Invariant differential equations and the Adler–Gel'fand–Dikii bracket

20 Pages·2015·0.8 MB·English

Invariant differential equations and the
Adler–Gel’fand–Dikii bracket
Artemio Gonza´lez-Lo´pez and Rafael Herna´ndez Heredero
DepartamentodeF´ısicaTeo´ricaII,FacultaddeCienciasF´ısicas,Universidad
Complutense,28040Madrid,Spain
Gloria Mar´ı Beffa
DepartmentofMathematics,UniversityofWisconsin,Madison,Wisconsin53706
(cid:126)Received 24 March 1997; accepted for publication 30 June 1997(cid:33)
In this paper we find an explicit formula for the most general vector evolution of
curves on RPn(cid:50)1 invariant under the projective action of SL(n,R). When this
formulaisappliedtotheprojectivizationofsolutioncurvesofscalarLaxoperators
with periodic coefficients, one obtains a corresponding evolution in the space of
such operators. We conjecture that the formula we have found gives another alter-
native definition of the second KdV Hamiltonian evolution under appropriate con-
ditions. In other words, both evolutions are identical provided that the vector dif-
ferential invariant characterizing the SL(n,R)-invariant evolution on the space of
projectivized curves is identified with the coefficients of the Hamiltonian pseudo-
differential operator. We prove the above facts for n(cid:60)6, and further simplify both
evolutions in appropriate coordinates so that one can attempt to prove the equiva-
lence for arbitrary n. © 1997 American Institute of Physics.
(cid:64)S0022-2488(cid:126)97(cid:33)02111-7(cid:35)
I. INTRODUCTION
In an attempt to generalize the bi-Hamiltonian character of the Korteweg–deVries (cid:126)KdV(cid:33)
equation, Adler1 defined a family of second Hamiltonian structures with respect to which the
generalized higher-dimensional KdV equations could also be written as Hamiltonian systems.
Jacobi’s identity for these brackets was proved by Gel’fand and Dikii in Ref. 2. These Poisson
structuresarecalledsecondHamiltonianKdVstructuresorAdler–Gel’fand–Dikiibrackets.Since
theoriginaldefinitionofAdlerwasquitecomplicatedandnotveryintuitive,alternativedefinitions
have been subsequently offered by several authors, most notably by Kupershmidt and Wilson in
Ref. 3, and by Drinfel’d and Sokolov in Ref. 4. Once higher-dimensional KdV equations were
proved to be bi-Hamiltonian, their integrability was established via the usual construction of a
sequence of Hamiltonian structures with commuting Hamiltonian operators. The second Hamil-
tonianStructureinthehierarchyofKdVbracketscoincideswiththeusualsecondPoissonbracket
for the KdV equation, that is, the canonical Lie–Poisson bracket on the dual of the Virasoro
algebra. This is the only instance in which the second KdV bracket is linear.
AsubjectapparentlyunrelatedtotheHamiltonianstructuresofpartialdifferentialequationsis
the theory of Klein geometries and differential and geometric invariants. This theory had its high
point in the last century before the appearance of Cartan’s approach to differential geometry, and
it is closely related to equivalence problems. Namely, one poses the question of equivalence of
twogeometricalobjectsundertheactionofacertaingroup,thatis,whencanoneofthoseobjects
betakentotheotheroneusingatransformationbelongingtothegivengroup?Forexample,given
twocurvesontheplane,whenaretheyequivalentunderanEuclideanmotion?Or,whenarethey
the same curve, up to parametrization?, etc. One answer can be given in terms of invariants,
namely, expressions that depend on the objects under study and that do not change under the
action of the group. If two objects are to be equivalent, they must have the same invariants. If
these invariants are functions on some jet space (cid:126)for example, if they depend on the curve and its
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5720 J.Math.Phys.38(11),November1997 ©1997AmericanInstituteofPhysics
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Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket 5721
derivatives with respect to the parameter(cid:33), then they are called differential invariants. In the case
of curves on the Euclidean plane under the action of the Euclidean group, the basic differential
invariantisknowntobetheEuclideancurvature.Withinthenaturalscopeofthestudyofequiva-
lence problems and their invariants lies also the description of invariant differential equations,
symmetries, relative invariants, etc. For example, recently Olver etal.5 used these ideas to char-
acterize all scalar evolution equations invariant under the action of a subgroup of the projective
groupintheplane,aproblemofinterestinthetheoryofimageprocessing.SeeOlver’sbook6for
an account of the state of the subject.
In this paper we establish a connection between the foregoing two theories while trying to
answerthefollowingquestion.LetL(t,(cid:117)) beafamilyofscalardifferentialoperatorswithperiodic
coefficients following an evolution (cid:126)in t(cid:33) which is Hamiltonian with respect to the Adler–
Gel’fand–Dikiibracket.Considerafamilyofsolutioncurves(cid:106)(t,(cid:117)) associatedtoL(t,(cid:117)). Isthere
a simple and explicit way to describe the evolution of (cid:106)(t,(cid:117))? The importance of studying the
spaceofsolutionsofL waspointedoutbyWilsoninRef.7.AlsoinRef.8,fromadifferentpoint
of view than the one presented here, a description of this evolution was given and proven to be
SL(n,R) invariant. These curves are also used to provide a discrete invariant of the Poisson
bracket,oneofthetwoinvariantswhichclassifythesymplecticleaves.9Here,weaimtoshowthat
theevolutionofthesolutioncurvesisofrelevanceinitself,andcanbedescribedusingthetheory
of differential invariants. We will see that the evolution of the projectivization (cid:102)(t,(cid:117)) of a
solution curve is invariant under the projective action of SL(n,R). Following Olver’s approach,6
we will write explicitly the most general(cid:126)vector(cid:33) evolution of curves on real (n(cid:50)1)-dimensional
projective space RPn(cid:50)1 of the form
(cid:102)t(cid:53)F(cid:126)(cid:102),(cid:102)(cid:117),(cid:102)(cid:117)(cid:117),...(cid:33)
which is invariant under the SL(n,R) projective action. Moreover, under certain conditions that
we will state precisely in the paper, we conjecture that every SL(n,R)-invariant evolution of
curves on RPn(cid:50)1 must correspond to an Adler–Gel’fand–Dikii Hamiltonian evolution in the
space of time-dependent nth order scalar differential operators with periodic coefficients. This
correspondence, which provides an alternative definition of the Adler–Gel’fand–Dikii bracket,
will be described in detail and shown to be true for many fixed values of n. Unfortunately, we
haven’t succeeded in proving the general case, which is considerably more involved. We will
nevertheless guide the reader in simplifying the proof in the general case, so that he or she can
attempt to prove the conjecture for any particular value of n.
