(4!

 !

)it

takes the form (  )@





V þ @



V þ @



V = 0 solved

by V(, ) = ( þ )

[f () þ g()] which is separable

in the parabolic coordinates.

Effective Criterion of Separability

for Arbitrary Dimension

For n > 2, a similar theorem characterizing separ-

ability in generalized elliptic coordinates has been

formulated by Rauch-Wojciechowski (1986).

Theorem 8 (Elliptic Bertrand–Darboux). For a

natural Hamiltonian H = (1=2)p

þ V(q), the

following statements are equiva lent:

(i) H has n global, functionally independent and

involutive integrals of motion {H, K

} = 0,

, K

} = 0, i, j = 1, ..., n, having the form

r¼1;r6¼i

ð

 

1

þ p

þ k

ðqÞ

¼ q

 q

½18

Separation of Variables for Differential Equations 531