This page contains additional material from the paper

M. Fenucci, À. Jorba: Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

Click here to download the preprint. We take into account the Coulomb (N+1)-body problem, which is a system composed by a positively charged particle fixed at the origin (i. e. the nucleus), surrounded by N negatively charged particles (i. e. electrons). These particles move under the effect of the Coulomb force, without gravitation. This system remids the Rutherford model of the atom. Units are chosen so that
  • the charge of the electron is -1,
  • the mass of the electron is 1,
  • the Coulomb constant is 1.
Setting N = 12, 24, 60, we compute periodic orbits which are invariant under the rotations of the symmetry group of one of the five Platonic polyhedra. Such orbits depend only on the trajectory of one of the electrons, which is called generating particle. Moreover, its trajectory cannot pass through the axes of the rotations of the group: due to this fact, we compute orbits in different homotopy classes of the space minus the union of the rotation axes.

Below are listed the periodic orbits we computed for the three different rotation groups (the period is always set to be 1). The first column is just a number to identify the orbit. The second column represents the homotopy class in which the orbit lies. These classes are coded using a path on the edges of an Archimedean solid, and the sequence of numbers refers to the enumeration of the vertexes we used. For a deepen explanation, please see references [1] and [2] at the end of this page. The number M indicates how many particles move on the trajectory of the generating particle, following this curve with a constant shift in time. The number in the column labeled with R identifies a rotation of the group, which keeps unchanged the trajectory of the generating particle. This number refers to the enumeration we used to represent the rotations in our software. The fifth column contains the minimum integer value of the central charge for which we were able to compute the corresponding orbits.

The last column contains a link to display videos and additional information about the orbits. Each page contains
  • The orbit lying in the indicated homotopy class, with the corresponding value of the central charge.
  • Another orbit lying in the same homotopy class, with the same value of the central charge as the previous. This orbit is found following the curve of solutions depending on the central charge, using a continuation method. In these videos, the red electron is the generating particle and the red curve its trajectory.
  • The path on the Archimedean polyhedron indicated in the column "Sequence", which identifies the homotopy class.
  • A graphic representing the spectral radius of the monodromy matrix, in function of the value of the central charge.
Interested people can e-mail Marco Fenucci to obtain more information about the representations of homotopy classes and the rotation groups, or also to get the initial conditions for the orbits.

