Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlevé Series

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by  Christos Efthymiopoulos

Explicit Construction of First Integrals with Quasi-monomial
Terms from the Painlev´e Series
Christos Efthymiopoulos1, Tassos Bountis2, and Thanos Manos2
1Research Center for Astronomy and Applied Mathematics, Academy of Athens,
Soranou Efessiou 4, 115 27 Athens, Greece
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0 2Center for Research and Applications of Nonlinear Systems,
0
Department of Mathematics, University of Patras
2

n
a Abstract The Painlev´e and weak Painlev´e conjectures have been used widely to identify new in-
J
tegrable nonlinear dynamical systems. For a system which passes the Painlev´e test, the calculation
1
1 of the integrals relies on a variety of methods which are independent from Painlev´e analysis. The
present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed
]
I as ‘quasi-polynomial’functions, from the information providedsolely by the Painlev´e- Laurent series
S
solutionsofasystemofODEs. Restrictionsonthenumberandformofquasi-monomialtermsappear-
.
n
inginaquasi-polynomialintegralareobtainedbyanapplicationofatheorembyYoshida(1983). The
i
l integralsareobtainedbya properbalancingofthe coefficientsina quasi-polynomialfunctionselected
n
[ as initialansatz forthe integral,sothat alldependence onpowersofthe time τ =t t0 is eliminated.
−
BothrightandleftPainlev´eseriesare usefulin the method. Alternatively,the method canbe usedto
1
v showthenon-existenceofaquasi-polynomialfirstintegral. Examplesfromspecificdynamicalsystems
1
are given.
9
7
1
1 Introduction
.
1
0
The present paper deals with autonomous dynamical systems described by ordinary differen-
8
0 tial equations of the form
:
v
i x˙i = Fi(x1,x2,...,xn), i = 1,2,...,n. (1)
X
r where the functions F are of the form
a i
N
F = a xQij (2)
i ij
jX=1
with x (x ,x ...x ), Q (q ,q ,...,q ), xQij n xqijk, and the exponents q
≡ 1 2 n ij ≡ ij1 ij2 ijn ≡ k=1 k ijk
are assumed to be rational numbers, i.e., the r.h.s. of EQq.(2) is a sum of quasimonomials
(Goriely 1992).
A question of particular interest concerns the existence of first integrals of the system of
ordinary differential equations (1). A first integral is a function I(x ) satisfying the equality
i
dI/dt = I x˙(t) = 0, where x(t) (x (t),...,x (t) is any possible solution of (1). First
x 1 n
∇ · ≡
integrals are important because they allow one to constrain the orbits on manifolds of di-
mensionality lower than n. In particular, a system (1) is called completely integrable if it
admits n 1 independent and single-valued first integrals. Integrable systems exhibit regular
−
dynamics, while the lack of a sufficient number of firstintegrals very often results in complex,
chaotic dynamics.
1
The question of the existence of an algorithmic method which can determine all the first
integrals of (1) is an important open problem in the theory of ordinary differential equations.
Most relevant to this question are the methods of a) direct search or method of undetermined
coefficients (e.g. Hietarinta 1983, 1987) b) normal forms and formal integrals (see Arnold
1985, Haller 1999 and Gorielly (2001) for a review), and c) Lie group symmetry methods (e.g.
Lakshmanan and Senthil Velan (1992a,b), Marcelli and Nucci (2003)).
Even more difficult is the question of an algorithmic method probing the integrability of
Eqs.(1). The most relevant method here is singularity analysis. According to the ‘Painlev´e
conjecture’ (Ablowitz, Ramani and Segur 1980), a system possessing the Painlev´e property
should be integrable. The Painlev´e property means that any global solution of (1) in the
complex time plane should be free of movable critical points other than poles. According
to the ‘weak-Painlev´e conjecture (Ramani et al. 1982, Grammaticos et al. 1984, Abenda et
al. 2001), however, certain types of movable branch points are compatible with integrability.
AlgorithmsprovidingnecessaryconditionsforasystemtobePainlev´earea)theclassicalARS
test (Ablowitz et al. 1980) b) the perturbative Painlev´e - Fuchs test (Fordy and Pickering
1991) which examines the role of negative resonances, and c) the generalized Painlev´e test
of Goriely (1992) which introduces coordinate transformations clarifying the nature of the
singularity structure in the complex time domain. On the other hand, there is no algorithm,
to the present, determining sufficient conditions for a system to be Painlev´e. The most
important obstacle is the search for essential singularities, which are not detectable by any of
the above Painlev´e tests.
