From global dynamical models to the Hénon–Heiles potential

From global dynamical models to the H�eenon–Heiles potential

N.D. Caranicolas

Department of Physics, Section of Astrophysics, Astronomy and Mechanics, University of Thessaloniki, 540 06 Thessaloniki, Greece

Received 15 February 2002

Abstract

We present a global non-galactic dynamical model reducing to the H�eenon–Heiles potential. Expanding the globalmodel in the vicinity of a circular orbit, we find the potential of a two-dimensional perturbed harmonic oscillator which

can give the H�eenon–Heiles potential for certain values of the parameters of the global model. Our numerical calcu-lations suggest that the properties of motion in the global model are similar of those displayed by the local model for

small as well as for large values of energy. Comparison to previous work is also made.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Global dynamical model; Local potential; Circular orbit; Harmonic oscillators; Properties of motion

1. Introduction

In an earlier paper (Caranicolas and Innanen, 2001), we presented outcomes indicating that the potential

introduced by H�eenon and Heiles (1964) does not come from an expansion of global galactic potentials. Inthe present work we shall give results suggesting that non-galactic dynamical models can give the H�eenon–Heiles potential, when expanded in a Taylor series near an equilibrium point.

Let us consider the dynamical model

Uðr; zÞ ¼ �12lnðr2 þ az2 þ c2Þ þ kr þ kz2; ð1Þ

where r, z are the usual cylindrical coordinates while a, c, k, k are parameters. Potential (1) consists of threeparts: a logarithmic part with a minus sign plus two terms; one proportional to r while the other term is an

harmonic term. Our choice is justified by the following lines of arguments: (i) Using Poisson equation one

can see that this model gives negative values for the density near the centre, therefore it is not a galacticpotential. (ii) It represents a simple mathematical model and can give, by expansion of the corresponding

effective potential (see below), the H�eenon–Heiles potential for a suitable choice of the parameters.The Hamiltonian to the potential (1) is

HG ¼ 12ðp2r þ p2z Þ þ

L2z2r2

þ Uðr; zÞ ¼ 12ðp2r þ p2z Þ þ Veffðr; zÞ ¼ E; ð2Þ

Mechanics Research Communications 29 (2002) 291–298

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where pr, pz are the momenta per unit mass conjugate to r and z, Lz is the third component of the angular

momentum, that is constant since the Hamiltonian is invariant through rotations about the third axis, while

E is the numerical value of HG. The potential

Veffðr; zÞ ¼L2z2r2

þ Uðr; zÞ; ð3Þ

describing the motion in the ðr; zÞ plane, is called the effective potential.Expanding the effective potential (3) in the vicinity of a circular orbit (r ¼ r0, z ¼ 0), we find

Veffðr0 þ Dr;DzÞ ¼ Veffðr0; 0Þ þ 12AðDrÞ2 þ 1

2BðDzÞ2 þ ½a1DrðDzÞ2 þ a2Dr3 þ � � �� ð4Þ

where

A ¼ 3L2z

r40þ o2U

or2

� �r0;0

; B ¼ o2Uoz2

� �r0;0

; a1 ¼1

2

o3Uoroz2

� �r0;0

; a2 ¼ � 2L2z

r50þ 16

o3Uor3

� �r0;0

ð5Þ

and we have taken into account the condition for the circular orbit

L2zr30

¼ oUor

� �r0;0

; ð6Þ

as well as the fact that Veff is symmetric about z ¼ 0. The expressions for the coefficients (5) in terms of theparameters entering the potential (3) are

A ¼ 3L2z

r40þ r20 � c2

ðr20 þ c2Þ2; B ¼ �a

r20 þ c2þ 2k; a1 ¼

ar0ðr20 þ c2Þ2

; a2 ¼ � 2L2z

r50þ r0ð3c2 � r20Þ3ðr20 þ c2Þ3

: ð7Þ

Dividing both sides of Eq. (4) by A and writing V ¼ ½Veffðr0 þ Dr;DzÞ � Veffðr0; 0Þ�=A, B=A ¼ x2, b ¼ a1=A,c ¼ a2=A, x ¼ Dr, y ¼ Dz, we obtain

V ðx; yÞ ¼ 12ðx2 þ x2y2Þ þ bxy2 þ cx3; ð8Þ

where 1, x are the unperturbed frequencies of oscillation along the x- and y-axis respectively.In the present paper we shall study the properties of motion in the two dynamical models (3) and (8) in

the case, where the parameters of the model (8) are x ¼ 1, b ¼ 1, c ¼ �1=3. In other words the localpotential (8) is the potential used by H�eenon and Heiles (1964). As it was mentioned, the results obtained inCaranicolas and Innanen (2001) suggest that the H�eenon–Heiles potential does not come from the Taylorexpansion of global models relevant for actual galaxies. Furthermore, it was observed in Caranicolas and

Innanen (2001) that the properties of orbits in the global and local potential were the same and no resonant

periodic orbits were present.