II. NOTATION AND BASIC FACTS
In this Section we will set the notation used in the rest of the paper, and recall some elemen-
tary properties of the projectivized solution curves of scalar Lax operators. Denote by A the
n
infinite-dimensional manifold of scalar differential operators (cid:126)or Lax operators(cid:33) with T-periodic
coefficients of the form
dn dn(cid:50)2 d
L(cid:53)d(cid:117)n(cid:49)un(cid:50)2 d(cid:117)n(cid:50)2(cid:49)•••(cid:49)u1 d(cid:117)(cid:49)u0, (cid:126)2.1(cid:33)
and let (cid:106) (cid:53)((cid:106) ,...,(cid:106)) be a solution curve associated to L the Wronskian of whose components
L 1 n
equals one. Due to the periodicity of the coefficients of L, there exists a matrix M (cid:80)SL(n,R),
L
called the monodromy of L, such that
(cid:106) (cid:126)(cid:117)(cid:49)T(cid:33)(cid:53)M (cid:106) (cid:126)(cid:117)(cid:33), for all (cid:117)(cid:80)R.
L L L
(cid:126)M isdefinedbytheFloquetmatrixofthedifferentialequation.(cid:33)Thissamepropertyholdsforits
L
(cid:126)non-degenerate(cid:33) projection on the n(cid:50)1 sphere Sn(cid:50)1 (cid:126)(cid:106)ˆ (cid:53)((cid:106) /(cid:117)(cid:106) (cid:117)), where (cid:117) (cid:117) represents the
L L L (cid:149)
J.Math.Phys.,Vol.38,No.11,November1997
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5722 Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket
norm on Rn(cid:33), and is also shared by the projective coordinates of this projection, whenever we
considertheactionsofSL(n,R) onthesphereandonprojectivespace,respectively.Observethat
the monodromy is not completely determined by the operator L, but by its solution curves.
Namely, if one chooses a different solution curve, its monodromy won’t be equal to M in
L
general,butitwillbetheconjugateofM byanelementofGL(n,R). Thatis,L onlydetermines
L
theconjugationclassofthemonodromy.Ofcourse,thisproblemdoesnotexistoncethesolution
curve has been fixed.
Conversely, let (cid:102):R!RPn(cid:50)1 be a curve on RPn(cid:50)1. Assume that the curve (cid:102)is non-
degenerate and right-hand oriented, that is the Wronskian determinant of the components of its
derivative (cid:102)(cid:56) is positive. (cid:126)This is equivalent to the Wronskian of the components of (cid:126)1,(cid:102)(cid:33) being
positive; for example, the curve would be convex and right-hand oriented in the case n(cid:53)3.(cid:33)
Assume also that (cid:102)satisfies the following monodromy property:
(cid:102)(cid:126)(cid:117)(cid:49)T(cid:33)(cid:53)(cid:126)M(cid:102)(cid:33)(cid:126)(cid:117)(cid:33), for all (cid:117)(cid:80)R, (cid:126)2.2(cid:33)
for a given M(cid:80)SL(n,R). Here M(cid:102)represents the usual action of SL(n,R) on RPn(cid:50)1, induced
by the action of SL(n,R) on Rn. One can associate to (cid:102)a differential operator of the form (cid:126)2.1(cid:33)
inthefollowingmanner:Welift(cid:102)toacurveonRn, sayto f((cid:117))(1,(cid:102)). Wechoosethefactor f so
that the Wronskian of the components of the new curve equals 1. There is a unique choice of f
with such a property (cid:126)up to perhaps a sign(cid:33), namely
f(cid:53)W(cid:126)1,(cid:102)1,...,(cid:102)n(cid:50)1(cid:33)(cid:50)1/n(cid:53)W(cid:126)(cid:102)1(cid:56),...,(cid:102)n(cid:56)(cid:50)1(cid:33)(cid:50)1/n,
where (cid:102)(cid:53)((cid:102)1,...,(cid:102)n(cid:50)1) and W represents the Wronskian determinant.