The present paper explores the following question: Is it possible to recover the first inte-
grals of a system by the information provided solely by singularity analysis of its differential
equations? Our answer is partially affirmative. We cannot circumvent the difficulty con-
cerning the choice of initial ansatz for the functional form of the integral. The most natural
choice for quasi-polynomial equations (2) is to consider also a quasi-polynomial ansatz for the
integral, with undetermined quasi-monomial coefficients.
This freedom in the initial ansatz notwithstanding, we show in the present paper that
singularity analysis gives indeed the remaining information needed to recover the integrals.
First, as proposed by Roekaerts and Schwarz (1987), a theorem by Yoshida (1983) on
the relation between Kowalevski exponents and weights of weighted-homogeneous integrals,
can be used to impose restrictions on the degree of the quasi-monomial terms in the integral
by analysing the resonances found by the Painlev´e method. Now, Yoshida’s theorem for
weighted-homogeneous integrals is applicable on two conditions: If b τ λi is a balance of the
i −
system (τ = t t is the time around the singularity t ), and I is a weighted homogeneous
0 0
−
integral, the theorem holds if a) I(b ) is finite, and b) I(b ) = 0. These conditions shall be
i i
∇ ∇ 6
refered to as ‘Yoshida’s conditions’. The latter impose a severe restriction in the search for
first integrals, because one cannot specify in advance whether these conditions are satisfied
until the integrals are determined. In conclusion, the theorem of Yoshida has only indicative
power as regards restrictions on the degree of quasimonomial terms in the integral. On the
other hand, an analysis of the integrable Hamiltonian systems of two degrees of freedom
with a polynomial potential given by Hiterinta (1983) shows that they all satisfy Yoshida’s
condition when the balance is taken equal to the ‘principal’ balance, in which all the b are
i
different from zero. Yet, the extent of applicability of this result to other types of systems
is unknown. In fact, we were able to find also a counterexample concerning a Hamiltonian
proposed by Holt (1982).
Assuming a quasi-polynomial functional form of the integral, say Φ(x;c ,c ,...,c ),as
1 2 M
2
above, with undetermined quasi-monomial coeficients c ,i = 1,...M, the question now is
i
whether it is possible to determine the coefficients c by the series derived via singularity
i
analysis. The answer to this question is affirmative. Namely, by the usual Painlev´e tests, the
Painlev´e-type series solutions around movable singularities are first identified
x (τ) = 1 ∞ b τm, i = 1,...n (3)
i τλi m
mX=0
Then, the series (3) is substituted into the quasi-polynomial function Φ(x;c ,c ,...,c ). The
1 2 M
resulting expression is a Puisseux series, i.e., a series in rational powers of τ
Φ(x (τ),...,x (τ);c ,c ,...,c )= τq/p ∞ d (b ;c )τm/p (4)
1 n 1 2 M m i i
mX=0
with q,p,m integers. The coefficients d (b ;c ) are nonlinear functions of the coefficients b
m i i i
(determined by (3), i.e., by singularity analysis), and linear functions of the undetermined
coefficients c . But the function Φ is an integral of the system if it is constant along all the
i
solutions of the system, including (3). As a result, the functions d satisfy the set of linear
m
equations
d (b ;c ) =0, i= 0,1,..., i = q (5)
m i i
6 −
This is an infinite number of homogeneous linear equations with a finite number of unknowns
(the coefficients c ). If M is the number of unknown coefficients, Eq.(5) can be written as
i
A(b ) C = 0 (6)
i
·
where C = (c ,...,c )T and A(b ) is a matrix with M columns and an infinite number of
1 M i
lines. The entries of A depend only on the coefficients b which were previously determined
i
by singularity analysis. In this representation, the first integrals are functions with quasi-
monomial coefficients given by the basis vectors of ker(A) (or linear combinations of them).
