Are the geometries of the first and second laws of thermodynamics compatible?
Gerardo Hernández Ernesto A. Lacomba
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 1113-1116 doi: 10.3934/dcds.2013.33.1113
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.
keywords: Boothby fibration classical thermodynamics. Riemannian contact geometry Hessian geometry
Oscillatory motions in the rectangular four body problem
Ernesto A. Lacomba Mario Medina
Discrete & Continuous Dynamical Systems - S 2008, 1(4): 557-587 doi: 10.3934/dcdss.2008.1.557
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.
keywords: symbolic dynamics Poincaré section. Four body problem invariant manifolds
Symbolic dynamics of the elliptic rectilinear restricted 3--body problem
Samuel R. Kaplan Ernesto A. Lacomba Jaume Llibre
Discrete & Continuous Dynamical Systems - S 2008, 1(4): 541-555 doi: 10.3934/dcdss.2008.1.541
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.
keywords: collinear restricted problem 3-body problem symbolic dynamics.

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