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DCDS

First and second laws of thermodynamics are naturally associated, respectively, to contact and Hessian geometries.
In this paper we seek for a unique geometric setting that might account for both thermodynamic laws.
Using Riemannian metrics that are compatible with the contact structure, we prove that the Hessian
manifold of thermodynamic states cannot isometrically be embedded as Legendre submanifold of a contact
manifold. Well known fibrations suggest the nature of the obstruction for such embedding.

DCDS-S

In this paper we describe a symbolic dynamics for the rectangular four body problem by applying blow ups at total collisions and at infinity, studying the homoclinic or heteroclinic orbits obtained as intersection of corresponding two dimensional invariant submanifolds in a 3 dimensional energy level plus a convenient Poincaré map. With this tool we show the existence of a very rich dynamics and obtain the Main Theorem of this article. It gives the transition matrix for the symbolic dynamics of the images of conveniently chosen rectangles in the Poincaré section of the flow.

DCDS-S

We apply symbolic dynamics to continue our previous study of a
symmetric collinear restricted 3--body problem, where the equal
mass primaries perform elliptic collisions, while a third massless
body moves in the line between the primaries. Based on properties
of the homothetic orbit, which is a transversal heteroclinic orbit
beginning and ending in triple collision hyperbolic equilibria and
using a global Poincaré section, we describe the possible
itineraries of binary collisions an orbit can have.

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