In computer algebraic implementations of the method, we work on a sub-matrix A defined
f
by a finitenumberof lines, which is equal or larger thanM. A basis for the subspaceker(A )
f
is determined by the singular value decomposition algorithm. Then, it is checked with direct
differentiation that the resulting expression is an integral. This completes the determination
of all quasi-polynomial first integrals for the given system.
This method was implemented in a number of examples presented below. Following some
preliminary notions exposed in section 2, the results are presented in section 3, along with
variousdetailsandimplicationsintheimplementationofthealgorithm. Section4summarizes
the main conclusions of the present study.
2 Preliminary notions
Following Yoshida (1983), a system of the form (1) is called scale-invariant if the equations
remain invariant under the scale transformation x aλix , t a 1t for some λ . Then, the
i i − i
→ →
system (1) has exact special solutions of the form
b
i
x = (7)
i τλi
3
where b ,i = 1,...n is any of the sets of roots of the system of algebraic equations
i
F (b ,b ,...,b )+λ b = 0
i 1 2 n i i
and the time τ = t t is considered around any movable singularity t in the complex time
0 0
−
plane. Any solution of the form (7) is called a ‘balance’. If the system (1) is scale-invariant
and the functions F are of the form (2), the exponents λ associated with any of the balances
i i
are rational numbers. It should be stressed that the definition above does not require that
all the b be non-zero. Balances of the form x 0/τλi, for some i, are also considered. The
i i
∼
latter remark is essential in order to avoid the confusion which is sometimes made between
‘balances’ and ‘dominant terms’ in the ARS test (see e.g. the discussion between Steeb et
al. (1987) and Ramani et al. (1988)), and the associated difference between ‘Kowalevski
exponents’ and ‘resonances’. In the standard ARS algorithm (Ablowitz et al. 1980) the
solutions (7) arise by the definition of the dominant behaviors, i.e., which is the first step in
the implementation of the algorithm.
The second step in the ARS algorithm is to look for series solutions that we call ’Painlev´e
series’. These are expansions of the form (3) starting with dominant terms of the form (7).
They are Laurent (Taylor) series when the λ ’s are integers (positive integers), otherwise
i
they can be series in rational powers of τ, which are called ‘Puisseux series’. To build up
the series, one first specifies the resonances, i.e. the values of r for which the coefficients of
the terms τr λi in the series are arbitrary. In the case when the coefficients of the balance
−
b are all non-zero, the resonances are equal to the eigenvalues of the Kowalevski matrix
i
K = (∂F /∂x +δ λ ) which are called ‘Kowalevski exponents’. If, however, some of
ij i j ij i |xi=bi
the b s are equal to zero, then the resonances are not equal one to one to the Kowalevski
i
exponents, but some resonances differ from the corresponding Kowalevski exponents by a
quantity equal to the difference between the exponent λ in the balance and the exponent of
i
the first non-zero dominant term in the Laurent-Puisseux series of x (τ), as specified in the
i
first step of the ARS algorithm (Ramani et al. 1988).
At this point, we are not interested in whether the system passes the generalized Painlev´e
(orweakPainlev´e) test. Thismeansthatwedonotrequirethatallthesolutionsofthesystem
(1) can be written locally (around a movable singularity) in the form (3), or that there is at
least one solution of the form (3) which contains n arbitrary constants (including t ). On the
0
other hand, we do check the compatibility conditions to ensure that no logarithms enter in
the series. As regards positive resonances, compatibility is fulfilled automatically for scale-
invariant systems. In summary, we are interested only that the system have special solutions
of the form (3), but no other claim on it being Painlev´e or not is required. Thus, the results
are valid also for partially integrable systems, i.e., systems with a number of first integrals
smaller than n.
Let us now assume that (1) possesses a weighted - homogeneous first integral Φ of weight
M, i.e. an integral function Φ which satisfies the relation:
Φ(aλ1x ,aλ2x ,...,aλnx ) =aMΦ(x ,x ,...,x ) (8)
1 2 n 1 2 n
for some M. Then we have the following
Theorem 1 (Yoshida 1983): If, for a particular balance (7) the following conditions hold:
a) Φ(b ) is finite, and b) Φ(b ) = 0, then M is equal to one of the Kowalevski exponents
i i
∇ ∇ 6
associated with that balance.
