Periodic three-body orbits in the Coulomb potential

TL;DR

This paper reports the discovery and classification of approximately 100 nonrelativistic, collisionless, periodic three-body orbits in the Coulomb potential, with potential applications in ion trap experiments.

Contribution

It introduces a new set of Coulomb three-body orbits, classifies them by symmetry and topology, and establishes a linear relation between their periods and topologies.

Findings

Discovered around 100 distinct orbits in Coulomb potential.

Classified orbits based on symmetry and topology.

Established a linear relation between period, energy, and topology.

Abstract

We numerically discovered around 100 distinct nonrelativistic collisionless periodic three-body orbits in the Coulomb potential in vacuo, with vanishing angular momentum, for equal-mass ions with equal absolute values of charges. These orbits are classified according to their symmetry and topology, and a linear relation is established between the periods, at equal energy, and the topologies of orbits. Coulombic three-body orbits can be formed in ion traps, such as the Paul, or the Penning one, where one can test the period vs topology prediction.

Tables

Table 1. Table 1: Initial conditions of six orbits, depicted in Fig. 2 , that belong to the sequence described by the free-group elements [ ( 𝙰𝙱 ) 𝟸 ​ ( 𝚊𝚋 ) 𝟸 ] 𝚔 superscript delimited-[] superscript 𝙰𝙱 2 superscript 𝚊𝚋 2 𝚔 {\tt[(AB)^{2}(ab)^{2}]^{k}} , with k = 1 , 2 , 3 , ⋯ 𝑘 1 2 3 ⋯ k=1,2,3,\cdots , and four linearly stable orbits, Table 2 . The columns correspond to: solution label, name of the sequence that the solution belongs to, initial velocities [ x 1 ˙ ( 0 ) [\dot{x_{1}}(0) , and y 1 ˙ ( 0 ) ] \dot{y_{1}}(0)] , period, negative energy, scaled period, free group element, number of letters in free group element (equal to number of asymmetric syzygies), and the total number of syzygies over a period. For initial conditions of all other found solutions, see suppm .

Label	Seq.	$\dot{x_1}\left(0\right)$	$\dot{y_1}\left(0\right)$	$T$	$-E$	$T{\left|E\right|}^{3/2}$	Free group element	$N_w$	$N_e$
A.4	I	0.191764	0.330958	13.4332	1.06108	14.6826	${\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}$	8	14
A.12.a	I	0.147917	0.323693	37.1599	1.12003	44.0473	${\left[{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\right]}^{\mathrm{𝟹}}$	24	42
A.12.b	I	0.246251	0.335527	45.3784	0.980345	44.0472	${\left[{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\right]}^{\mathrm{𝟹}}$	24	42
B.4	I	0.111427	0.305087	11.3981	1.18352	14.6755	${\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}$	8	14
B.12.a	I	0.327539	0.337033	57.4554	0.83738	44.0266	${\left[{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\right]}^{\mathrm{𝟹}}$	24	42
B.12.b	I	0.345214	0.344247	63.0644	0.786962	44.0266	${\left[{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\right]}^{\mathrm{𝟹}}$	24	42
A.15.b	II	0.108065	0.323579	44.7536	1.15086	55.2534	${\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\mathrm{𝙰𝙱𝙰}{\left(\mathrm{𝚋𝚊}\right)}^{\mathrm{𝟸}}\mathrm{𝚋𝙰𝙱𝙰}$
							$\times {\left(\mathrm{𝚋𝚊}\right)}^{\mathrm{𝟸}}\mathrm{𝙱𝙰𝙱}{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\mathrm{𝚊𝙱𝙰𝙱}$	30	52
A.18	III	0.105224	0.336995	55.6513	1.12609	66.5019	${\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}\mathrm{𝙰𝙱𝙰}{\left(\mathrm{𝚋𝚊}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝙱𝙰}\right)}^{\mathrm{𝟸}}\mathrm{𝚋𝚊𝚋}$
							$\times {\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}\mathrm{𝚊𝚋𝚊}{\left(\mathrm{𝙱𝙰}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝚋𝚊}\right)}^{\mathrm{𝟸}}\mathrm{𝙱𝙰𝙱}$	36	62
A.20.b	I	0.126494	0.315968	59.3293	1.15249	73.4049	${\left[{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}\right]}^{\mathrm{𝟻}}$	40	70
A.24.a	II	0.249577	0.291337	80.2223	1.0585	87.364	${\left[{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}{\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}𝙰\right]}^{\mathrm{𝟸}}{\left(\mathrm{𝚋𝚊}\right)}^{\mathrm{𝟸}}𝚋$
							$\times {\left(\mathrm{𝙰𝙱}\right)}^{\mathrm{𝟸}}{\left[{\left(\mathrm{𝚊𝚋}\right)}^{\mathrm{𝟸}}𝚊{\left(\mathrm{𝙱𝙰}\right)}^{\mathrm{𝟸}}𝙱\right]}^{\mathrm{𝟸}}$	48	86

