The 2D Euler equations is the basic example of fluid models for which a microcanical measure can be constructed from first principles. This measure is defined through finitedimensional approximations and a limiting procedure. Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz's algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory). We found full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz's algorithm, a first-order phase transition never observed previously, and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and contrasting usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with less symmetries) when energy is increased. We explain this paradox. arXiv:1210.4351v3 [cond-mat.stat-mech]

Context. The dwarf planet Ceres and the asteroid Vesta have been studied by the Dawn space mission. They are the two heaviest bodies of the main asteroid belt and have different characteristics. Notably, Vesta appears to be dry and inactive with two large basins at its south pole. Ceres is an ice-rich body with signs of cryovolcanic activity. Aims. The aim of this paper is to determine the obliquity variations of Ceres and Vesta and to study their rotational stability. Methods. The orbital and rotational motions have been integrated by symplectic integration. The rotational stability has been studied by integrating secular equations and by computing the diffusion of the precession frequency.Results. The obliquity variations of Ceres over [−20 : 0] Myr are between 2 and 20 • and the obliquity variations of Vesta are between 21 and 45 • . The two giant impacts suffered by Vesta modified the precession constant and could have put Vesta closer to the resonance with the orbital frequency 2s 6 − s V . Given the uncertainty on the polar moment of inertia, the present Vesta could be in this resonance where the obliquity variations can vary between 17 and 48 • . Conclusions. Although Ceres and Vesta have precession frequencies close to the secular orbital frequencies of the inner planets, their long-term rotations are relatively stable. The perturbations of Jupiter and Saturn dominate the secular orbital dynamics of Ceres and Vesta and the perturbations of the inner planets are much weaker. The secular resonances with the inner planets also have smaller widths and do not overlap, contrary to the case of the inner planets.

We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments of inertia of the integrated body, we can construct symplectic dedicated integrators with fewer stages than in the general case for a splitting in three parts of the Hamiltonian. We perform numerical tests to compare the developed dedicated 4th-order integrators to the existing reference integrators for the water molecule. We also estimate analytically the accuracy of these new integrators for the set of the rigid bodies and conclude that they are more accurate than the existing ones only for very asymmetric bodies.

The motion of a satellite can experience secular resonances between the precession frequencies of its orbit and the mean motion of the host planet around the star. Some of these resonances can significantly modify the eccentricity (evection resonance) and the inclination (eviction resonance) of the satellite. In this paper, we study in detail the secular resonances that can disturb the orbit of a satellite, in particular the eviction-like ones. Although the inclination is always disturbed while crossing one eviction-like resonance, capture can only occur when the semi-major axis is decreasing. This is, for instance, the case of Phobos, the largest satellite of Mars, that will cross some of these resonances in the future because its orbit is shrinking owing to tidal effects. We estimate the impact of resonance crossing in the orbit of the satellite, including the capture probabilities, as a function of several parameters, such as the eccentricity and the inclination of the satellite, and the obliquity of the planet. Finally, we use the method of the frequency map analysis to study the resonant dynamics based on stability maps, and we show that some of the secular resonances may overlap, which leads to chaotic motion for the inclination of the satellite.