4
There has been a number of theorems in the literature linking Kowalevski exponents with
the weights of homogeneous or quasi-homogeneous integrals of a system. Some characteris-
tic papers on this subject are Llibre and Zhang (2002), Tsygvinsev (2001), Goriely (1996)
and Furta (1996). However, the application of Yoshida’s original theorem in the search for
weighted-homogeneous integrals seems to be the most practical. Furthermore, Yoshida’s the-
orem can be reformulated in an interesting way: consider the Painlen´e series starting with
one of the balances (7). It follows that the series can be written as:
b
x = x +R = i +...+A τr λi +R (9)
i iE i τλi i − i
where r is the maximum Kowalevski exponent associated with this balance and A is the
i
corresponding arbitrary coefficient entering in the Painlev´e series (9) for the variable x . The
i
sum x = bi + ...+ A τr λi will be called essential part of the series and the remaining
iE τλi i −
part R remainder of the series. The remainder R starts with terms of degree r λ +1/p
i i i
−
where p is the denominator of λ written as a rational λ = q/p with q,p coprime integers. A
i i
quasi-polynomial integral Φ has the form
n
Φ = c xki (10)
k1,k2,...,kn i
X iY=1
where the exponents k are rational numbers. If the integral Φ is weighted-homogeneous of
i
weight M, we have
n
k λ = M (11)
i i
Xi=1
due to (8). Taking into account the fact that the remainder R starts with terms of degree
i
O(τr λi+1/p),itfollowsthatthecontributionofR inxki isintermsofdegreeO(τ λiki+r+1/p)
− i i −
or higher. This means that if the series (9) is substituted in the integral Φ(x ), the contribu-
i
tion of R in Φ(x ) is of degree O(τ λiki+r+1/p) = O(τ M+r+1/p) or higher. But r M.
i i − −
Thus, the remainder R contributes onPly to terms of positive degree in τ. On the other ≥hand,
i
since Φ is an integral, the time τ as a denominator must be eliminated in Φ(x ). But since
i
there are no negative powers of τ generated in Φ by R , it follows that all the negative powers
i
of τ are already eliminated by substituting the expression x alone into Φ. Hence, we have
iE
the following
Proposition 2: If the conditions of Yoshida’s theorem hold for a weighted-homogeneous
integral of (1) and a particular balance (7), then the expression Φ(x ), where x are the
iE iE
essential parts of Painlev´e series x (τ) initiated with the same balance, does not contain sin-
i
gular terms in τ.
Consider next the case when the functions F in (1) are not homogeneous. By the restrictions
i
imposed by (2), it follows that the functions F can be decomposed in sums of the form
i
F = F(mi0)+F(mi0+1/p)+...+F(mi0+q/p) (12)
i i i i
(j)
where the functions F are homogeneous of degree j, with j,m rational, p,q integer, and
i i0
p is the denominator in the simplest fraction giving m . In this case, if the system (1) has
i0
5
Painlev´e type solutions, the dominant behaviors and associated resonances of these solutions
are determined by the homogeneous term of the highest degree F(mi0+q/p). On the other
i
(j)
hand, the functions F , j < m +q/p must have a special form to ensure that compatibility
i i0
conditions are fulfilled and the series solution is of the Painlev´e type. Finally, as regards
potential first integrals, the assumption that they consist of a sum of quasi-monomial terms
implies that they can also be written as sums of the form (12). The selection of terms in
the quasi-polynomial integral can be determined by Yoshida’s theorem (or proposition 2)
implemented in the scale-invariant systems
x˙ = F(mi0)(x ) (13)
i i i
and
x˙ = F(mi0+q/p)(x ) (14)
i i i
respectively (Nakagawa 2002).
3 Explicit construction of integrals with quasi-monomial terms
3.1 An elementary example
Consider the two-dimensional nonlinear system
x˙ = x , x˙ = x 3x2 (15)
1 2 2 − 1− 1
The only first integral of this system
Φ = x2+x2+2x3 (16)
1 2 1
can be recovered by elementary means. However, we will use this example to illustrate the
steps used by the present method. The corresponding homogeneous system containing the
terms of maximum degree
x˙ = x , x˙ = 3x2 (17)
1 2 2 − 1
has the unique balance x = 2/τ2, x = 4/τ3, with Kowalevski exponents (equal to reso-
1 2
−
nances) 1 and 6. Combatibility conditions are fulfilled for the system (15), which admits
−
the Laurent series solution
2 1 1
x (τ) = − τ2+a τ4+O(τ6)
1 τ2 − 6 − 120 4
4 1
x (τ) = τ +4a τ3+O(τ5) (18)
2 τ3 − 60 4
where a is an arbitrary parameter.