Table 2. Table 2: Stability coefficients λ j , ν j subscript 𝜆 𝑗 subscript 𝜈 𝑗 \lambda_{j},\nu_{j} , with j = 1 , 2 𝑗 1 2 j=1,2 , of linearly stable (double elliptic) orbits, where λ j = exp ⁡ ( 2 ​ π ​ i ​ ν j ) subscript 𝜆 𝑗 2 𝜋 𝑖 subscript 𝜈 𝑗 \lambda_{j}=\exp(2\pi i\nu_{j}) .

name	$\mathrm{Re}\left({\lambda }_j\right)$	$\mathrm{Im}\left({\lambda }_j\right)$	${\left|{\lambda }_j\right|}^2$	${\nu }_j$	$N_w$
A.15.b	0.510145	0.860102	1.000023	0.164797	30
	-0.11507	0.993357	0.999999	0.268355
A.18	-0.002025	0.999961	0.999926	0.250322	36
	-0.820340	0.571882	1.000007	0.403107
A.20.b	0.009875	0.998966	0.998031	0.248427	40
	0.94189	0.339728	1.002572	0.055094
A.24.a	-0.988601	0.174067	1.007631	0.472261	48
	0.993975	0.116863	1.001643	0.018627

Equations

$$T|E{|}^{3/2}\sim N,$$

Full text

three-body problem, Coulomb potential

The Newtonian three-body problem is one of the outstanding classical open questions in science. After more than 300 years of observation, only two topologically distinct types of periodic three-body systems, or orbits, have been observed in the skies Tokovinin:2007 :

1. the so-called hierarchical systems, such as the Sun-Earth-Moon one, to which type belong more than 99% of all observed three-body systems; 2) Lagrangian three-body systems, such as Jupiter’s Trojan satellites, to which the remaining $\leq 1\%$ belong.

There has been some significant theoretical progress on the subject over the past few years: several hundred new, topologically distinct families of periodic solutions have been found by way of numerical simulations Moore1993 ; Chenciner:2000 ; Martynova2009 ; Suvakov:2013 ; Suvakov:2013b ; Suvakov:2014 ; Iasko2014 ; Dmitrasinovic:2015 ; Dmitrasinovic:2017 ; site ; Jankovic:2015 ; Rose:2016 ; Shibayama:2015 ; Li:2017 ; Liao:2017 , and unexpected regularities have been observed among them Dmitrasinovic:2015 ; Dmitrasinovic:2017 ; Liao:2017 ; Li:2017 relating the periods, topologies and linear stability of orbits.

Of course, one would like to observe at least some of the new orbits and test their properties in an experiment, but such a test would be impeded by a number of obstacles: (1) only stable orbits have a chance of actually existing for a sufficiently long time to be observed; (2) stability depends on the ratio(s) of masses, and on the value of angular momentum, neither of which can be controlled in astronomical settings; (3) even if an orbit is stable in a wide range of mass ratios and angular momenta, there is no guarantee that such a system will have been formed sufficiently frequently and sufficiently close to Earth, that it may be observed by our present-day instruments.

All of the above prompted us to look for alternative three-body systems that share (at least) some of the same properties with Newtonian three-body systems. The Coulombic potential shares one basic similarity with the Newtonian gravity – its characteristic $1/r$ (homogeneous) spatial dependence – as well as several important differences: (1) the (much) larger coupling constant; (2) both attractive and repulsive nature; (3) naturally identical (quantized) electric charge(s); (4) ions with opposite charges may have masses equal to one part in a few thousand; (5) ions have a finite probability of elastic scattering in head-on collisions; and (6) Coulombic bound states can be formed in table-top ion-trap experiments Major:2005 . For these reasons we turn to the study of periodic three-body orbits bound by Coulombic potential. The application of only the Coulomb interaction amounts to a non-relativistic approximation, which is good only in the low-velocity limit LandauII .

