# Journal of Nonlinear Mathematical Physics

Volume 28, Issue 3, September 2021, Pages 303 - 308

# A New Case of Separability in a Quartic Hénon-Heiles System

Authors
Nicola Sottocornola*
Department of Mathematics and Statistics, Zayed University, Abu Dhabi, UAE
Corresponding Author
Nicola Sottocornola
Received 17 November 2020, Accepted 9 April 2021, Available Online 25 April 2021.
DOI
10.2991/jnmp.k.210419.002How to use a DOI?
Keywords
Integrable systems; separation of coordinates; integration in quadratures
Abstract

There are four quartic integrable Hénon-Heiles systems. Only one of them has been separated in the generic form while the other three have been solved only for particular values of the constants. We consider two of them, related by a canonical transformation, and we give their separation coordinates in a new case.

Open Access

## 1. INTRODUCTION

Hénon-Heiles (HH) systems are Hamiltonian systems in ℝ4 endowed with the standard symplectic form dP1dQ1 + dP2dQ2. The Hamiltonian function has the form

H=12(P12+P22)+V(Q1,Q2)
where V is a polynomial function. There are four nontrivial integral cases with quartic potential whose name in the literature is HH4 followed by three numbers giving the ratios of the coefficients of the quartic monomials: 1:2:1, 1:6:1, 1:6:8 and 1:12:16. The generalized HH systems are obtained adding inverse terms to the potential V, without destroying the integrability of the system.

The problem of the integration in quadratures of these systems has been extensively studied in the last decades. The most efficient and elegant method for this purpose, is to find canonical coordinates that separate the Hamilton-Jacobi equation. In this paper we will deal with the delicate task of characterizing such coordinates. The difficulty of the task is well known so that, despite decades of efforts, only one of these four systems has been separated in the generic form: HH4 1:2:1. For the other three systems, the separation coordinates are known only in some degenerate cases. For HH4 1:12:16, the best available results can be found here [7]. In this paper we deal with HH4 1:6:1 and HH4 1:6:8 only.

Let’s now introduce them.

## 2. THE LINK BETWEEN HH4 1:6:1 AND HH4 1:6:8

The generalized Hamiltonian function has the form:

H161=12(P12+P22)-12ω(Q12+Q22)-Q1432-3Q12Q2216-Q2432-k122Q12-k222Q22 (1)
and depends on three arbitrary constants, ω, k1 and k2. The last two terms are the inverse terms and the ratios of the coefficients of the quartic terms are 1:6:1 as expected. This Hamiltonian system possesses an integral of motion that we call K:
K161=(P1P2-Q1Q2(Q128+Q228+ω))2-k12(P22Q12-Q224)-k22(P12Q22-Q124)+k12k22Q12Q22 (2)

The reader can easily check that these two functions are in involution with respect to the standard Poisson bracket hence the system is Liouville integrable. The separation coordinates for this system are unknown.

The canonical change of coordinates [1]:

Q1=R+2Q2=R-2P1=R+2(-p2q2-q12-k1-k2q22)+2k1R+P2=R-2(-p2q2+q12-k1-k2q22)-2k2R- (3)
where
R±2=-4q12±8p1-2q22+16p22q2216q1p2q2+32(k1-k2)p2q2316(k1-k2)q1q22+16(k1-k2)2q24+16ω
changes HH4 1:6:1 into HH4 1:6:8:
h168=12(p12+p22)+ω(4q12+q22)2-q142-3q12q228-q2416-γq1+β2q22k168=14(p22-q22(2q12+q22-8ω)8+βq22)2-q22(q2p1-2q1p2)216-γ4(2γq22-4q2p1p2+q1q242+q13q22+4p22q1-4ωq1q22+4q1βq22). (4)

These functions are usually written in a slightly different form in the literature. It’s easy to pass from one form to the other with a simple change of coordinates. The relationships between the coefficients of the two systems are

γ=12(k1+k2)β=-(k1-k2)2. (5)

The separation coordinates of (4), in the case γ = ω = 0, were found using Painlevé analysis in 1994 [6]:

2q12+q22-8p22±8Rq22 (6)

where R is the polynomial obtained replacing β = 0 in k168. The case ω ≠ 0 is treated in [8].

