On the Drach superintegrable systems

16 Pages·0.19 MB·English

by  A. V. Tsiganov

On the Drach superintegrable systems.
A.V. Tsiganov
Department of Mathematical and Computational Physics, Institute of Physics,
St.Petersburg University, 198 904, St.Petersburg, Russia,
e-mail: [email protected]
0
0
0 Cubicinvariantsfortwo-dimensionaldegenerateHamiltoniansystemsareconsideredbyusingvariablesofsepa-
2 rationoftheassociated St¨ackelproblemswithquadraticintegralsofmotion. ForthesuperintegrableSt¨ackelsystems

n thecubic invariant is shown to admit newalgebro-geometric representation that is far more elementary than the all
a the known representations in physical variables. A complete list of all known systems on the plane which admit a
J cubic invariant is discussed.

5
2

] 1 Introduction
I
S
n. In 1935 Jules Drach applied direct method for search of the integrable Hamiltonian systems of two degrees
i of freedom, which admit a cubic second integral [2]. Recall, the direct approach leads to a complicated set
l
n ofnonlinearequations,whosenonlinearityhasnoa priory restriction. We canattemptto solvethesesecond
[ invariant differential equations using various simplifying assumptions. The Drach ansatz for the Hamilton

1 function H and for the second cubic invariant K
v
H = p p +U(x,y),
3 x y
5 (1.1)
0
∂H ∂H
1 K = 6w(x,y) p p P(p ,p ,x,y)
y x x y
0 ∂x − ∂y −
(cid:18) (cid:19)
0
yields ten new integrable systems. Recall, for the the fixed polynomial P(p ,p ,x,y) the potential U(x,y)
0 x y
and function w(x,y) be solution of some differential equations [2].
/
n For the other natural Hamilton functions
i
nl H =p2x+p2y+V(x,y) (1.2)
:
v the similar approach has been used by Fokas and Lagerstrom [3], by Holt [5] and by Thompson [9]. Note,
i that the some Drach results have been rediscoveredin these papers.
X
The most complete classifications of known results was later brought together by Hietarinta in 1987
r
a [4]. Recently [6], two-dimensional hamiltonian systems with the cubic integrals were investigated using the
Jacobi change of the time. In [6] a complete list of all known systems was extended in comparison with [4].
TheaimofthisnoteistostudytheDrachsystemsandsomeotherdegeneratesystemsontheplanewith
the cubic in momenta integrals of motion. We prove that eight Drach hamiltonians belong to the Stackel
family of integrals [8] and, moreover,seven of them are degenerate systems.
Recall,thesystemiscalledsuperintegrableordegenerateiftheHamiltonfunctionH isintheinvolution
with two integrals of motion I and K, such that
H,I = H,K =0, I,K =J(H,I,K). (1.3)
{ } { } { }
Initial integrals and the constant of motion J(H,I,K) are generators of the polynomial associative algebra
[1, 7], whose defining relations are polynomials of certain order in generators.
Belowweshallconsidertwo-dimensionalsystemswithtwoquadraticintegralsofmotionI =H, I =I
1 2
andonequbic integralK. So,by the Bernard-Darbouxtheorem[17]the systemwithintegralsI , I belong
1 2
to the St¨ackel family of integrable systems [8]. Therefore, let us begin with remaining of some necessary
results about the St¨ackel systems [8].
1
2 The St¨ackel systems
ThesystemsassociatedwiththenameofSt¨ackel[8]areholonomicsystemsonthephasespaceR2n equipped
with the canonical variables p ,q n . The nondegenerate n n St¨ackel matrix S, with entries s de-
{ j j}j=1 × kj
pending only on q
j
∂s
detS=0, kj =0, j =m
6 ∂q 6
m
defines n functionally independent integrals of motion
n skj
I = c p2+U , c = , (2.1)
k jk j j jk detS
j=1
X (cid:0) (cid:1)
which are quadratic in momenta. Here C = [c ] denotes inverse matrix to S and skj be cofactor of the
jk
element s . The common level surface of these integrals
kj
M = z R2n :I (z)=α , k =1,...,n
α k k
∈
is diffeomorphic to the n-dimensional r(cid:8)eal torus and one immediately gets(cid:9)
2 n
∂
p2 = S = α s (q ) U (q ). (2.2)
j ∂q i ij j − j j
(cid:18) j(cid:19) i=1
X
Here (q ...,q )isareducedactionfunction[8]. FortherationalentriesofSandrationalpotentialsU (q )
1 n j j
S
one gets
k (q e )
p2 = i=1 j − i , (2.3)
j ϕ2(q )
Q j j
wheree areconstantsofmotionandfunctionsϕ (q )dependoncoordinateq andnumericalconstants[11].
i j j j