In this Rapid Communication we present the results of a search that led to around 100 distinct collisionless orbits, only four of which are stable, and around 80 isosceles quasi-colliding (free-fall, or “brake”) ones. We use the collisionless orbits to display a new regularity, akin to Kepler’s third law, in the form of a linear dependence

[TABLE]

between the scale-invariant period $T|E|^{3/2}$, where $T$ is the period, and $E$ is the energy of an orbit, on one hand, and the orbit’s topological complexity $N$, expressed as the number of collinear configurations (“syzygies”) encountered during one cycle, see the text below, on the other. This prediction ought to be tested in ion trap experiments.

We used the same search method as in the Newtonian gravity three-body problem Suvakov:2013 . There are 12 independent variables that define the initial state of this system, for each body there are the $x$ and $y$ coordinates of the body, and the $v_{x}$ and $v_{y}$ components of their velocity. Adopting the center-of-mass reference frame reduces this number (12) to eight. Fixing the value of angular momentum ($L=0$) reduces this further to six. Using the scaling rules Landau for the solutions and the fact that periodic solution must pass through at least one syzygy (collinear configuration) during one period, yields a four-dimensional search space for all zero-angular-momentum periodic solutions. We search for solutions in the two-dimensional subspace of orbits that pass through the Euler configuration, defined as the symmetric collinear configuration wherein the positively charged particle with velocity $(-2v_{x},-2v_{y})$ passes through the origin $(0,0)$, i.e., exactly between the two negatively charged particles, which, in turn, pass through the points $(-1,0)$ and $(1,0)$, both with velocity equal to $(v_{x},v_{y})$.

In order to search for periodic solutions numerically, we have discretized the search window in this two-dimensional subspace and calculated the return proximity function (RPF) $d_{T_{0}}(v_{x},v_{y})$ that measures how close to the initial condition the trajectory returns (see suppm ) up to some pre-defined upper limit on the integration time $T_{0}$, at each grid point. The negative logarithm of the computed RPF is shown in Fig. 1. Local minima of RPF are used as candidates for periodic solutions. After applying the gradient descent algorithm starting at each candidate point, we have declared as periodic the solutions with RPF of ${\cal O}(10^{-8})$ and smaller.

We have followed the example set by three-body orbits in Newtonian gravity Suvakov:2013 ; Dmitrasinovic:2017 , and classified the newly found Coulombic orbits according to their topologies, studied their stability and organized them into sequences some of which, though fewer in number, appear very similar to the Newtonian ones. Each orbit has a well defined topology which can be algebraized in at least two different ways, see suppm and Refs. Montgomery:1998 ; Tanikawa:2008 . Here we use Montgomery’s method Montgomery:1998 wherein each solution is associated with (the conjugacy class, see Suvakov:2013 , of) an element of the two-generator (${\tt a}$, ${\tt b}$, ${\tt A=a^{-1}}$, ${\tt B=b^{-1}}$) free group $F_{2}({\tt a,b})$.

There are important distinctions among the 100-odd orbits:

1. the orbits can be separated into two classes, using their symmetry: class (A) consists of orbits that are symmetrical under two perpendicular reflections, and class (B) of orbits with a point reflection symmetry;
2. each of the classes can be further separated into sequences, defined by their free-group elements, as follows. For both class A and class B, sequence (I): $w_{n,k}^{\rm{(I)}}={\tt[(AB)^{n}(ab)^{n}]^{k}}$ with integers $n,k=1,2,\ldots$; and for class A only, sequence (II): $w_{m,n,k}^{\rm{(A.II)}}={\tt[(AB)^{m}(ab)^{n}]^{k}A[(BA)^{m}(ba)^{n}]^{k}B}$, with $m,n,k=1,2,3,\cdots$; and sequence (III): $w_{n}^{\rm{(III)}}={\tt[(ab)^{2}ABA(ba)^{2}BAB]^{n}}$, with $n=1,2,3,\cdots$.