Inverting the change of coordinates (3), they provide the separation coordinates for HH4 1:6:1 in the symmetric case k12=k22. As far as we know, no other cases have been separated to this day. In this paper we solve the case k1k2 = 0. Before that, let’s turn our attention to an alternative method to see the process of separation of coordinates.

## 3. THE KOWALEWSKI CONDITIONS

In 2005 F. Magri published a paper [3] revisiting the famous problem solved by S. Kowalewski in 1888 [2]: the so called Kowalewski top. The method adopted in the paper is general and can be applied even in the non-Hamiltonian case, provided that a convenient number of commuting vector fields and first integrals are present. It was subsequently refined in several publications over the years and finally presented in a complete form in [4,5], where the reader will find all the proofs that are omitted here.

Let’s now summarize the key ideas in the case of a symplectic system in ℝ4 with Hamiltonian functions H and K.

The method assumes the presence of a second Poisson tensor P2 compatible with the tensor P1 associated to the symplectic structure:

[P1,P2]=[P2,P2]=0
where [...] is the Schouten bracket. We also assume that the two Hamiltonian functions H and K are in involution with respect to the Poisson bracket associated to P2:
P2(dH,dK)=0.

At this stage one can built the torsionless, recursive operator N = P2P1−1. If N has maximal rank, the two distinct eigenvalues provide (half of) the separation coordinates. The explicit determination of the compatible Poisson tensor P2, that requires the calculation of six unknown functions, can result quite cumbersome even in relatively simple cases. The number of unknown functions can be reduced to four, the components of a vector field X, looking for tensors P2 = LX(P1). Using this method the bi-Hamiltonian structure of cubic Hénon-Heiles systems can be calculated directly [9].

However, in the present case, the explicit determination of a bi-Hamiltonian structure in natural coordinates seems definitely too complicated. The good news is that one does not have to build up N (and, before this, P2) in order to calculate its eigenvalues. To understand this point, we start observing that N acts on the vector fields tangent to the Lagrange foliation defined by the level surfaces of H and K [9]. This bi-dimensional foliation is spanned by the Hamiltonian vector fields XH and XK so that

NXH=m1XH+m2XKNXK=m3XH+m4XK

for some functions m1,..., m4. It is now clear that the so called control matrix M=(m1m2m3m4) is nothing but the restriction of N to the leaves of the foliation, written in the basis associated with XH and XK. Furthermore the tensor M is also torsionless since it is the restriction of a torsionless tensor to an invariant surface. This is the first of the two properties that characterize M. The second one is that the vector fields XH and XK must commute with respect to the modified commutator

[X,Y]M[MX,Y]+[X,MY]-M[X,Y]
defined on the vector fields tangent to the Lagrangian foliation.

F. Magri proved that these two properties are necessary and sufficient conditions for the system to be separable and for the eigenvalues of M to be separation coordinates [5]. The point of interest in all this discussion is that these two conditions, T(N) = 0 and [X, Y]M = 0, can be reduced to four differential constraints, on the entries of M, called Kowalewski Conditions (KC):

{m3,H}={m1,K}{m4,H}={m2,K}{m1m3+m3m4,H}={m12+m2m3,K}{m2m3+m42,H}={m1m2+m2m4,K} (7)

We also need an extra condition for the new coordinates to be canonical: the trace and the determinant of M must be in involution

{m1+m4,m1m4-m2m3}=0. (8)

In order to solve the KC one has to solve four differential equations in four unknown functions m1,..., m4; this can be quite challenging. A first step could be to select a particular class of solutions that is easier to calculate but general enough to include most of the significant examples in the literature. The experience suggests this form for M:

m1=aF2+bFG+cG2+dF+eG+fm2=gF+vm3=pF2+qFG+rG2+sF+tG+um4=gG+w (9)
where all the coefficients a, b, c... are constants of the motion while F and G are unknown functions.