4
Following Yoshida’s theorem, weshalllook foranintegral ofthesystem (15) byrequesting
that this integral be a sum of weighted- homogeneous functions of weight not higher than
M = 6. Sincetheequations(15)arepolynomial,theintegralwillalsobeassumedpolynomial.
According to the definition of the weighted-homogeneous functions (8), the undetermined
integralcontainstermsoftheformxq1xq2 theexponentsofwhicharerestrictedbytherelation
1 2
2q +3q 6. This leaves only six possibilities, namely (q = 1,q = 0), (q = 0,q = 1),
1 2 1 2 1 2
≤
6
(q = 2,q = 0), (q = 1,q = 1), (q = 0,q = 2), (q = 3,q = 0). Thus the integral is
1 2 1 2 1 2 1 2
assumed to have the form
Φ = c x +c x +c x2+c x x +c x2+c x3 (19)
10 1 01 2 20 1 11 1 2 02 2 30 1
Up tonow, the stepsare exactly as proposedby Roekaerts andSchwarz (1987). At this point,
however, we do not proceed by the ‘direct method’; instead, the series (18) is substituted into
(19). Then, terms of equal power in τ are separated and their respective coefficients are set
equal to zero. We must determine at least as many equations as the number of unknown
coefficients c , i.e., six equations. These are:
ij
Order O(1/τ6) : 16c 8c = 0
02 30
−
Order O(1/τ5) : 8c = 0
11
−
Order O(1/τ4) : 4c 2c = 0
20 30
−
2
Order O(1/τ3) : 4c c = 0
01 11
− 3
2 2 4
Order O(1/τ2): 2c + c c c = 0
10 20 02 30
− 3 − 15 − 15
Order O(1/τ) : 0 = 0
These equations can be written in matrix form:
0 0 0 0 16 8 c 0
10
−
 0 0 0 8 0 0  c01   0 
−
0 0 4 0 0 2 c 0
 −  20  =   (20)
 0 4 0 2/3 0 0  c   0 
  11   
 −    
 2 0 2/3 0 2/15 4/15  c02   0 
 − − −    
 0 0 0 0 0 0  c   0 
  30   
or simply
A C = 0 (21)
f
·
where C is a six-dimensional vector and A is a 6 6 matrix with constant entries.
f
×
The singular value decomposition of A yields a one-dimensional null space:
f
ker(A )= λ(0,0,1,0,1,2,0) (22)
f
The basis vector (0,0,1,0,1,2) corresponds to the first integral Φ = x2+x2+2x3.
1 2 1
A few remarks are here in order:
a) the last line of A has only zero entries, since it corresponds to the identity 0 = 0 for
f
the O(1/τ) terms. This is not a problem, because lines with zero elements are allowed by the
singular value decomposition algorithm which determines the subspace ker(A ).
f
b) The entries of A are constant numbers which depend only on the coefficients of the
f
Laurent series (18). This is the crucial remark; it implies that the information on the first
integral is contained in the Painlev´e series built by singularity analysis.
c) The arbitrary parameter a in (18) does not appear in A . This phenomenon is not
4 f
generic. In general, all the arbitrary parameters of the Painlev´e series appear in A . In the
f
computer implementation of the algorithm, we proceed by giving fixed values to the arbitrary
parameters. Although the choice of values affects the convergence of the Painlev´e series, it
does not influence the present algorithm which is based only in the formal properties of the
series.