Note that the 100-odd collisionless Coulombic orbits are substantially fewer than roughly 200 collisionless Newtonian orbits with similar search parameters, and that there are only four linearly stable solutions in contrast to more than 20 in the Newtonian case.

All of this is a consequence of just one sign change in the potential: one pair of charged particles must experience repulsion, contrary to Newtonian gravity, where all pairs are attractive. Therefore, no choreographic solution, i.e. permutationally symmetric solution with all three particles following the same trajectory, such as the famous “figure-8” orbit, may exist in the Coulombic case. Moreover, at least one orbit, similar to Orlov’s Martynova2009 colliding “$S$-orbit” (in Newtonian gravity) still exists in the Coulombic case, but it is not stable any more, and consequently does not produce an infinite sequence of periodic orbits, see Dmitrasinovic:2017 .

The initial conditions of all 100-odd orbits and their corresponding topological and kinematical properties can be found in suppm ; in Fig. 2 and Table 1 we have shown six representative solutions.

Next we show that Eq. (1), the (striking) property of orbits that was first observed in Newtonian three-body systems Dmitrasinovic:2015 , also features in the Coulombic three-body systems. This relation between topological and kinematical properties of Newtonian three-body systems was first reported in Dmitrasinovic:2015 and later studied in more detail in Refs. Jankovic:2015 ; Liao:2017 ; Li:2017 ; Dmitrasinovic:2017 . Equstion (1) is a (simple) linear dependence of the scale-invariant period $T|E|^{3/2}$ on the topological complexity $N$. The topological complexity $N$ can be measured in at least two different ways: (1) we used the length $N_{w}$ of the free-group element (word) describing the orbit’s topology, which, due to symmetry in our case, is equal to the number of asymmetric syzygies, i.e., collinear configurations wherein the two equal-charge particles are next to each other, over one period; (2) the number $N_{e}$ of all syzygies (collinear configurations)) was considered in Refs. Dmitrasinovic:2015 ; Dmitrasinovic:2017 as the measure of topological complexity $N$ of Newtonian orbits.

In Fig. 3 one can see that Eq. (1) holds for three-body orbits in the Coulomb potential: (1) with $N=N_{w}$, a linear fit yields a slope equal to 1.8252, with asymptotic standard error of 0.08% and an average relative deviation of points from fit values that equals 0.63%; (2) with $N=N_{e}$ the number of all syzygies (collinear configurations), in the Coulomb case, the situation is slightly different (see inset in Fig. 3): the slope of this fit is $1.0208$, the asymptotic standard error is 0.36%, but with a significantly larger (2.6%) average relative deviation of points from fit values.

As mentioned earlier, only four solutions are linearly stable. We solved the equations for an infinitesimal deviation from the exact periodic solution along each periodic orbit to find the eigenvalues of the monodromy matrix, see suppm . Due to the symmetry of the equations of motion, these eigenvalues appear as two ($i=1,2$) quadruples ($\lambda_{i}$, $\lambda_{i}^{*}$, $1/\lambda_{i}$, $1/\lambda_{i}^{*}$). For only four orbits listed in Table 2 both eigenvalues $\lambda_{i}$ have moduli equal to unity $|\lambda_{i}|=1$, within their respective margins of error, which means that the corresponding three-body orbit is linearly stable.

Thus we have shown that some of the phenomena first observed in Newtonian three-body orbits, such as the linear dependence of the scale-invariant period on the topology of the orbit, and the emergence of sequences 222from a linearly stable orbit, in agreement with the Birkhoff-Lewis theorem Birkhoff_Lewis:1933 exist in Coulombic three-body orbits, and are not features of Newtonian gravity alone. The homogeneity Landau of the Coulombic, Newtonian, and the strong Jacobi-Poincaré potentials is common to all three known cases of manifestation of this regularity Landau ; Dmitrasinovic:2017 ; Dmitrasinovic:2017b . This supports indirectly the explanation offered in Refs. Dmitrasinovic:2017 ; Dmitrasinovic:2017b .