The Riemann surfaces (2.3) are isomorphic to the canonical hyperelliptic curves
k
: µ2 = (λ e ), µ =ϕ(q )p , (2.4)
Cj j − i j j j
i=1
Y
wheretheseniordegreek ofpolynomialfixesthegenusg =[(k 1)/2]ofthealgebraiccurve . Considered
j j
together, these curves determine an n-dimensional Lagrangian−submanifold in R2n C
(n) : (p ,q ) (p ,q ) (p ,q ).
1 1 1 2 2 2 n n n
C C ×C ×···×C
The Abel transformation linearizes equations of motion on (n) by using first kind abelian differentials on
C
the correspondingalgebraiccurves[12]. Thebasisoffirstkindabeliandifferentialsisuniquely relatedtothe
St¨ackel matrix S [11, 12].
Now let us turn to the superintegrable or degenerate systems in the classical mechanics. One of the
main examples of the two-dimensional superintegrable systems is the isotropic harmonic oscillator, which
has many common properties with the Drach degenerate systems. Recall, for the oscillator the Hamilton
function and the second integral of motion look like
H =p2+p2+q2+q2, I =p2+q2 p2 q2.
1 2 1 2 1 1 − 2− 2
Obviously, the angular momentum
1 dp dp
2 1
K =q p p q = p p . (2.5)
1 2 1 2 1 2
− 2 dt − dt
(cid:18) (cid:19)
is the third integral of motion. Two pairs of quadratic integrals I = H, I = I and I˜ = H, I˜ = K2 are
1 2 1 2
associated with the following St¨ackel matrices
1 1 1 0
S= , and S˜ = , r2 =x2+y2
1 1 r−2 1
(cid:18) − (cid:19) (cid:18) (cid:19)
2
respectively. So, the corresponding equations of motion may be separated in the different curvilinear coor-
dinate systems.
For this degenerate St¨ackel system and for all other known superintegrable St¨ackel systems with quad-
ratic integrals of motion the number degree of freedom n>g is always more then sum of genuses g of the
j
corresponding algebraic curves. In this case the number of independent first kind abelian differentials be
insufficient for the inversion of the Abel-Jacobi map on (n).
C
To construct inversion of this map for the degenerate systems one has to complete a given basis of the
differentials to the set of n differentials. We have some freedom in a choice of complimentary differentials
and,therefore,wecanassociatethe differentSt¨ackelmatricestoone givenHamiltonfunction[12]. By using
first kind abelian differentials one gets superintegrable St¨ackel system with quadratic integrals only. Of
course,wecantrytoaddthesecondandthirdkindabeliandifferentials,butwedonotknowsuchexamples.
Below we prove that for all the known superintegrable systems with a qubic integral K the number
degree of freedom n = 2 is more then sum g = g +g = 0 of genuses g = 0 of the associated Riemann
1 2 j
surfaces (2.4) too. The corresponding dynamics is splitting on two spheres
: µ2 =α λ2+β λ+γ , g =0, (2.6)
1,2 1,2 1,2 1,2 1,2
C
where α ,β and γ be the constants of motion.
j j j
Invariablesµ (2.4)theadditionalcubic integralofmotionforallthe degenerateDrachsystemslooks
1,2
like
detS dµ dµ
2 1
K = µ µ . (2.7)
1 2
s s dt − dt
21 22 (cid:18) (cid:19)
This generalized ”angular momentum” gives rise to the first order integrals (2.5) or the third order poly-
nomials in momenta depending on the St¨ackel matrices S and potentials U . In our case it will be qubic
j
integral, which coincides with the initial Drach integral up to numerical factor.
To consider nonlinear algebra of integrals of motion for the Drach systems we shall introduce new
generators N,a,a† instead of the two quadratic integrals I = H, I = I, one qubic integrals K and the
1 2
{ }
constant of motion J (1.3). Similar to oscillator these new generators have the following properties
N,a =a, N,a† = a†,
{ } { } −
(2.8)
a,a† =Φ(I ,I ), aa† =Ψ(I ,I ).
1 2 1 2
{ }
Here generator N(I ,I ), functions Ψ(I ,I ) and Φ(I ,I ) depend on the quadratic St¨ackel integrals only.
1 2 1 2 1 2
Two other generators a and a† be functions on the all three constants of motion I ,I ,K, such that
1 2
a+a†
K =ρ(I ,I )(a a†), J(I ,I ,K)= .
1 2 1 2
− 2
The relations (2.8) remind the deformed oscillator algebra, which is widely used for the superintegrable
systemswithquadraticintegralsofmotion[1,7]. However,insteadoftheusualquadraticalgebraofintegrals
we shall get more complicated algebras of integrals.
3 The Drach systems
Let us reproduce corrected Drach results in his notations
α