If we agree to denote by

f˙=XH(f)f=XK(f)
the derivatives of a function f along the given Hamiltonian fields, and replacing (9) into (7), we obtain the following

### Proposition 3.1.

If the functions F and G are solutions of the equations

F=G˙G=(μF+τ)G˙-(μG+ν)F˙ (10)
where μ, τ and ν are constants of the motion, then the functions
m1-e(μF2+τF-G)-v(μF+τ)+wm2eF+vm3-e(μFG+νF)-v(μG+ν)m4eG+w (11)
verify the KC.

Therefore, if we limit our search to solutions of the form (9), the problem reduces to two differential equations (10) in two unknown functions F and G.

In [8] we suggested the method of the vector field Z, in order to reduce the task to the search of one single function V (the potential function) and a few constants.

Let’s outline the method in the case of HH4 1:6:1. The idea is to extend the phase space including the constants of the problem as new coordinates, so turning our system into a Poisson one. In our example, we can extend the phase space to ℝ6 with coordinates (P1, P2, Q1, Q2, k1, k2). The Poisson tensor is obtained adding two extra columns and two extra lines of zeros to P1. We now consider the vector field Z so defined:

Z=XV+w1k1+w2k2,w1,w2.
V, w1, w2 are unknown and XV is the Hamiltonian vector field associated to V. Sometimes it may be useful to look for a potential function of the form V = ln f (some examples are given in [8]). The next step is to define the “Fundamental Functions” F and G of Proposition 3.1 in this way:
F=Z(H161)G=Z(K161). (12)

A simple calculation proves that the first of equations (10) is automatically verified with any choice of the potential function [8]. This means that the problem is finally reduced to the determination of a single function V (plus, eventually, the constants w1, w2) verifying the second equation in (10). It should be stressed that the involutivity condition (8) has to be checked independently from the KC.

It’s time now to see how the method of the vector field Z can provide the separation coordinates for HH4 1:6:1 in the case k1k2 = 0.

## 4. HH4 1:6:1 IN THE CASE k1k2 = 0

The system (1)(2) is invariant under the symmetry

(P1,P2,Q1,Q2,k1,k2)(P2,P1,Q2,Q1,k2,k1)
so it’s enough to solve the case k2 = 0:
H=12(P12+P22)-12ω(Q12+Q22)-Q1432-3Q12Q2216-Q2432-k22Q12K=(P1P2-Q1Q2(Q128+Q228+ω))2-k2(P22Q12-Q224) (13)

In order to apply the method of the field Z we extend the phase space to ℝ5 with coordinates (P1, P2, Q1, Q2, k). A first remark is that the system is homogeneous with respect to the following gradation:

P1,P2,ω2Q1,Q21k3. (14)

We detail now the steps of the algorithm.

1. 1.

We look for a vector field of the form Z = XV + w∂/∂k with V = ln f as suggested in Section 3. Our problem is now to find the unknown function f and the constant w.

2. 2.

Because the system is homogenous we look for homogeneous Fundamental Functions F and G. For that purpose, the presence of the term ∂/∂k forces f ~ 4.

3. 3.

Replacing f with the general homogeneous polynomial of degree 4 with respect to the gradation (14) and adding inverse terms like kP2/Q1 suggested by the form of the Hamiltonian functions, we are now able to calculate F and G with (12).

4. 4.

We can choose the coefficients of f and w in such a way that the second equation in (10) is verified:

V=1kln(P1P2-Q1Q2(Q128+Q228+ω)+kP2Q1+kQ12) (15)
and Z=XV+1kk.

5. 5.