It is not very hard to see that the coordinate functions of the lifted curve are solutions of a
unique differential operator of the form (cid:126)2.1(cid:33). Such an operator defines an equation for an un-
known y of the fo(cid:85)rm (cid:85)
y f(cid:102)0 ... f(cid:102)n(cid:50)1
y(cid:56) (cid:126)f(cid:102)0(cid:33)(cid:56) ... (cid:126)f(cid:102)n(cid:50)1(cid:33)(cid:56) (cid:53)0; (cid:102)(cid:53)1, (cid:56)(cid:53) d . (cid:126)2.3(cid:33)
(cid:65) (cid:65) (cid:65) 0 d(cid:117)
(cid:29)
y(cid:126)n(cid:33) (cid:126)f(cid:102)0(cid:33)(cid:126)n(cid:33) ... (cid:126)f(cid:102)n(cid:50)1(cid:33)(cid:126)n(cid:33)
Equation (cid:126)2.3(cid:33) can be written in the usual manner as a system of first order differential equations
(cid:83) (cid:68)
dX/d(cid:117)(cid:53)NX, where
0 1 0 ... 0
0 0 1 ... 0
N(cid:53) (cid:65) (cid:65) (cid:29) (cid:29) (cid:65)
0 0 ... 0 1
(cid:50)u0 (cid:50)u1 ... (cid:50)un(cid:50)2 0
and X is a fundamental matrix solution associated to the differential equation (cid:126)2.3(cid:33). From this
formulation and the monodromy condition it is trivial to see that N(cid:53)(dX/d(cid:117))X(cid:50)1 is a periodic
matrix and so are the coefficients of the operator defining (cid:126)2.3(cid:33).
Ashortcommentisdueatthispoint:ifM isthemonodromymatrixassociatedto(cid:102),foreven
n the monodromy matrix associated to L could be either M or (cid:50)M, depending on whether the
first component of M(1,(cid:102)) is positive or negative. Hence, it would be more correct to talk about
theactionofPSL(n,R), thespaceobtainedfromSL(n,R) byidentifying M and(cid:50)M. Sincethis
choice makes no difference in what follows, we will keep SL(n,R) for the sake of simplicity.
J.Math.Phys.,Vol.38,No.11,November1997
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Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket 5723
III. THE EVOLUTION EQUATIONS ON A
n
TheAdler–Gel’fand–Dikiibracket.WestartbydescribingoneoftheHamiltonianevolutions
onthemanifoldA , thewellknownAdler–Gel’fand–Dikiibracket,orsecondKdVHamiltonian
n
structure.
Given a linear functional H on A , one can associate to it a pseudo-differential operator
n
n
(cid:40) d
H(cid:53) h (cid:93)(cid:50)1, (cid:93)(cid:53) ,
i(cid:53)1 i d(cid:117)
such that
(cid:69)
H(cid:126)L(cid:33)(cid:53) res(cid:126)HL(cid:33)d(cid:117),
S1
whereresselectsthecoefficientof(cid:93)(cid:50)1 andiscalledtheresidueofthepseudo-differentialoperator
(cid:126)see Ref. 1 or 2(cid:33). To any H we can associate a (cid:126)Hamiltonian(cid:33) vector field V defined as
H
VH(cid:126)L(cid:33)(cid:53)(cid:126)LH(cid:33)(cid:49)L(cid:50)L(cid:126)HL(cid:33)(cid:49),
whereby((cid:149))(cid:49) wedenotethenon-negative(cid:126)ordifferential(cid:33)partoftheoperator.ThemapH!VH
ˆ
isastructuremapdefiningaPoissonbracketonthemanifoldA . Ifl isthematrixofdifferential
n
operators defining the structure map, the Poisson bracket is defined as
(cid:69)
(cid:36)H,F(cid:37)(cid:126)L(cid:33)(cid:53) res(cid:126)ˆl(cid:126)H(cid:33)F(cid:33)d(cid:117), (cid:126)3.1(cid:33)
S1
cf. Refs. 1, 2 or 10. The original definition of the bracket was given by Adler,1 in an attempt to
make generalized KdV equations bi-Hamiltonian systems. Gel’fand and Dikii proved Jacobi’s
identity in Ref. 2. In the case n(cid:53)2, this bracket coincides with the Lie–Poisson structure on the
dual of the Virasoro algebra. Two other equivalent definitions of the original bracket were found
in Refs. 3 and 4. The original definition is rather complicated, so we will explain and use the one
in Ref. 3.
The Kupershmidt-Wilson bracket. In a very interesting paper,3 Kupershmidt and Wilson gave
anequivalentbutrathersimplerdefinitionofthebracket(cid:126)3.1(cid:33).ConsiderL tobeanoperatorofthe
form (cid:126)2.1(cid:33). Assume that the operator L factors into a product of first-order factors
L(cid:53)(cid:126)(cid:93)(cid:49)yn(cid:50)1(cid:33)(cid:126)(cid:93)(cid:49)yn(cid:50)2(cid:33)•••(cid:126)(cid:93)(cid:49)y1(cid:33)(cid:126)(cid:93)(cid:49)y0(cid:33),
where
yk(cid:53)(cid:118)kv1(cid:49)(cid:118)2kv2(cid:49)•••(cid:49)(cid:118)(cid:126)n(cid:50)1(cid:33)kvn(cid:50)1, 0(cid:60)k(cid:60)n(cid:50)2; (cid:118)(cid:53)e2(cid:112)i/n, (cid:126)3.2(cid:33)
and yn(cid:50)1(cid:53)(cid:50)(cid:40)in(cid:53)(cid:50)02yi. The variables vi, 1(cid:60)i(cid:60)n(cid:50)1, are what Kupershmidt and Wilson called
‘‘modified’’ variables. Even though the factorization is not unique (cid:126)and so some reduction had to
be involved in the proof of Ref. 3(cid:33), one can find a unique factorization once a solution curve has
been fixed, as we will see later.