7
3.2 Further examples
Of particular interest in nonlinear dynamics are autonomous Hamiltonian systems of two
degrees of freedom of the form
1
H (p2 +p2)+V(x,y) (23)
≡ 2 x y
whereV(x,y)isoftheform(2). Theeasiestexamplesaresystemswithapolynomialpotential
(e.g. Hietarinta 1983, 1987). For example:
1
H (p2 +p2 +x2+y2) x2y 2y3 (24)
≡ 2 x y − −
This system passes the Painlev´e test and it is integrable (Bountis et al., 1982). Keeping
only the highest order terms of the potential ( x2y 2y3) yields the principal balance
− −
x = 6i/τ2,y = 3/τ2 with resonances r = 3, 1,6,8, which are equal to the correspond-
− −
ing Kowalevski exponents. The Painlev´e series generated by the Hamiltonian (24) and the
above principal balance satisfies the compatibility conditions of the ARS test. Assuming now
a polynomial first integral Φ of (24), only the monomial terms of weight less or equal to 8
will be included to it. Since the leading terms of the momenta are p p O(1/τ3), the
x y
∼ ∼
selected monomial terms are:
x4, x3y, x2y2, xy3, y4, p2x, p p x, p2x, p2y, p p y, p2y (weight 8)
x x y y x x y y
p x2, p xy, p y2, p x2, p xy, p y2, (weight 7)
x x x y y y
x3, x2y, xy2, y3, p2, p p , p2,x3, x2y, xy2 ,y3 (weight 6)
x x y y
p x, p y, p x, p y, (weight 5)
x x y y
x2, xy, y2, p , p , x, y (weights 4,3,2)
x y
Itshouldbestressedthatthereareseveralotherrestrictionsthatreducethenumberofeligible
terms. For example, the integral Φ is either even or odd in the momenta (Nakagawa and
Yoshida2001). Furthermore, thelinear termscan beomitted byappropriatetransformations.
However, in the practical implementation of the algorithm these restrictions only introduce
a complication, because, if an integral Φ exists, the singular value decomposition algorithm
selects the basis for the corresponding null space of the matrix A without needing any extra
f
information on restrictions which are specific for the system under study, e.g. hamiltonian or
other.
Following the selection of monomial terms, the algorithm proceeds in building the ho-
mogeneous system (6) as well as the finite restriction A of the matrix A. In this case, the
f
dimension of A should be M 38, with M 38, since there are 38 unknown coefficients
f
× ≥
of the above monomial terms. At this point, it does not matter which balance and Laurent-
generated series are used to build the matrix A . In the above example, the principal balance
f
leads to a ’special solution’, since there are only two positive resonances (r = 6,8) mean-
ing that there are three arbitrary parameters in total entering in the series. On the other
hand, the general Laurent series solution with four arbitrary constants is given by a different
8
balance, namely x= 0/τ2,y = 1/τ2 so that x starts with dominant terms of O(1/τ), i.e.
A
x(τ) = +a τ +Bτ2+a τ3+a τ4a τ5+O(τ6)
1 3 4 5
τ
1
y(τ) = +b +b τ2+b τ3+Cτ4+O(τ5) (25)
τ2 0 2 3
A
p (τ) = +a +2Bτ +3a τ2+4a τ35a τ4+O(τ5)
x −τ2 1 3 4 5
2
y(τ) = +2b τ +3b τ2+4Cτ3+O(τ4)
−τ3 2 3
whereA,B,C together with t = t τ are arbitrary, and b = (1 A2)/12, a = A(1 2b )/2,
0 0 1 0
− − −
b = (b 6b2 2Aa )/10,a = (2b a a +2Ab )/4,b = AB/3,a = (2Ab B 2Bb )/10,
2 0− 0− 1 3 0 1− 1 2 3 − 4 3− − 0
a = (2AC +2a b +2b a a )/18. In this solution, the resonances r = 1,0,3,6 do not
5 1 2 0 3 3
− −
coincide one by one to the Kowalevski exponents ( 1,1,4,6) deduced by the Kowalevski
−
matrix associated with the above balance. Nevertheless, the general solution (25), as well as
anyothersolutionworkequallywellindeterminingthematrixA . Inourcase, byperforming
f
thesingularvaluedecompositionofA ,thesubspaceker(A )yieldstwoindependentintegrals
f f
in involution, with coefficients (up to the computer precision)
Φ = 0.211944988455(y2 +p2 4y3)+0.049612730813(x2 +p2)
1 y − x
1
+0.216443010189(xp p p2y x2y2 x4) (26)
x y − x − − 4
0.207446966722x2y
−
Φ = 0.035960121414(y2 +p2 4y3)+0.342416478352(x2 +p2)
2 y − x
1
0.408608475918(xp p p2y x2y2 x4) (27)
− x y − x − − 4
0.480528718746x2y
−
in terms of which we can express the Hamiltonian
H = 2.1645645936295Φ +1.146586289822Φ (28)
1 2
and find a second integral orthogonal to the hamiltonian
I = 2.1645645936295Φ +1.146586289822Φ (29)
2 2 1
−
The integral I is a linear combination of the hamiltonian and of the integral I given by
2 b
Bountis et al. (1982)
12
I = 0.1651752123636227(2H I ) (30)
2 b
− 7
Let us note that an alternative way to obtain the matrix A is by considering only the
f
essential parts of the Laurent series (25), for as many different sets of values of the arbitrary
parameters as requested in order to have a complete determination of the M 38 elements of
×
A , with M 38. Thisapproach is preferablein computer implementations of the algorithm,
f
≥
becauseonedoesnotneedtocalculate thetermsoftheLaurentseries(25) beyondthehighest
positive resonance. Furthermore, Proposition 2, instead of Yoshida’s theorem, can be used to
select the quasiminomial basis set.