(a) U = +βxr1yr2 +γxr2yr1, where r2+3r +3=0, (3.1)
xy j j
P = (xp p y)3, w=x2y2/2,
x y
−
α β γ(y+µx)
(b) U = + + , (3.2)
√xy (y µx)2 √xy(y µx)2
− −
P = 3(xp p y)2(p +µp ), w =xy(y µx),
x y x y
− −
3
β γ
(c) U = αxy+ + , (3.3)
(y ax)2 (y+ax)2
−
P = 3(xp p y)(p2 a2p2), w =(y2 a2x2)/2,
x− y x− y −
α β γx
(d) U = + + , (3.4)
y(x a) y(x+a) √x2 a2
− −
P = 3pp (xp p py)2 a2p2 , w= y(x2 a2),
y x− y − x − −
(cid:2) (cid:3)
α β γ
(e) U = + + , (3.5)
√xy √x √y
P = 3p p (xp p y), w = 2xy,
y x x y
− −
2x2+c γx
(f) U = αxy+βy + , (3.6)
√x2+c √x2+c
P = 3p2(xp yp ), w =(x2+c)/2,
y x− y
α
(g) U = +β(y 3mx)+γ(y mx)(y 9mx), (3.7)
(y+3mx)2 − − −
P = (p +3mp )2(p 3mp ), w = m(y+3mx),
x y x y
− −
mx 14mxy m2x2
(h) U = (y+ )−2/3 α+β(y mx/3)+γ(y2 + ) , (3.8)
3 − − 3 9
(cid:20) (cid:21)
mp 10mp p m2p2 mx
P = (p y) p2 + x y + y , w = m(y+ ),
x− 3 x 3 9 ! − 3
(k) U = αy−1/2+βxy−1/2+γx, (3.9)
P = 3p2p , w= y,
x y −
ρx
(l) U = α y +βx−1/2+γx−1/2(y ρx), (3.10)
− 3 −
(cid:16) (cid:17)
P = 3p p2+ρp3, w=x.
x y y
Here α, β, γ, µ, ρ, a, c, and m be arbitrary parameters. In compare with [2] we corrected function w in
the case (g) (3.7) and revisedpotential U in the case (k) (3.9). Namely this correctedHamiltonian is in the
involution with the initial Drach cubic integral K (1.1).
With an exception of three cases (a) (3.1), (h) (3.8) and (l) (3.10), other Drach systems are degenerate
or superintegrable St¨ackel systems. The separation variables associated with the pair of quadratic integrals
I =H,I arethe St¨ackelvariables. Equationsofmotionmaybeintegratedinquadratures[11],butthese
1 2
{ }
quadratures depend on the value of quadratic integral I . Thus, instead of the solution of initial Drach
2
problem related to integrals H,K we shall solve the associated problem with quadratic integrals H,I .
2
{ } { }
In the case (h) (3.8) we also have the St¨ackel systems [13]. Only in this case (h) (3.8) dynamics is
splitting on two tori and the number degrees of freedomis equal to the sum of genuses g =n=2, suchthat
the corresponding system is non-degenerate.
In the case (l) (3.10) the Hamilton function coincides with the hamiltonian of the previous St¨ackel
system (3.9) at ρ=0. Here we shall not consider this generalized St¨ackel system at ρ=0.
6
Below we shall consider the Drach integrals (1.1) up to linear transformations of the coordinates and a
rescaling of these integrals. It allows us to remove some parameters in the Hamilton functions without loss
4
of generality. To associatethe degenerateDrachhamiltonians with the St¨ackelmatrices S we can join these
systems into the four pairs of the systems with a common matrices S.
3.1 Case (a)
In our previous paper [14], the first Drach system (3.1) has been related to the three-particle periodic Toda
latticeinthecenter-of-massframe. Namely,aftercanonicalchangeofthetimet=q andtheHamiltonian
n+1
H =p at the extended phase space
n+1
dt=(xy)−1 dt, H =xy (H +δ),
· ·
and after further canonical transformation of other variables
e e
x=eq1+2iq2 , px =(p1 ip2)e−q1+2iq2 , y =eq1−2iq2 , py =(p1+ip2)e−q1−2iq2 ,
−
the Hamilton function (3.1) becomes
1 √3 1 √3
q q q + q
H =p2+p2+βe−2 1− 2 2 +γe−2 1 2 2 +δeq1 +α.
1 2
It is the Hamiltonian of the tree-particle periodic Toda lattice in the center-of-mass frame. The separation
e
variables survive at the change of the time. Thus, for the first Drach system we can separate variables and
integrate equations of motions in quadratures repeating the calculations for the Toda chain [15].
Later in [9] Thompson considered this system too. In fact, after point transformation
e−iφ eiφ
x=reiφ, p = p ip r−1 , y =re−iφ, p = p +ip r−1
x r φ y r φ
2 − 2
(cid:0) (cid:1) (cid:0) (cid:1)
the Drach hamiltonian H (1.1) looks like
p2
H =p2+ φ +U(r,φ),
r r2
up to numerical factor. Namely this Hamilton function was studied in [9] and [6]. The special substitution
of the potential U(r,φ) into the Drach equations leads to the following equation
f(φ)+f′′(φ)
U(r,φ)= , f′′′f′′ 2f′′f′ 3f′f =0,
r3 ⇒ − −
introduced in [6]. Of course, the same equation follows from the functional equation on the Toda potential
[4].
3.2 Cases (b) and (e)
Put µ=1 in (3.2). Let us introduce the St¨ackel matrix
q2 q2
1 2
S = , (3.11)
be
 