Assume that the coefficients u , 0(cid:60)i(cid:60)n(cid:50)2, of L evolve following a Hamiltonian evolution
i
with respect to the second KdV Hamiltonian structure. The result in Ref. 3 then states that the
corresponding ‘‘modified’’ coordinates v evolve following a Hamiltonian evolution with respect
i
to a Poisson bracket defined by the structure map
J.Math.Phys.,Vol.38,No.11,November1997
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5724 Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket
1
l(cid:53)(cid:50) (cid:93)J, (cid:126)3.3(cid:33)
n
where (cid:83) (cid:68)
0 ... 0 1
0 ... 1 0
J(cid:53) .
(cid:65) (cid:65)
(cid:28) (cid:28)
1 0 ... 0
That is,
(cid:83) (cid:68)
Du Du *
l (cid:53)ˆl, (cid:126)3.4(cid:33)
Dv Dv
where Du/Dv(cid:53)(Du /Dv ),
i j
Dui(cid:53)n(cid:40)(cid:50)1 (cid:93)ui (cid:93)k
Dvj k(cid:53)0 (cid:93)v(cid:126)jk(cid:33)
being the Fre´chet derivative of u with respect to v . Also, by * we denote the adjoint matrix
i j
operator, the transposed of the matrix whose entries are the adjoint operators of the entries of the
original matrix. Thus, the original Adler–Gel’fand–Dikii bracket arises from a very simple
bracket defined on the space of ‘‘modified’’ variables v.
Many facts are known about this Hamiltonian structure. Since it is Poisson (cid:126)degenerate(cid:33), the
manifold A foliates into symplectic leaves, maximal submanifolds where the Hamiltonian flow
n
alwayslies.Theseleavesareclassifiedlocallybytheconjugationclassofthemonodromymatrix
associated to the operators lying on the leaf. In other words, if two operators are close and have
conjugate monodromies, there is a Hamiltonian path joining them. There exists another discrete
invariant that classifies the leaves globally, cf. Ref. 9, based on topological properties of the
projection of the solution curves on the sphere Sn.
IV. INVARIANT EVOLUTION EQUATIONS ON C
n
The duality between A and C described in the previous sections makes it natural to study
n n
evolution equations on the space C whose associated flow leaves the Adler–Gel’fand–Dikii
n
symplectic leaves invariant. In other words, we are interested in partial differential equations of
the form
(cid:102)t(cid:53)F(cid:126)(cid:117),(cid:102),(cid:102)(cid:117),(cid:102)(cid:117)(cid:117),...(cid:33), (cid:102):R2!RPn(cid:50)1, (cid:126)4.1(cid:33)
for a function (cid:102)((cid:117),t), with the property that, if the initial condition has a monodromy property
(cid:126)2.2(cid:33), then every solution (cid:102)( ,t) of (cid:126)4.1(cid:33) has also a monodromy property, and the conjugation
(cid:149)
class of the monodromy matrix is independent of t. The simplest evolution equations having this
property are those of the form (cid:126)4.1(cid:33) with F independent of (cid:117)which are also invariant under the
standard projective action of SL(n,R) on the dependent variables (cid:102)(cid:53)((cid:102)1,...,(cid:102)n(cid:50)1). In other
words, we are dealing with equations of the form
(cid:102)t(cid:53)F(cid:126)(cid:102),(cid:102)(cid:117),(cid:102)(cid:117)(cid:117),...(cid:33), (cid:102):R2!RPn(cid:50)1, (cid:126)4.2(cid:33)
such that whenever (cid:102)((cid:117),t) is a solution of (cid:126)4.2(cid:33) so is (M(cid:102))((cid:117),t), for all M(cid:80)SL(n,R). To see
thatthemonodromyclassofthesolutions(cid:102)( ,t) ofanequation(cid:126)4.2(cid:33)invariantundertheactionof
(cid:149)
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Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket 5725
SL(n,R) is indeed preserved under the evolution, note that (cid:126)4.2(cid:33) is also invariant under transla-
tions of the independent variable (cid:117). Hence, if the initial condition (cid:102)(cid:126) ,0(cid:33) of (cid:126)4.2(cid:33) has a matrix
(cid:149)
M(cid:80)SL(n,R) as monodromy, and we consider a different curve in the flow (cid:102)( ,t), we have that
(cid:149)
(cid:102)((cid:117)(cid:50)T,t) isalsoasolution.If(cid:126)4.2(cid:33)isSL(n,R)-invariant,M(cid:102)((cid:117)(cid:50)T,t) willalsobeasolutionof
(cid:126)4.2(cid:33). Applying uniqueness of solutions of (cid:126)4.2(cid:33) (cid:126)whenever possible(cid:33), M(cid:102)((cid:117)(cid:50)T,t)(cid:53)(cid:102)((cid:117),t), so
that (cid:102)( ,t) has the same monodromy as (cid:102)(cid:126) ,0(cid:33). If there is no uniqueness of solutions, both
(cid:149) (cid:149)
Hamiltonian and invariant evolutions are obviously much more complicated; we won’t deal with
those cases in this paper.
Remark: note that the evolution associated to an SL(n,R)-invariant equation (cid:126)4.2(cid:33) preserves
exactly the monodromy (cid:126)not just the monodromy class(cid:33) of its solutions.
In this paper we conjecture that the Adler–Gel’fand‘–Dikii evolution on A and the
n
SL(n,R)-invariant evolution (cid:126)4.2(cid:33) on C are identical under the identification described in the
n
Introduction,providedthatthecoefficientsoftheHamiltonianH (cid:126)thepseudo-differentialoperator
describing the differential of the functional H(cid:33) are equal to a vector differential invariant of the
projective action. We will find the most general SL(n,R)-invariant evolution of the form (cid:126)4.2(cid:33),
showing then how the conjecture can be proved for a number of values of n and where the main
problem lies in the proof of the general case.