We choose the integrals of motion in M in such a way that the involutivity condition (8) is verified (see (16)).

The results of this discussion can be summarized in the following

### Theorem 4.1.

Consider

• the vector field Z=XV+1kk, where V is the function in (15);

• the functions F and G defined in (12);

• the Control Matrix

M=(-2k2F+2H1-2k2G+4K+8k2ω2H). (16)
Then the eigenvalues of M are separation (canonical) coordinates for (13).

Proof. A straightforward calculation gives

F=2Q12Q2+4Q1P1+4kQ14Q2+(Q23+8ωQ2-4k)Q12-8P1P2Q1-8kP2G=2ω-Q124-Q224-P2-k(Q14+(3y2+8ω+4P2)Q12+4Q1Q2P1+4kQ2)Q14Q2+(Q23+8ωQ2-4k)Q12-8P1P2Q1-8kP2
and (16) provides the four entries m1,..., m4 of the Control Matrix. With these functions (7) and (8) are verified so that the eigenvalues of M are separation coordinates.

## 5. FINAL REMARKS

• If we replace K with KH2 the Control Matrix can be written in the simplified form:

M=(-2kF+4H1-2kG+4K0).

F. Magri already pointed out a similar behavior of the Kowalewski top [5].

• Another set of separation coordinates can be obtained using quadratic functions in F and G:

M=(-2k2F2+GF-2k2FG+4KFG) (17)

• HH4 1:6:8 has been solved only in the particular cases βγ = 0 [10]. The eigenvalues of (16) or (17), through the change of coordinates (3), provide the separation coordinates for the case β = −4γ2.

• HH4 1:6:1 with k1 = k2 = k has already been solved using the method of the field Z and a potential function V = ln f [8]. The functions f for the cases k1 = k2 = k and (k1, k2) = (k, 0) are, respectively,

P1P2-Q1Q2(Q128+Q228+ω)+kP2Q1+kP1Q2+k2Q1Q2
and
P1P2-Q1Q2(Q128+Q228+ω)+kP2Q1+kQ12.

The idea is to guess, from these particular examples, the form of f for the generic case. The first part of the function is independent from the constants so it is reasonable to expect that it remains unchanged but for the last terms the situation is uncertain. For instance it is not clear why the term kQ1/2 does not appear in the case k2 = k1 = k.

The generic case remains unsolved.

## CONFLICTS OF INTEREST

The author declares no conflicts of interest.

## ACKNOWLEDGEMENT

This work was funded by Zayed University, Abu Dhabi, through the RIF Grant R18050.

## REFERENCES

[1]S Baker, VZ Enolskii, and AP Fordy, Integrable quartic potentials and coupled KdV equations, Phys. Lett. A, Vol. 201, 1995, pp. 167-174.
[4]F Magri, The Kowalewski’s top revisited, 2018. Available from: https://arxiv.org/abs/1809.01879v1.
[5]F Magri, The Kowalewski separability conditions, I Krichever, S Novikov, O Ogievetsky, and S Shlosman (editors), Volume Integrability, Quantization, and Geometry, AMS, in Proceedings of Symposia in Pure Mathematics (2021).
[10]C Verhoeven, M Musette, and R Conte, General solution for Hamiltonians with extended cubic and quartic potentials, Theor. Math. Phys., Vol. 134, 2003, pp. 128-1.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 3
Pages
303 - 308
Publication Date
2021/04/25
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.2991/jnmp.k.210419.002How to use a DOI?
Open Access

TY  - JOUR
AU  - Nicola Sottocornola
PY  - 2021
DA  - 2021/04/25
TI  - A New Case of Separability in a Quartic Hénon-Heiles System
JO  - Journal of Nonlinear Mathematical Physics
SP  - 303
EP  - 308
VL  - 28
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.210419.002
DO  - 10.2991/jnmp.k.210419.002
ID  - Sottocornola2021
ER  -