In this talk, I will focus on a class of 4d mechanical systems on a surface Σ
with a potential V a finite number of singularities $C := {c_1, . . . , c_n}$ of
the form
$$ V (q) \simeq C_i d(c_i, q)^{−α_i},\quad C_i > 0,\ \alpha_i ≥ 1, $$
and $q \in O(c_i)$. The first result I will present is about existence: there are periodic solutions
in many conjugacy classes of $\Pi_1(\Sigma, C)$. Using this fact, I will construct an
invariant set for the system which admits a semi-conjugation with the Bernoulli
shift. The second result I will discuss investigates when the semi-conjugation
is actually a conjugation and the invariant set constructed displays a chaotic
behaviour. This happens, for instance, under some negativity condition on the
curvature of $\Sigma$ and for large values of the energy. Much emphasis will be put
on the interplay between geometry, topology and variational methods. This is
a joint work with Gian Marco Canneori.