The most general evolution equation of the form (cid:126)4.2(cid:33) invariant under the projective action
(cid:102)(cid:126)(cid:117),t(cid:33)(cid:176)(cid:126)M(cid:102)(cid:33)(cid:126)(cid:117),t(cid:33)
ofSL(n,R) canbefoundusingthegeneralinfinitesimaltechniquesdescribedinRefs.10,6.First
of all, the infinitesimal generators of the projective SL(n,R) action are easily found to be the
following vector fields on R(cid:51)R(cid:51)RPn(cid:50)1:
(cid:93) (cid:93) n (cid:93)
(cid:40)
v(cid:53) , v (cid:53)(cid:102) , w(cid:53)(cid:102) (cid:102) ; 1(cid:60)i, j(cid:60)n(cid:50)1. (cid:126)4.3(cid:33)
i (cid:93)(cid:102) ij i (cid:93)(cid:102) i i j (cid:93)(cid:102)
i j j(cid:53)1 j
The vector fields (cid:126)4.3(cid:33) are a basis of a realization of the Lie algebra sl(n,R). Note that all these
vector fields are independent of the variables ((cid:117),t), and they are also ‘‘vertical,’’ i.e., their (cid:117)and
t components vanish.
If v(cid:53)(cid:83)n(cid:50)1(cid:104)((cid:117),t,(cid:102))(cid:93)/(cid:93)(cid:102) is a vertical vector field, its prolongation is the vector field prv
i(cid:53)1 i i
defined by
n(cid:50)1 (cid:93)
(cid:40) (cid:40)
pr v(cid:53)v(cid:49) (cid:126)DkDj(cid:104)(cid:33) , (cid:126)4.4(cid:33)
t i (cid:93)(cid:126)(cid:93)k(cid:102)(cid:126)j(cid:33)(cid:33)
j(cid:62)1 i(cid:53)1 t i
k(cid:62)0
where (cid:102)(j)(cid:53)(cid:93)j(cid:102), D is the total derivative operator with respect to (cid:117)
i i
n(cid:50)1 (cid:93)
(cid:40) (cid:40)
D(cid:53)(cid:93)(cid:49) (cid:102)(cid:126)j(cid:49)1(cid:33) ,
i (cid:93)(cid:102)(cid:126)j(cid:33)
j(cid:62)0 i(cid:53)1 i
and
n(cid:50)1 (cid:93)
(cid:40) (cid:40)
D (cid:53)(cid:93)(cid:49) (cid:126)(cid:93)(cid:102)(cid:126)j(cid:33)(cid:33) (cid:126)4.5(cid:33)
t t t i (cid:93)(cid:102)(cid:126)j(cid:33)
j(cid:62)0 i(cid:53)1 i
is the total derivative operator with respect to t. In general, the vector field prv is defined on the
infinite-dimensional jet space J(cid:96)(R(cid:51)R,RPn(cid:50)1) with local coordinates (cid:117),t,(cid:93)k(cid:102)(j) (1(cid:60)i(cid:60)n
t i
(cid:50)1;k,j(cid:62)0). However,whenprvisappliedtoafunction(cid:126)likeF(cid:33)independentofthecoordinates
(cid:93)k(cid:102)(j) (k(cid:62)1) involving explicitly t-derivatives, (cid:126)4.4(cid:33) reduces to the vector field
t i
J.Math.Phys.,Vol.38,No.11,November1997
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5726 Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket
n(cid:50)1 (cid:93)
(cid:40) (cid:40)
pr v(cid:53)v(cid:49) (cid:126)Dj(cid:104)(cid:33) , (cid:126)4.6(cid:33)
i (cid:93)(cid:102)(cid:126)j(cid:33)
j(cid:62)1 i(cid:53)1 i
defined on the infinite-dimensional jet space J(cid:96)(cid:91)J(cid:96)(R,RPn(cid:50)1) with local coordinates (cid:117), (cid:102)(j)
i
(1(cid:60)i(cid:60)n(cid:50)1, j(cid:62)0). FollowingRef.10,wecanexpressthenecessaryandsufficientconditionfor
(cid:126)4.2(cid:33) to be invariant under the action of SL(n,R) ‘‘infinitesimally’’ as follows:
n(cid:50)1 (cid:93)
(cid:40)
pr v(cid:126)F(cid:33)(cid:53)Dt(cid:104)(cid:117)(cid:102)(cid:53)F, for all v(cid:53) (cid:104)i(cid:126)(cid:102)(cid:33) (cid:93)(cid:102)(cid:80)sl(cid:126)n,R(cid:33). (cid:126)4.7(cid:33)
t i(cid:53)1 i
Notethat,althoughbothprv,D andD areformallydefinedoninfinite-dimensionaljetspaces,in
t
practice they will always act on functions depending on a finite number of the local coordinates.
Finally, using the fact that (cid:104)is a function of (cid:102)only and (cid:126)4.5(cid:33), equation (cid:126)4.7(cid:33) can be further
simplified as follows:
(cid:93)(cid:104)
pr v(cid:126)F(cid:33)(cid:53) F, (cid:126)4.8(cid:33)
(cid:93)(cid:102)
where (cid:93)(cid:104)/(cid:93)(cid:102)is the (n(cid:50)1)(cid:51)(n(cid:50)1) matrix with (i,j) entry (cid:93)(cid:104)/(cid:93)(cid:102). In other words, F is a
i j
relative vector differential invariant of the Lie algebra sl(n,R) given by (cid:126)4.3(cid:33), whose associated
weight is the matrix (cid:93)(cid:104)/(cid:93)(cid:102). Using standard techniques (cid:126)cf. Ref. 6(cid:33), we can give the following
characterization of the general solution of (cid:126)4.8(cid:33):
4.1 Theorem: The most general solution F of equation (4.8) is of the form
F(cid:53)(cid:109)I,
where the (n(cid:50)1)(cid:51)(n(cid:50)1) matrix (cid:109)(cid:53)((cid:109)1(cid:109)2•••(cid:109)n(cid:50)1) is any matrix with non-vanishing deter-
minant and whose columns (cid:109)i are particular solutions of (4.8), and I(cid:53)(Ik)kn(cid:53)(cid:50)11 is an arbitrary
absolute (cid:126)vector(cid:33) differential invariant of the algebra (4.3), i.e., a solution of
pr v(cid:126)I (cid:33)(cid:53)0, for all v(cid:80)sl(cid:126)n,R(cid:33), i(cid:53)1,..., n(cid:50)1.
