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### Session S03 - Geometric and Variational Methods in Celestial Mechanics

Monday, July 12, 14:15 ~ 14:55 UTC-3

## Symbolic dynamics for the anisotropic $N$-centre problem at negative energies

### Susanna Terracini

#### University of Turin, Italy   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak54a48d742afc69d7c8ce5d4073213025').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy54a48d742afc69d7c8ce5d4073213025 = 's&#117;s&#97;nn&#97;.t&#101;rr&#97;c&#105;n&#105;' + '&#64;'; addy54a48d742afc69d7c8ce5d4073213025 = addy54a48d742afc69d7c8ce5d4073213025 + '&#117;n&#105;t&#111;' + '&#46;' + '&#105;t'; var addy_text54a48d742afc69d7c8ce5d4073213025 = 's&#117;s&#97;nn&#97;.t&#101;rr&#97;c&#105;n&#105;' + '&#64;' + '&#117;n&#105;t&#111;' + '&#46;' + '&#105;t';document.getElementById('cloak54a48d742afc69d7c8ce5d4073213025').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy54a48d742afc69d7c8ce5d4073213025 + '\'>'+addy_text54a48d742afc69d7c8ce5d4073213025+'<\/a>';

The 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: $\ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j)$

In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-\alpha_j$: $V_j(x)=|x|^{-\al_j}V_j\left(\frac{x}{|x|}\right).$ 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.

Joint work with Vivina Barutello (University of Turin) and Gian Marco Canneori (University of Turin).