This page contains additional material from the paper
M. Fenucci, G.
F. Gronchi: Symmetric constellations of satellites moving around a
central body of large mass
Journal of Dynamics and Differential Equations,
DOI: https://doi.org/10.1007/s10884-021-10083-5
We take into account a (1+N)-body problem, composed by a massive particle fixed at the
center of mass, which can be thought as a planet, surrounded by N equal satellites.
The satellites move in the three-dimensional space, under the force generated by a
potential of the form 1/r
α, where r is the distance between two
particles and α is a real number in [1,2). Units are chosen so that
Periodic orbits are found as minimizers of the
Lagrangian action. Indeed, imposing both
symmetry and tolopogical constraints on the space of loops, the coercivity is
recovered. Moreover, as the central mass goes to infinity, the Lagrangian action
Γ-converges
to the action of a Kepler problem, defined on a suitable loop set.
We impose different kind of symmetry
- the Z2N symmetry for 2N satellites,
- the Z2×Z2 symmetry for 4 satellites,
- the symmetry of the Tetrahedron, the Cube (or the Octahedron) and the Dodecahedron
(or the Icosahedron) for 12, 24 or 60 satellites, respectively.
Videos of the orbits we computed can be found in the links below.
For the last case (namely, the symmetry of the
Platonic polyhedra), we compute orbits in different homotopy classes
of the space minus the union of the rotation axes.
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, see Subsection 6.1 of the paper or, alternatively, 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 exponent α for which
the orbit has been computed.
Interested people can
contact me to obtain
more information about the representations of homotopy classes and the rotation groups,
or
also to get the initial conditions for the orbits.