i
The problem of calculating the most general absolute differential invariant I of a given Lie
algebra of vector fields is a classical one,11–13 whose solution in a modern formulation can be
found in Ref. 6. The general result asserts that their exist n functionally independent differential
fundamentalinvariantsJ0,J1,...,Jn(cid:50)1, suchthatanydifferentialinvariantisafunctionoftheJi’s
and their ‘‘covariant derivatives’’ D kJ , where D(cid:53)(DJ )(cid:50)1D. Since in our case the generators
i 0
(cid:126)4.3(cid:33) are independent of (cid:117), we can take J (cid:53)(cid:117), so that the operator D reduces to D in this case.
0
Therefore, we can state the following Theorem:
4.2Theorem:Themostgeneral((cid:117)-independent)absolutedifferentialinvariantofthesl(n,R)
Lie algebra (4.3) is a function of n(cid:50)1 fundamental differential invariants J ((cid:102),...,(cid:102)(m)) and
i
their total derivatives with respect to (cid:117).
Forn(cid:53)2,itisstraightforwardtocomputethefundamentalsl(2,R) invariantJ . Theresultis
1
the classical Schwartzian derivative S((cid:102)) of (cid:102):
(cid:102)(cid:45) 3 (cid:102)(cid:57)2
J (cid:53) (cid:50) . (cid:126)4.9(cid:33)
1 (cid:102)(cid:56) 2 (cid:102)(cid:56)2
Inthiscase,thematrix(cid:93)(cid:104)/(cid:93)(cid:102)isjustafunction,whichmakesasimplemattertofindaparticular
vector differential invariant of weight (cid:93)(cid:104)/(cid:93)(cid:102). The simplest such invariant is (cid:102)(cid:56)(cid:91)(cid:102)(cid:117); therefore,
Theorem 4.2 implies the following:
J.Math.Phys.,Vol.38,No.11,November1997
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Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket 5727
4.3 Theorem: For n(cid:53)2, the most general evolution equation (4.2) invariant under the pro-
jective action of SL(n,R) is
(cid:102)t(cid:53)(cid:102)(cid:117)I(cid:126)S,DS,...,DlS(cid:33),
where S is the Schwartzian derivative of (cid:102)((cid:149),t), and I is an arbitrary (smooth) function.
Even for the case n(cid:53)3, it is not an easy matter to find the n(cid:50)1 fundamental differential
invariants of (cid:126)4.3(cid:33) and a particular matrix relative differential invariant of weight (cid:93)(cid:104)/(cid:93)(cid:102)from
scratch.Fortunately,however,thedifferentialinvariantsoftheprojectiveactionofSL(n,R) have
been the object of considerable study in classical projective differential geometry.14 From this
viewpoint, the differential invariants of a projective curve describe the properties of the curve
invariant under the group of motions of projective space, or in other words the properties of the
curve independent of the particular system of projective coordinates used to represent it. An
intrinsic description of a projective curve must therefore be done in terms of its sl(n,R) differ-
ential invariants. It is not hard to see (cid:126)as we will explain in the following section(cid:33) that the
coefficients of the operator L defined by a projective curve (cid:102)as in (cid:126)2.3(cid:33) are a set of functionally
independent differential invariants. Obviously, they determine the curve up to a projective trans-
formation; this was already known to Wilczynski14 and it is a generalization of the well known
result in Euclidean geometry that the curvatures of a curve in Euclidean space, expressed as
functionsoftheEuclidean-invariantarclength,uniquelycharacterizethecurveuptoanEuclidean
motion. We shall explain in the following sections how this equivalence between fundamental
differential invariants and coefficients of the operator L is the key to the duality of evolutions.
V. THE EXPLICIT FORMULA FOR THE SL n,R INVARIANT EVOLUTION
(cid:132) (cid:133)
In this section we will describe a complete set of independent differential invariants for the
projective action of SL(n,R), and we will give the explicit expression of the relative invariant
(cid:126)4.8(cid:33) with the required weight, for arbitrary n. The complete set of differential invariants was
alreadyfoundbyRef.14andispreciselygivenbythecoefficientsoftheoperatorL determinedby
the curve (cid:102), as mentioned in the Introduction.
5.1 Theorem: Let (cid:102)be a non-degenerate and right-hand oriented curve on RPn(cid:50)1, and let
dn dn(cid:50)2 d
L(cid:53)d(cid:117)n(cid:49)un(cid:50)2 d(cid:117)n(cid:50)2(cid:49)•••(cid:49)u1 d(cid:117)(cid:49)u0
be the differential operator determined by (cid:102)through the relation (2.3). Then the coefficients u ,
i
0(cid:60)i(cid:60)n(cid:50)2, form a complete set of functionally independent differential invariants for the
SL(n,R) action on RPn(cid:50)1.