12 electrons (Tetrahedron Symmetry)
Index Sequence M R
Min integer value
of the central charge
 Videos
1 [1 5 10 2 7 9 1] 2 2
11
 Link
2 [1 5 2 6 11 3 12 9 1] 2 11
9
 Link
3 [1 5 8 3 12 4 9 7 1] 2 12
10
 Link
4 [1 5 10 11 4 9 1] 3 10
13
 Link
5 [1 5 2 7 6 4 9 12 8 1] 3 7
9
 Link
6 [1 5 2 6 4 9 7 2 10 11 6 7 1] 3 4
10
 Link
7 [1 5 8 3 10 11 3 12 4 9 12 8 1] 3 10
9
 Link
8 [1 5 8 12 3 10 11 4 6 2 7 9 1] 3 3
13
 Link
9 [1 5 10 11 4 9 1 5 10 11 4 9 1] 3 4
13
 Link
24 electrons (Cube Symmetry)
Index Sequence M R
Min integer value
of the central charge
 Videos
1 [ 1 3 8 10 16 5 1] 2 10
16
 Link
2 [ 1 3 10 6 4 9 22 16 1] 2 4
15
 Link
3 [ 1 3 7 14 11 19 2 5 1] 2 11
31
 Link
4 [ 1 3 10 6 4 9 17 19 11 14 1] 2 9
29
 Link
5 [ 1 3 8 10 6 4 9 2 22 16 1] 2 4
25
 Link
6 [ 1 3 8 15 6 4 9 2 5 16 1] 2 4
27
 Link
7 [ 1 3 7 20 23 11 19 2 22 16 1] 2 11
31
 Link
8 [ 1 3 7 14 23 11 19 2 5 16 1] 2 11
29
 Link
9 [ 1 3 7 18 20 7 14 1 5 16 1] 2 7
23
 Link
10 [ 1 3 8 15 13 12 4 6 10 16 1] 2 12
19
 Link
11 [ 1 3 7 20 24 12 4 9 2 5 1] 2 12
18
 Link
12 [ 1 3 8 15 4 6 10 16 5 11 23 14 1] 2 10
30
 Link
13 [ 1 3 8 10 6 4 9 17 21 19 11 14 1] 2 9
28
 Link
14 [ 1 3 7 18 13 12 4 9 17 19 11 14 1] 2 4
20
 Link
15 [ 1 3 8 15 6 4 9 17 21 23 11 14 1] 2 9
28
 Link
16 [ 1 3 8 18 13 12 4 9 2 19 11 14 1] 2 4
24
 Link
17 [ 1 3 8 15 13 12 4 9 2 5 11 14 1] 2 4
24
 Link
18 [ 1 3 7 20 24 12 4 9 17 21 23 14 1] 2 4
29
 Link
19 [ 1 3 10 6 4 12 13 18 20 23 11 5 1] 2 13
17
 Link
20 [ 1 3 8 10 6 15 4 9 2 22 16 5 1] 2 4
24
 Link
21 [ 1 3 8 10 16 5 1 3 8 10 16 5 1] 2 1
16
 Link
22 [ 1 3 7 18 20 23 11 19 2 9 22 16 1] 2 11
31
 Link
23 [ 1 3 10 8 18 13 12 4 15 6 22 16 1] 2 12
19
 Link
24 [ 1 3 8 15 4 12 13 18 7 14 11 5 1] 2 13
17
 Link
25 [ 1 3 7 18 8 15 6 10 16 1] 3 18
23
 Link
26 [ 1 3 8 15 13 24 21 23 14 1] 3 15
30
 Link
27 [ 1 3 7 20 18 8 15 4 6 10 16 5 1] 3 18
30
 Link
28 [ 1 3 10 6 15 13 12 17 21 23 11 5 1] 3 15
30
 Link
29 [ 1 3 8 15 6 10 3 7 18 8 10 16 1] 3 6
19
 Link
30 [ 1 3 10 8 15 13 12 24 21 23 11 14 1] 3 15
27
 Link
31 [ 1 3 8 15 4 6 10 3 7 20 18 8 10 16 5 1] 3 6
20
 Link
32 [ 1 3 7 14 1 5 11 19 2 5 16 22 6 10 16 1] 3 5
22
 Link
33 [ 1 3 8 10 6 15 13 24 12 17 21 23 14 11 5 1] 3 15
15
 Link
34 [ 1 3 10 8 15 4 6 10 8 3 7 20 18 8 3 10 16 5 1] 3 6
26
 Link
35 [ 1 3 10 6 22 2 5 11 14 7 3 10 16 22 2 19 11 14 1] 3 5
28
 Link
36 [ 1 3 10 16 22 2 5 11 14 1 3 10 16 22 2 5 11 14 1] 3 5
23
 Link
37 [ 1 3 8 18 13 15 6 10 3 1 14 7 18 8 10 6 22 16 1] 3 6
