Abstract
AbstractThe planar N-centre problem describes the motion of a particle moving in the plane under the action of the force fields of N fixed attractive centres: $$\begin{aligned} \ddot{x}(t)=\sum _{j=1}^N\nabla V_j(x(t)-c_j). \end{aligned}$$
x
¨
(
t
)
=
∑
j
=
1
N
∇
V
j
(
x
(
t
)
-
c
j
)
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In this paper we prove symbolic dynamics at slightly negative energy for an N-centre problem where the potentials $$V_j$$
V
j
are positive, anisotropic and homogeneous of degree $$-\alpha _j$$
-
α
j
: $$\begin{aligned} V_j(x)=|x|^{-\alpha _j}V_j\left( \frac{x}{|x|}\right) . \end{aligned}$$
V
j
(
x
)
=
|
x
|
-
α
j
V
j
x
|
x
|
.
The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis’ functional. Compared with the classical N-centre problem with Kepler potentials, a major difficulty arises from the lack of a regularization of the singularities. We will consider both the collisional dynamics and the non collision one. Symbols describe geometric and topological features of the associated trajectory.
Funder
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
Reference42 articles.
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3. Barutello, V., Ferrario, D.L., Terracini, S.: On the singularities of generalized solutions to $$n$$-body-type problems. Int. Math. Res. Not. IMRN, pages Art. ID rnn 069, 2008
4. Barutello, V., Hu, X., Portaluri, A., Terracini, S.: An index theory for asymptotic motions under singular potentials. Adv. Math. 370, 107230, 57, 2020
5. Barutello, V., Secchi, S.: Morse index properties of colliding solutions to the $$N$$-body problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 539–565, 2008
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