1 1
 
and take the following potentials
β 2γ β+2γ
(b) U =2α − , U = 2α ,
1 − q2 2 − − q2
1 2
(e) U =2α+2(β+γ)q , U = 2α 2(β γ)q .
1 1 2 2
− − −
ThecorrespondingHamiltonfunctionsI (2.1)coincidewiththeHamiltonfunctionsH fortheDrachsystems
1
(3.2) and (3.5), after the following canonical point transformation
(q q )2 p p (q +q )2 p +p
1 2 1 2 1 2 1 2
x= − , p = − , y = , p = .
x y
4 q q 4 q +q
1 2 1 2
−
5
The second integrals of motion I (2.1) are second order polynomials in momenta. The third independent
2
integrals of motion K are defined by (2.7), where
(b) µ =q p , µ =q p , (e) µ =p , µ =p .
1 1 1 2 2 2 1 1 2 2
Fromthe abovedefinitions we can introduce generatorsof the nonlinearalgebraof integrals(2.8)andverify
properties of this algebra
I
(b) N = 2 , a=J +4√HK, a† =J 4√HK,
4√H −
aa† =16 4H(β+2γ) (2α+I )2 4H(β 2γ) (2α I )2 ,
2 2
− − − −
(cid:0) (cid:1)(cid:0) (cid:1)
a,a† = 256√H (I (I 2α)(I +2α) 4H(βI 4αγ)) ,
2 2 2 2
{ } − − − −
and
I
(e) N = 2 , a=J +2√HK, a† =J 2√HK,
2√H −
aa† = 16 H(2α+I ) (β γ)2 H(2α I )+(β+γ)2 ,
2 2
− − − −
(cid:0) (cid:1)(cid:0) (cid:1)
a,a† = 64H3/2(I H β2 γ2).
2
{ } − − −
3.3 Cases (c) and (g)
Put a=1 in (3.3) and m=1/3 in (3.7). Let us introduce the St¨ackel matrix
1 1
S = 2 −2 , (3.12)
cg  
1 1
 