The aim of the present paper is to show that the mathematical model (3) can give a local potential which

is the H�eenon–Heiles potential. We shall also show that the properties of motion in the global model (3) aresimilar to those of the H�eenon–Heiles local model. The comparison of the properties of motion for the twopotentials is made in the Section 2. In Section 3 we present a discussion and the conclusions of this work.

2. Properties of motion

If we choose a ¼ 1:02, c ¼ 0:01, k ¼ 1:0024, k ¼ 1:01355, then potential (1) has a circular orbit at r0 ¼ 1with Lz ¼ 0:05, which is the value of the circular angular momentum. For the above values of the pa-rameters we obtain x ¼ 1, b ¼ 1, c ¼ �1=3. Then the Hamiltonian to the local potential (8) reads

292 N.D. Caranicolas / Mechanics Research Communications 29 (2002) 291–298

H ¼ 12ðp2x þ p2y þ x2 þ y2Þ þ xy2 � x3

3¼ h; ð9Þ

where px, py are the momenta per unit mass conjugate to x and y while h is the numerical value of H that isthe value of the local energy.

Let us first try to connect the global energy E, associated to the potential Veff , with the local energy h,associated to potential V ðx; yÞ. It is clear that E0 ¼ Veffðr0; 0Þ is the energy of the circular orbit of a radius r0.

Fig. 1. (a)–(d) The r–pr phase plane for the global Hamiltonian (2) when E ¼ 1:053 (a) and E ¼ 1:145 (c). The x–px phase plane for thelocal Hamiltonian (9) when h ¼ 0:049 (b) and h ¼ 0:14 (d). The similarity between the patterns (a)–(b) and (c)–(d) is evident.

N.D. Caranicolas / Mechanics Research Communications 29 (2002) 291–298 293

The energy E ¼ Veffðr0 þ Dr;DzÞ is the energy in the vicinity of the circular orbit. In other words E0 ¼ const.represent a point in the r–z plane while E ¼ const. represents a curve defining an area in the same plane.

Fig. 2. The 1:1 resonant periodic orbits in the global Hamiltonian (2).

Fig. 3. The 1:1 resonant periodic orbits in the local Hamiltonian (9).


The motion takes place inside this curve, which is better known as the equipotential curve or the curve of

zero velocity. The difference E � E0 defines the local energy. In order to avoid large numbers we havedivided all parameters by A. Thus

h ¼ E � E0A

: ð10Þ

Increasing Dr, Dz, E increases, therefore h also increases. We consider only bounded global and localmotion. This means that we have always closed zero velocity curves Veffðx; yÞ ¼ E, V ðx; yÞ ¼ h in the r–z or

Fig. 4. (a)–(d) Velocity spectra for the global Hamiltonian (2) (a)–(c) and for the local Hamiltonian (9) (b)–(d). Details are given in the

text.


the x–y plane, that is E6Eesc, h6 hesc where hesc, Eesc is the energy of escape for the global and localHamiltonian respectively (Caranicolas and Varvoglis, 1984).

In order to visualise and compare the properties of motion we shall use the r–pr (z ¼ 0, pz > 0) and the x–px (y ¼ 0, py > 0) Poincar�ee phase plane for the Hamiltonians (2) and (9) respectively. The results are shownin Fig. 1(a)–(d). Fig. 1(a) shows the r–pr phase plane for the Hamiltonian (2) when E ¼ 1:053 while Fig. 1(b)shows the x–px phase plane for the local Hamiltonian (9) when h ¼ 0:049 which comes from (10) forA ¼ 1:007 and E0 ¼ 1:004.The motion for both systems is regular. It is also important to observe that thetwo patterns look very similar. One can see four stable and one unstable invariant points in both cases. The

corresponding periodic orbits for the Hamiltonian (2) are shown in Fig. 2 while the periodic orbits of

system (9) are shown in Fig. 3. The outermost curve is the curve of zero velocity in both cases.

Fig. 1(c) shows the r–pr phase plane for the global Hamiltonian (2) when E ¼ 1:145. Here, one observes alarge chaotic sea with small regular regions. There are also some smaller islands corresponding to secondary

resonances. The x–px phase plane for the local Hamiltonian (9) is shown in Fig. 1(d). The correspondingenergy is h ¼ 0:14. Here again we can see a large chaotic sea, some small regular regions and secondaryislands as well. The above results suggest, that the properties of motion of the local system look very similarto those of the global system for small as well as for large energies.