Proof: Using the form of equation (cid:126)2.3(cid:33) one can easily see that the coefficient of dk/d(cid:117)k is
given by u (cid:53)(cid:50)(cid:68) , where (cid:68) is the determinant obtained from the Wronskian determinant
k k k
W(f,f(cid:102)1,...,f(cid:102)n(cid:50)1)(cid:53)1 when we substitute the (k(cid:49)1)th row by the nth derivative row
(f(n),(f(cid:102)1)(n),...,(f(cid:102)n(cid:50)1)(n)). Thus, the coefficients of L are functions of the components of the
curve (cid:102)and their derivatives. From this it follows that the coefficients of the operator L are
functionally independent functions. Indeed, if there were a functional relation among these func-
tions one could choose an operator whose coefficients did not satisfy this relation. The projectiv-
ization (cid:102)of the solution curve of such an operator would then have coefficients u ((cid:102)), k
k
(cid:53)0,...,n(cid:50)2, not satisfying the functional relation, and we would get a contradiction.
Thecoefficientsu areeasilyshowntobeinvariants.Indeed,letM(cid:80)SL(n,R) andletM(cid:102)be
i
the image of the curve (cid:102)under the projective action of M. If we lift (cid:102)to a solution curve of L,
say (f,f(cid:102)), and we also lift the curve M(cid:102), we see that the latter is simply M (f,f(cid:102)) (cid:126)the dot
(cid:149)
denoting matrix multiplication(cid:33). Since M (f,f(cid:102)) represents a non-degenerate linear combination
(cid:149)
J.Math.Phys.,Vol.38,No.11,November1997
Copyright ©2001. All Rights Reserved.
5728 Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket
of the solution curve (f,f(cid:102)), both lifted curves are solutions of the same operator and hence
u ((cid:102))(cid:53)u (M(cid:102)) for all k. Q.E.D.
k k
Next, we will find the explicit expression for n independent relative vector invariants, solu-
tionsof(cid:126)4.8(cid:33)forallvectorfieldsv(cid:53)(cid:83)n(cid:50)1(cid:104)((cid:102))(cid:93)/(cid:93)(cid:102)(cid:80)sl(n,R). Thatis,wewanttofindamatrix
i(cid:53)1 i i
(cid:109)(cid:53)(cid:126)(cid:109)1(cid:109)2...(cid:109)n(cid:50)1(cid:33) (cid:126)5.1(cid:33)
eachofwhosecolumns(cid:109)i isasolutionofequation(cid:126)4.8(cid:33),andsuchthatthedeterminantof(cid:109)does
not vanish.
Before going into the details of how one finds this matrix, we need several preliminary
definitions and results:
5.2 Definition: For i ,...,i (cid:62)0 and(cid:85)1(cid:60)k(cid:60)n(cid:50)1, let us deno(cid:85)te
1 k
(cid:102)(cid:126)i1(cid:33) (cid:102)(cid:126)i1(cid:33) ... (cid:102)(cid:126)i1(cid:33)
1 2 k
(cid:102)(cid:126)i2(cid:33) (cid:102)(cid:126)i2(cid:33) ... (cid:102)(cid:126)i2(cid:33)
w (cid:53) 1 2 k
i1i2...ik (cid:65) (cid:65) (cid:65)
(cid:29)
(cid:102)(cid:126)ik(cid:33) (cid:102)(cid:126)ik(cid:33) ... (cid:102)(cid:126)ik(cid:33)
1 2 k
and
W (cid:53)w .
k 12...k
We define the homogeneous variables q by
i i ...i
12 k
w
i i ...i
q (cid:53) 12 k.
i1i2...ik W
k
Finally, for k(cid:53)1,2,...,n the variables qk are defined as follows:
n
qnk(cid:53)q12...kˆ...n,
ˆ
where the notation k means that the index k is to be omitted.
The following statements follow easily from elementary properties of determinants:
5.3 Lemma:
(cid:126)i(cid:33) For any k, i ,...,i (cid:62)0 and 1(cid:60)s(cid:44)r(cid:60)n(cid:50)1 we have the following identities:
1 r
q q (cid:53)q q (cid:49)q q (cid:49)•••(cid:49)q q ,
k i1i2...ir i1 ki2...ir i2 i1ki3...ir ir i1...ir(cid:50)1k (cid:126)5.2(cid:33)
q q (cid:53)q q (cid:49)q q k (cid:49)•••(cid:49)q q .
i1i2...isk i1i2...ir i1...isis(cid:49)1 i1...iskis(cid:49)2...ir i1...isis(cid:49)2 i1...is(cid:49)1 is(cid:49)3...ir i1...isir i1...ir(cid:50)1k
(cid:126)ii(cid:33) If we define q0(cid:53)0 for all m(cid:62)2, then the following identity holds:
m
qk(cid:53)qk qn(cid:50)1(cid:50)qk qn(cid:50)2(cid:50)(cid:126)qk (cid:33)(cid:56)(cid:49)qk(cid:50)1, 1(cid:60)k(cid:44)n.
n n(cid:50)1 n n(cid:50)1 n(cid:50)1 n(cid:50)1 n(cid:50)1
Note that qn(cid:53)1 by definition. The affine algebra is the subalgebra of the sl(n,R) algebra (cid:126)4.3(cid:33)
n
generatedbythevectorfields v andv , 1(cid:60)r,s(cid:60)n(cid:50)1.Thecorrespondinggroupoftransforma-
r rs
tionsistheaffinegroup,i.e.,thesemidirectproductofthetranslationgroupwiththegenerallinear
group in the variables ((cid:102)1,...,(cid:102)n(cid:50)1).