 
and take the following potentials
αq2 γ αq β
(c) U = 1 + , U = 2 ,
1 4 q2 2 4 − q2
1 2
(3.13)
γq2 α 4γq2
(g) U = 1 + , U = 2 βq .
1 − 3 q2 2 − 3 − 2
1
ThecorrespondingHamiltonfunctionsI (2.1)coincidewiththeHamiltonfunctionsH (3.3)and(3.7),after
1
the following canonical point transformation
q q q +q
1 2 1 2
x= − , p =p p , y = , p =p +p .
x 1 2 y 1 2
2 − 2
ThesecondintegralsofmotionI (2.1)arethesecondorderpolynomialsinmomenta. Thethirdindependent
2
integrals K are defined by (2.7), where
(c) µ =q p µ =q p , (g) µ =q p , µ =p .
1 1 1 2 2 2 1 1 1 2 2
Generators and defining relations of the nonlinear algebra of integrals (2.8) look like
I
(c) N = 2 , a=J +2√ αK, a† =J 2√ αK,
2√ α − − −
−
aa† = H2+4HI +4I2 4αγ H2 4HI +4I2+4αβ ,
2 2 − − 2 2
(cid:0) (cid:1)(cid:0) (cid:1)
a,a† = 32√a (I (H 2I )(H +2I ) αβ(H +2I ) αγ(H 2I )) ,
2 2 2 2 2
{ } − − − − −
6
and
I 3 γ γ
(g) N = 2 , a=J +4 K, a† =J 4 K,
4 γ 3 − 3
r r r
aa† =1/9 8γH 16γI +3β2 3H2+12HI +12I2+16αγ ,
− 2 2 2
(cid:0) (cid:1)(cid:0) (cid:1)
γ 3/2 (2I +H)(4γH 24γI +3b2) 16αγ
a,a† =64 2 − 2 .
{ } 3 4γ − 3
(cid:16) (cid:17) (cid:18) (cid:19)
3.4 Cases (d) and (f)
Put a=1 in (3.4) and c=1 in (3.6). Let us introduce two the St¨ackel matrices
1 1
1 1
q q
1 2
Sd == 1 1  , Sf =  (3.14)
1 1
 q12 q22   q12 q22 
and takes the following potentials
2√2(α+β) 2√2(α β)
(d) U =2γ+ , U = 2γ − ,
1 2
q − − q
1 2
γ (α+2β) γ (α 2β)
(f) U = + , U = + − .
1 2
2q 4 2q 4
1 2
The corresponding Hamilton functions I (2.1) coincide with the Hamilton functions H (3.4) and (3.6) up
1
to numerical factor, after the following explicit canonical transformations
q2+q2 (p q p q )q q p q +p q
(d) x= 1 2, p = 1 1− 2 2 1 2, y =q q , p = 1 1 2 2,
2q q x q2 q2 1 2 y 2q q
1 2 1 − 2 1 2
(f) x= q1−q2, p = 2(p1q1−p2q2)√q1q2, y =√q q , p = p1q1+p2q2.
x 1 2 y
2√q1q2 q1+q2 √q1q2
The second integrals of motion I (2.1) are the quadratic polynomials in momenta. The third independent
2
integrals K are defined by (2.7), where for the both systems one gets
µ =q p , µ =q p .
1 1 1 2 2 2
Generators of the nonlinear algebra of integrals (2.8) are given by
N = I , a=J +2 I K, a† =J 2 I K,
2 2 2
−
p p p
which have the following properties
(d) aa† =16 I (2γ H)+2(α+β)2 I (2γ H) 2(α β)2 ,
2 2
− − − −
(cid:0) (cid:1)(cid:0) (cid:1)
a,a† =64 I I (2γ H)(2γ+H)+2(α2+β2)H +8αβγ ,
2 2
{ } −
p (cid:0) (cid:1)
1
(f) aa† = 4I (α+2β)+(γ 2H)2 4I (α 2β)+(γ+2H)2 ,
2 2
16 − −
(cid:0) (cid:1)(cid:0) (cid:1)
αγ2
a,a† = 4 I (2β α)(2β+α)I +αH2+2βγH+ .
2 2
{ } − − 4
(cid:18) (cid:19)
p
7
3.5 Cases (h) and (k)
Put m=3 in (3.8). Let us introduce two the St¨ackel matrices
q q q q
1 2 1 2
S == , S = − (3.15)
h k
   