9
A final remark concerns the applicability of Yoshida’s condition Φ(b ) = 0. We checked
i
∇ 6
whether this condition is satisfied in Hientarinta’s table (1983) of integrable Hamiltonian
systems of two degrees of freedom with a polynomial homogeneous potential. There are six
non-trivial cases:
(a) V(x,y) = 2x3+xy2
with integral Φ = yp p xp2 +x2x + 1y4
x y − y 2 4
16
(b) V(x,y) = x3+xy2
3
with integral Φ = p4 +4xy2p2 4y3p p 4x2y4 2y6
y y − 3 x y − 3 − 9
√3
(c) V(x,y) = 2x3+xy2+i y3
9
with integral Φ = p4 + 2 ip p3 + 2 iy3p2 (2y3 +2i√3xy2)p p +(4i√3x2y +2i√3y3 +
y √3 x y √3 x − x y
4xy2)p2 + 4 ix3y3+ 2 ixy5 x2y4 5y6
y √3 √3 − − 9
4 1
(d) V(x,y) = x4+x2y2+ y4
3 12
with integral Φ = yp p xp2 +(12x3 +xy2)y2
x y − y 3
4 1
(e) V(x,y) = x4+x2y2+ y4
3 6
with integral Φ = p4 + 2y4p2 8xy3p p +(4x2y2+ 2y4)p2 + 1(y8+4x2y6+4x4y4) and
y 3 x− 3 x y 3 y 9
3
(f) V(x,y) = x5+x3y2+ xy4
16
with integral Φ = yp p xp2 + 1x4y2+ 3x2y4+ 1 y6.
x y − y 2 8 32
In all these cases, the principal balance, with all b different from zero, satisfies Yoshida’s
i
condition. Note that case (f) the principal balance leads to ‘weak-Painlev´e’ solutions, but
the associated integral Φ is easily recoverable by the present algorithm. In fact, the role of
Yoshida’s condition is toexcludefromTheorem 1integrals which arecompositefunctions of a
simplerintegral. Forexample,considertheintegralI = Φ2,whereΦisthepolynomialintegral
in any of the above six Hamiltonian systems. In all of them Φ is weighted-homogeneous for
some weight M. Thus I is weighted-homogeneous of weight 2M. Substituting the special
solution (7) in the integral Φ yields Φ(b /τλi) = τ MΦ(b ). Since Φ is an integral, it should
i − i
be time-independent, thus Φ(b ) = 0. Similarily, I(b ) = 0. However, while Φ(b ) = 0, we
i i i
∇ 6
have I = 2Φ Φ, thus I(b ) = 0. Thus, while Φ satisfies the Yoshida’s condition, I does
i
∇ ∇ ∇
not. This ensures that while M is necessarily a Kowalevski exponent, the multiples of it, e.g.
2M are not necessarily Kowalevski exponents. Viewed under this context, it appears that the
condition Φ(b ) = 0 makes a ‘natural’ choice of the simplest integral among an infinity of
i
∇ 6
possible integrals which are composite functions of Φ.
However, there are interesting counterexamples which challenge this point of view. One
example is the homogeneous limit of the Holt (1982) Hamiltonian
1 3
H = (p2 +p2) x4/3 y2x 2/3 (31)
2 x y − 4 − −
10