Our next concern ought to be the observation of some of these orbits in an experiment. The trajectories of a number (ranging between 1 and 32) of positively charged particles moving in a Paul trap have been photographically recorded as early as 1959 Major:2005 ; Wuerker:1959 . The challenge to actually confine and photograph a few oppositely charged macroscopic particles in an ion trap has remained unanswered to the present day, to our knowledge. It is well known that Paul and/or Penning traps can lead to binding of pairs of identical ions, including periodic orbits as well as their chaotic motions Hoffnagle:1988 ; Totsuji:2002 , when the circumstances (such as the frequency and amplitudes of the applied electric and/or magnetic fields) are right. Such periodic orbits are impossible in free space, however, as there the identical ions experience only Coulomb repulsion 333Curiously, the case of two opposite-charged, equal-mass ions has not been examined in detail, thus far, see e.g. Trypogeorgos:2013 , although it also falls into the first Mathieu stability zone. So, before one observes any periodic three-body orbits in an ion trap, and declares them genuine Coulomb orbits, one must know which periodic three-body orbits exist in free space – information that we have provided here. With the present work we have prepared the terrain for future numerical, and we hope also experimental studies of three-ion motions in traps Major:2005 .

Naturally, the orbits that are (linearly) stable in free space are also expected to exist in a trap; that is not to say that the unstable orbits cannot be stabilized by appropriate trapping fields, or that new kinds of periodic orbits cannot be formed in a trap. Moreover, ions have a non-zero elastic head-on collision cross-section, unlike the stars and/or planets, so one may even observe some “colliding” orbits 444We put the word colliding into quotation marks here, because only zero-angular momentum orbits experience actual collisions, whereas ion traps generally impart angular momentum to ions. in ion traps. This gives one an opportunity to observe hitherto experimentally unobserved orbits and to study some of their unprecedented properties.

At any rate, trap-induced corrections will have to be calculated for each three-ion orbit in any trap where experiments are conducted, before an interpretation is given. With this Rapid Communication we hope to start a discussion of trap-induced corrections for periodic three-ion orbits: in order to calculate such corrections, one needs the (initial conditions of) free-space periodic orbits, of which we have provided around 100, that ought to suffice for a starting point.

There are no records, to our knowledge, of searches for periodic Coulombic three-body systems with equal masses and equal charges, which are the closest to the equal-mass Newtonian system that was studied in Refs. Martynova2009 ; Suvakov:2013 ; Suvakov:2013b ; Suvakov:2014 ; Iasko2014 ; Dmitrasinovic:2015 ; site ; Jankovic:2015 ; Rose:2016 ; Shibayama:2015 ; Li:2017 ; Liao:2017 . As we wished to compare the closest analogons of the Coulombic and Newtonian three-body systems, we had to repeat a search for periodic collisionless orbits at the present mass and charge ratios.

To be sure, we are not the first ones who have studied Coulombic periodic three-body motion: the subject has a long history, see e.g. Refs. Langmuir:1921 ; Wannier:1953 , with a revival in the 1980s, since when a number of studies have been published: DAVIES:1983 ; Klar:1986a ; Poirier:1989 ; Grujic:1988 ; Richter:1990 ; Richter:1993 ; Yamamoto:1993 ; Santander:1997 ; Stuchi:2000 ; Sano:2004 ; Rupp:2010 . Numerical discovery of more than 8000 collinear colliding periodic orbits with He atom mass ratios was reported in Ref. Richter:1993 , and of somewhat fewer collisionless ones in Ref. Yamamoto:1993 . The initial conditions were not published, so one could not simply retrieve these previously discovered orbits and use them here.

With this Rapid Communication we also hope to induce practitioners to consider experimental searches, particularly in view of the fact that, at least in the case of past periodic-orbit discoveries, the theory did not precede experiment Major:2005 .

The work of V. D. and M. Šuvakov was supported by the Serbian Ministry of Education, Science and Technological Development under Grants No. OI 171037 and No. III 41011. M. Šindik conducted her work during the summer breaks of 2016 and 2017, when she was supported by a Student scholarship from the Serbian Ministry of Education, Science and Technological Development. All numerical work was done on the Zefram cluster, Laboratory for gaseous electronics, Center for non-equilibrium processes, at the Institute of Physics, Belgrade. We thank Marija Janković, Ana Hudomal and Srdjan Marjanović for their help in the early stages of this work.