1 1 1 1
−
   
and take the following potentials
β2 β2
(h) U = αγ− 16 2γq13, U = αγ− 16 2γq23
1 2
4γ − 9 4γ − 9
γq2 γq2
(k) U =α+βq + 1, U = α+βq + 2.
1 1 2 2
2 − 2
The corresponding Hamilton functions I (2.1) coincide with the Hamilton functions H (3.8) and (3.9) up
1
to numerical factor, after the following explicit canonical transformations
p p (3q +3q )3/2 β p +p
2 1 1 2 1 2
(h) x= − + + , px =3 +√γ(q1 q2),
4√γ 54 16γ √3q1+3q2 −
p p (3q +3q )3/2 β p +p
2 1 1 2 1 2
y = − + , py =3 √γ(q1 q2),
− 4√γ 54 − 16γ √3q1+3q2 − −
and
q q (q +q )2 p +p
1 2 1 2 1 2
(k) x= − , p =p p , y = , p = .
x 1 2 y
2 − 4 q +q
1 2
Note,inthecase(h)(3.8)weusednon-pointcanonicaltransformationincontrastwithotherDrachsystems.
In the last case (k) (3.9) integral of motion I (2.1) is the second order polynomial in momenta. The
2
third independent integral K is defined by (2.7), where
µ =p and µ =p .
1 1 2 2
Generators and defining relations of nonlinear algebra of integrals (2.8) look like
I
(k) N = 2 , a=J + 2γK, a† =J 2γK,
√ 2γ − − −
−
p p
aa† = 2γ(I +α)+(H +β)2 2γ(I α)+(H β)2 ,
2 2
− −
(cid:0) (cid:1)(cid:0) (cid:1)
a,a† = 4γ√ 2c H2+2γI +β2 .
2
{ } − −
In the case (h) (3.8) the second integral of m(cid:0)otion I (2.1) is t(cid:1)he second order polynomial in momenta
2
p ,p . However, after the non-point transformation of variables this integral I becomes the qubic in
1 2 2
{ }
momenta p ,p Drach integral K (3.8). The corresponding dynamics is splitting on two tori and the
x y
{ }
third order polynomial (2.7) does not commute with the Hamilton function. Later this system has been
rediscoveredby Holt [5].
4 Other degenerate systems on the plane with a qubic integral of
motion
In this section we consider the St¨ackel systems on the plane with a qubic integral of motion defined by the
following Hamilton function
1
H = p2 +p2 +V(x,y).
2 x y
(cid:16) (cid:17)
8
As above, the corresponding qubic integral will be written at the Drach form (1.1).
On the plane we know four orthogonal systems of coordinates: elliptic, parabolic, polar and cartesian.
Thus, we reproduce all the known results [4, 6] in correspondence with the type of the associated St¨ackel
matrix [11, 12].
The systems whose Hamilton functions separable in cartesian coordinates:
γ
(A) V = α(4x2+y2)+βx+ , (4.1)
y2
y
P = p p2, w= , (4.2)
x y 6
β γ
(B) V = α(x2+y2)+ + , (4.3)
x2 y2
xy
P = (xp yp )p p , w= ,
y x x y
− 6
xy
(C) V = α(x2+y2)+β , (4.4)
(x2 y2)2
−
x2 y2
P = (xp yp ) p2 p2 , w = − ,
y− x x− y 6
(cid:0) (cid:1)
(D) V = α(9x2+y2), (4.5)
y2
P = (xp p y)p2, w = ,
y− x y −18
The systems whose Hamilton functions separable in parabolic coordinates:
β γ
(F) V = (α+ + )r−1, r= x2+y2, (4.6)
r+x r x
−
p
yr2
P = (xp p y)2p , w = ,
y x x
− 12
βx
(G) V = (α+ )r−1, (4.7)
y2
yr2
P = (xp p y)2p , w = ,
y x x
− 12
(H) V = (α+β√r+x+γ√r x)r−1, (4.8)
−
β γ r2
P = (xp p y) 2p2 +2p2 , w = ,
y − x x y− √r+x − √r x − 6
(cid:18) − (cid:19)
One system with the Hamilton function separable in polar coordinates:
β ρ γx δy
(I) V = α+ + + − , (4.9)
x2+y2 (δx+γy)2 x2+y2(δx+γy)2
p p (x2+y2)(δx+γy)
P = (p y p x)(γp δp ), w= .
x y x y
− − 12
It is new superintegrable system with a cubic integral of motion, which is a deformation of the degenerate
kepler model. An application of the direct method [3, 5, 4] or the Jacobi method [6] does not allows us to
obtain this system (4.9).
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Three exceptional systems whose qubic integral of motion K we can not rewrite in the ”generalized
angular momentum” form (2.7):
β α
(K) V = α√x β√y, P = p3 p3, w =√xy, (4.10)
± α x ∓ β y
√x
(L) V = α √x+βy , P =p3, w = , (4.11)
x −2β
(cid:0) (cid:1)
(M) V = f′(φ)r−2, K =p2 cosφp sinφr−1p + (4.12)
φ r − φ
(cid:0) (cid:1)
+ (2f′(φ)cosφ f(φ)sin(φ)) p +(3f′(φ)sinφ+f(φ)cosφ)r−1p .
r φ
−
At the case (M) (4.12) we used the standard polar coordinates r,p ,φ,p and the function f(φ) has to
r φ
{ }
satisfy the following equation
f′′ (3f′sinφ+fcosφ)+2f′ (2f′cosφ fsinφ)=0.
−
At these exceptional cases the Hamilton functions (4.10,4.11,4.12)are separable at the cartesian and polar
coordinates,respectively. However,theSt¨ackelpotentialsU arenotpolynomialsinvariablesofseparation.
1,2
4.1 Cartesian coordinates, cases A-D
Let us introduce the St¨ackel matrix
1 1
S = 2 2 , (4.13)
A−D  
1 1
 − 
 