28
 Link
38 [ 1 3 7 14 23 20 18 8 15 13 12 4 6 10 16 22 2 5 1] 3 18
29
 Link
39 [ 1 3 10 8 15 13 18 8 3 10 16 22 6 10 8 3 7 14 1] 3 18
25
 Link
40 [ 1 3 7 20 24 21 19 2 5 1 3 7 20 24 21 19 2 5 1] 3 19
20
 Link
41 [ 1 3 8 10 6 4 15 13 24 12 17 19 21 23 14 11 5 16 1] 3 15
27
 Link
42 [ 1 3 8 15 6 4 15 13 24 21 17 19 21 23 14 1 5 16 1] 3 15
27
 Link
43 [ 1 3 8 15 4 9 2 5 1] 4 8
19
 Link
44 [ 1 3 10 8 15 6 4 9 22 2 5 16 1] 4 8
17
 Link
45 [ 1 3 7 18 8 15 13 12 4 9 17 19 2 5 11 14 1] 4 8
20
 Link
46 [ 1 3 8 15 4 9 2 5 1 3 8 15 4 9 2 5 1] 4 4
19
 Link
47 [ 1 3 10 8 3 7 18 20 7 14 23 11 14 1 5 16 1] 4 3
19
 Link
48 [ 1 3 10 6 22 2 5 16 10 6 4 9 22 16 10 8 15 6 22 16 1] 4 2
12
 Link
49 [ 1 3 7 20 18 8 15 13 24 12 4 9 17 21 19 2 5 11 23 14 1] 4 8
18
 Link
50 [ 1 3 10 6 4 9 2 5 16 10 8 15 4 9 22 16 1 3 8 15 6 22 2 5 1] 4 2
13
 Link
51 [ 1 3 10 16 5 1 3 7 18 8 10 3 7 14 23 20 18 7 14 1 5 11 23 14 1] 4 3
28
 Link
52 [ 1 3 8 15 4 9 2 5 1 3 8 15 4 9 2 5 1 3 8 15 4 9 2 5 1] 4 2
19
 Link
53 [ 1 3 10 8 3 7 14 1 5 16 1 3 7 14 23 11 14 1 3 7 18 20 7 14 1] 4 14
18
 Link
54 [ 1 3 10 6 15 8 3 7 18 13 24 20 7 14 23 21 19 11 14 1 5 2 22 16 1] 4 3
20
 Link
55 [ 1 3 8 15 6 10 3 7 20 24 13 18 7 14 11 19 21 23 14 1 16 22 2 5 1] 4 3
17
 Link
56 [ 1 3 10 8 15 6 4 9 22 2 5 16 1 3 10 8 15 6 4 9 22 2 5 16 1] 4 4
18
 Link
57 [ 1 3 10 6 22 9 2 5 16 10 6 15 4 9 22 16 10 3 8 15 6 22 16 5 1] 4 2
30
 Link
60 electrons (Dodecahedron Symmetry)
Index Sequence M R
Min integer value
of the central charge
 Videos
1 [ 1 3 6 15 48 28 45 19 1] 1 1
64
 Link
2 [ 1 3 59 52 53 51 36 50 43 19 1] 1 1
62
 Link
3 [ 1 3 7 59 54 50 1 ] 2 59
78
 Link
4 [ 1 3 6 11 28 42 20 31 45 19 1 ] 2 42
93
 Link
5 [ 1 3 6 47 7 12 52 59 54 1 ] 3 47
39
 Link
6 [ 1 3 6 15 47 7 12 13 52 59 54 50 1 ] 3 25
95
 Link
7 [ 1 3 59 54 50 43 19 1 54 51 36 50 1 ] 3 50
52
 Link
8 [ 1 3 59 7 12 21 33 26 25 38 34 48 28 11 19 1 ] 3 21
66
 Link
9 [ 1 3 7 59 54 50 43 45 19 1 54 51 35 36 50 1 ] 3 50
93
 Link

References

  1. G. Fusco, G. F. Gronchi , P. Negrini: 2011. Platonic polyhedra, topological constraints and periodic orbits of the classical N-body problem, Invent. Math. Vol. 285, Num. 2, 283-332. DOI: https://doi.org/10.1007/s00222-010-0306-3.

  2. M. Fenucci, G. F. Gronchi: 2018. On the stability of periodic N-body motions with the symmetry of Platonic polyhedra, Nonlinearity, Vol. 31, Num. 11, pp 4935-4954. DOI: https://doi.org/10.1088/1361-6544/aad644.