5.4 Lemma: If a (cid:117)-independent function (cid:99): J(cid:96)(cid:91)J(cid:96)(R,RPn(cid:50)1)!R is invariant under the
action of the affine algebra, then (cid:99)necessarily depends only on the affine coordinates qr, r
n
(cid:53)1,...,n(cid:50)1, and their derivatives.
J.Math.Phys.,Vol.38,No.11,November1997
Copyright ©2001. All Rights Reserved.
Gonza´lez-Lo´pez,Heredero,andBeffa:InvariantequationsandtheAGDbracket 5729
Proof: Consider the prolonged action of the affine algebra on the kth jet space Jk
(cid:91)Jk(R,RPn(cid:50)1), whose infinitesimal generators are the kth prolongations (cid:126)i.e., the truncations of
the prolongations (cid:126)4.6(cid:33) at differential order k(cid:33)
k (cid:93)
(cid:40)
pr(cid:126)k(cid:33)v (cid:53)v , pr(cid:126)k(cid:33)v (cid:53) (cid:102)(cid:126)j(cid:33) , 1(cid:60)r,s(cid:44)n(cid:50)1. (cid:126)5.3(cid:33)
r r rs r (cid:93)(cid:102)(cid:126)j(cid:33)
j(cid:53)0 s
For k(cid:60)n(cid:50)1, at a generic point of Jk the n(n(cid:50)1) vector fields (cid:126)5.3(cid:33) span the
(k(cid:49)1)(n(cid:50)1)-dimensionalsubspaceofthetangentspaceofJk whoseelementsarethe‘‘vertical’’
vector fields (cid:126)whose component along (cid:93)/(cid:93)(cid:117)vanishes(cid:33). By Frobenius theorem, this implies that
there are no affine differential invariants of differential order between 1 and n(cid:50)1, and the only
zerothorderinvariantisclearly(cid:126)afunctionof(cid:33)thecoordinate(cid:117).Itisalsoimmediatetocheckthat
fork(cid:62)n(cid:50)1 thevectorfields(cid:126)5.3(cid:33)arelinearlyindependentatagenericpoint.Hencethemaximal
dimensionofthespanofthesevectorfieldsstabilizesfork(cid:53)n(cid:50)1.Olver’sgeneralresults,cf.Ref.
6, imply that the affine algebra has n(cid:50)1 fundamental invariants of order n, and that an arbitrary
differential invariant can be expressed as a function of (cid:117), the fundamental invariants, and their
derivatives with respect to the zeroth order invariant (cid:117). Since the n(cid:50)1 functions qr, 1(cid:60)r(cid:60)n
n
(cid:50)1, all have differential order n, and are clearly functionally independent and invariant under
generalaffinetransformationsofthevariables((cid:102)1,...,(cid:102)n(cid:50)1) bytheirdefinition,theycanbetaken
as the n(cid:50)1 fundamental invariants. Q.E.D.
5.5 Lemma: The variables qs(r(cid:46)s(cid:62)1) can be written in terms of the functionally indepen-
r
dent functions qk(cid:50)1(k(cid:62)2) and their derivatives. We will call the latter functions basic homoge-
k
neous variables.
Proof: We will prove the lemma by induction on r(cid:50)s. For r(cid:50)s(cid:53)1, the lemma holds trivi-
ally. Assume now that the functions qs(cid:56) with r(cid:56)(cid:50)s(cid:56)(cid:44)m can be expressed in terms of the func-
r(cid:56)
tions qk(cid:50)1 and their derivatives. Let qs be such that r(cid:50)s(cid:53)m. From (cid:126)ii(cid:33) of Lemma 5.3 we have
k r
that
qs(cid:53)qs qr(cid:50)1(cid:50)qs qr(cid:50)2(cid:50)(cid:126)qs (cid:33)(cid:56)(cid:49)qs(cid:50)1,
r r(cid:50)1 r r(cid:50)1 r(cid:50)1 r(cid:50)1 r(cid:50)1
so that by the induction hypothesis qs can be written in terms of the functions qk(cid:50)1 and their
r k
derivatives if, and only if, the same is true for qs(cid:50)1. Repeating this argument s(cid:50)2 times, we see
r(cid:50)1
thatqs willbeafunctionoftheqk(cid:50)1 andtheirderivatives,ifandonlyifthisisthecaseforq1 ,
r k m(cid:49)1
with m(cid:53)r(cid:50)s(cid:46)0. Again from (cid:126)ii(cid:33) in Lemma 5.3, we have that
q1 (cid:53)q1qm (cid:50)q1qm(cid:50)1(cid:50)(cid:126)q1(cid:33)(cid:56)
m(cid:49)1 m m(cid:49)1 m m m
which, by the induction hypothesis, proves the lemma. Q.E.D.
Wearenowgoingtomakeanansatzforthematrix(cid:109).Namely,wewilllookamongmatrices
(cid:109)of the form
(cid:109)(cid:53)(cid:70)(cid:126)Id(cid:49)A(cid:33), (cid:126)5.4(cid:33)
where (cid:83) (cid:68)
(cid:102)(cid:56) (cid:102)(cid:57) ••• (cid:102)(cid:126)n(cid:50)1(cid:33)
1 1 1
(cid:102)(cid:56) (cid:102)(cid:57) ••• (cid:102)(cid:126)n(cid:50)1(cid:33)
(cid:70)(cid:53) 2 2 2 , (cid:126)5.5(cid:33)
(cid:65) (cid:65) (cid:65)
(cid:29)
(cid:102)(cid:56) (cid:102)(cid:57) ••• (cid:102)(cid:126)n(cid:50)1(cid:33)
n(cid:50)1 n(cid:50)1 n(cid:50)1
J.Math.Phys.,Vol.38,No.11,November1997
Copyright ©2001. All Rights Reserved.