and take the following potentials
2γ
(A) U =8αq2+2βq , U =2αq2+ ,
1 1 1 2 2 q2
2
2β 2γ
(B) U =2αq2+ , U =2αq2+ ,
1 1 q2 2 2 q2
1 2 (4.14)
αq2 β αq2 β
(C) U = 1 , U = 2 + ,
1 2 − 4q2 2 2 4q2
1 2
(D) U =18αq2, U =2αq2.
1 1 2 2
The corresponding Hamilton functions I (2.1) coincide with the Hamilton functions H (4.1,4.3,4.5) and
1
(4.4) if
(A B,D) x=q , y =q
1 2
−
or after the following canonical transformation
q q q +q
1 2 1 2
(C) x= − , p =p p , y = , p =p +p .
x 1 2 y 1 2
2 − 2
ThesecondintegralsofmotionI (2.1)arethesecondorderpolynomialsinmomenta. Thethirdindependent
2
integrals K are calculated by (2.7), where variables
(A) µ =p µ =q p , (B C) µ =q p , µ =p .
1 1 2 2 2 1 1 1 2 2
−
determine the left hand side of the canonical algebraic curves (2.4). At the case (D) the variables
p q
1 2
(D) µ =p q , µ =p q
1 2 1 2 2 2
− 3
have not such natural algebro-geometricmeaning.
10