This page contains additional material from the paper
L. Asselle, M. Fenucci, A. Portaluri: Bifurcations of balanced
configurations for the Newtonian n-body problem in R4
We take into account the Newtonian n-body problem with masses
m1,...,mn, and we denote with q=(q1,...,qn) in
Rnd their positions. The masses are at a normalized S-balanced configuration if
q satisfies the algebraic system of equations
∇ U(q) + U(q)Ŝ(s)Mq = 0,
where U is the Newtonian potential,
M = diag(m1Idd,...,mnIdd) is the mass matrix,
and Ŝ = diag(S,...,S), S = diag(s1,...,sd).
Equal masses
m1 = m2 = m3 = 1
Figure 1: the branch connecting the Euler collinear configuration to the
Lagrange equilateral configuration, originating from the bifurcation point at s=2.4.
lol
One unitary mass and two equal smaller masses
m1 = 1, m2 = m3 = 0.9
Figure 2: the branch connecting the Euler collinear configuration with
m2 placed in the middle to the Euler collinear configuration with m3
placed in the middle.
Figure 3: the connection between the Euler collinear configuration with
m2 placed in the middle and the Lagrange equilateral configuration. This
connection is found following the non-trivial branch originating from the Euler central
configutaion, and a second branch originating from the turning point.
m1 = 1, m2 = m3 = 0.01
Figure 4: same as Figure 2, but with smaller masses.
Figure 5: same as Figure 3, but with smaller masses.
With the unitary mass in the center
Figure 6: the non-trivial branch of the Euler collinear configuration, in
the case that the larger mass is in the middle. The behaviour is similar to the case of
Figure 1.
Figure 7: same as Figure 6, but with smaller masses.
Two unitary masses and a smaller one
m1 = m2 =1, m3 = 0.9 and
m1 = m2 =1, m3 = 0.01
Figure 8: the non-trivial branch originating from the Euler collinear
configuration. As s goes to infinity, the masses approach a limit configuration.
Figure 9: same as Figure 8, but with smaller masses.
With the small mass in the center
Figure 10: the non-trivial branch in the case the smaller mass is placed in
the center. Also here, this branch connects the Euler collinear configuration to the
Lagrange equilateral configuration.
Figure 11: same as Figure 10, but with smaller masses.
Figure 12: the non-trivial branch in the case of three different masses. The
behaviour is similar to the case of Figure 8.
Figure 13: the non-trivial branch in the case of three different masses,
with the smallest one placed in the center of the Euler collinear configuration. The
behaviour is similar to the case of Figure 1.
Figure 14:
References
L. Asselle, A. Portaluri: 2020. Morse theory for
S-balanced configurations in the Newtonian n-body problem, arXiv preprint.
R. Moeckel: 2014. Central configurations, Scholarpedia.