In what follows we shall compare the velocity spectra derived from the local and global system. In this

spectrum we use the distribution of the velocities ti, where ti are the successive values of the velocities on thePoincar�ee phase plane. We call the dynamical spectrum of the parameter its distribution function, defined by

SðtÞ ¼ DNðtÞN dt

; ð11Þ

where DNt is the number of the parameters in the interval (t, t þ dt) after N iterations. We use the velocityspectrum because is faster than the spectrum of stretching numbers (see Karanis and Caranicolas, 2001 for

details). In this work we use the radial velocity tr for the global Hamiltonian (2) and the velocity tx for thelocal system (9).

Fig. 5. LCN for the global Hamiltonian (2) (dashed line) and the local system (9) solid line.


Fig. 4(a) shows the velocity spectrum for a regular orbit belonging to the global system. Initial condi-

tions r0 ¼ 0:85, z ¼ 0, tr0 ¼ 0 while the value of tz0 is found using the energy integral. The value of energy isE ¼ 1:053. As one can see, the spectrum is a typical U type spectrum. The velocity spectrum for a regularorbit belonging to the local system is shown in Fig. 4(b). Here again one observes the U type spectrumwhich is the characteristic spectrum of the regular motion. Initial conditions x0 ¼ �0:15, y ¼ 0, tx0 ¼ 0while the value of ty0 is found using the energy integral. The value of energy is h ¼ 0:049. In both spectrathe value of N is 104. The spectra shown in Fig. 4(c) and (d) are same to those shown in Fig. 4(a) and (b),

respectively, when the global energy E ¼ 1:145 while the local energy is h ¼ 0:14. Here we have two spectrawith a large number of small and large peaks indicating chaotic motion. The maximal Liapunov charac-

teristic number (LCN) (see Caranicolas and Vozikis, 1999) for the above two chaotic orbits are given in

Fig. 5. The dashed line gives the LCN of the global while the solid line is the LCN of the local system. As

one can see, in both systems, we have the LCN 1 which shows slow chaos (Caranicolas, 1990). Note thatin the case of fast chaos the LCN is of order of unity (Caranicolas and Vozikis, 1987).

3. Discussion

In the present work we have presented the non-galactic dynamical model (1) reducing to the H�eenon–Heiles potential. This potential has always a stable circular periodic at 0 < r06 1 when 1 < k6 1:5,0:01 < Lz 6 0:2, 0:01 < c6 0:1.This can be found using Eq. (6). For the values of the other parameters wecan use 1 < k6 1:5, 1 < a6 1:5. The existence of the stable circular periodic orbit is important because inthis stationary point the effective potential (3) has a minimum and can be expanded in a Taylor series giving

a potential of a two dimensional perturbed harmonic oscillator. The interesting properties of dynamical

systems made up of perturbed harmonic oscillators have been frequently studied using new modern ana-lytical methods (Elipe, 1999; Elipe and Deprit, 1999; Elipe, 2001) during the last years.

Analytical and numerical results given in Caranicolas and Innanen (2001), strongly suggest that the

H�eenon–Heiles potential does not come from the Taylor expansion of global models relevant for actualgalaxies. On this basis, one concludes that this potential can be obtained if we choose the particular values

for x, b and c. An alternative is to construct a mathematical model that can produce the desired values forthe parameters x, b and c. For our numerical calculations we have used the classical method of the Po-incar�ee phase plane. In order to look to the properties of orbits from another point of view we have also usedthe velocity spectrum and the LCN.Many numerical calculations suggest that the behavior of orbits in the local model is similar to the

behavior of the orbits in the global potential. The above results, together with the outcomes from Car-

anicolas and Innanen (2001), indicate that the behavior of orbits in some local potentials is very similar to

that of the corresponding global models, in cases where the resonant periodic orbits are present or absent.

This similar behavior of orbits is important not only for galactic dynamics but also for the study of general

dynamical systems. This arises from the fact that the local system has always a polynomial form, which is

easier to handle in order to produce analytical results. For instance, it is easy to find analytical outcomes for

a polynomial potential using the generalized lissajous transformation described in (Elipe and Deprit, 1999).Finally the author would like to make clear that the potential here presented is not unique.

Acknowledgement

The author would like to thank the anonymous referee for his useful comments.


References

Caranicolas, N.D., 1990. A&A 227, 54.

Caranicolas, N.D., Innanen, K.A., 2001, preprint.

Caranicolas, N.D., Varvoglis, H., 1984. A&A 141, 383.

Caranicolas, N.D., Vozikis, Ch, 1987. Cel. Mech. 40, 35.

Caranicolas, N.D., Vozikis, Ch, 1999. A&A 449, 70.

Elipe, A., 1999. Phys. Rev. E 61 (6), 6477.

Elipe, A., 2001. Math. Comp. Sim. 57, 217.

Elipe, A., Deprit, A., 1999. Mech. Res. Com. 26, 635.

H�eenon, M., Heiles, C., 1964. A. J. 69, 73.

Karanis, G.I., Caranicolas, N.D., 2001. A&A 367, 443.


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From global dynamical models to the Hénon–Heiles potential