A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic

Collective symplectic integrators

We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector space with a Hamiltonian action with momentum map $J$ whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by $J$. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on $\mathfrak{so}(3)^*$. The method specializes in the case that a sufficiently large symmetry group acts on the fibres of $J$, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented

Symplectic integrators for spin systems

We present a symplectic integrator, based on the canonical midpoint rule, for classical spin systems in which each spin is a unit vector in $\mathbb{R}^3$. Unlike splitting methods, it is defined for all Hamiltonians, and is $O(3)$-equivariant. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields an integrable discretization of the reduced motion of a free rigid body

Geometry of discrete-time spin systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $T^*\mathbf{R}^{2n}$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with K\"ahler geometry, this provides another geometric proof of symplecticity.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.333

Hausdorff dimension of some groups acting on the binary tree

Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T. Abert and Virag showed that there exist finitely generated (but not necessarily level-transitive) subgroups of AutT of arbitrary dimension in [0,1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then show examples of 3-generated spinal groups which have transcendental Hausdroff dimension, and exhibit a construction of 2-generated groups whose Hausdorff dimension is 1.Comment: 10 pages; full revision; simplified some proof

Equilibrium composition between liquid and clathrate reservoirs on Titan

Hundreds of lakes and a few seas of liquid hydrocarbons have been observed by the Cassini spacecraft to cover the polar regions of Titan. A significant fraction of these lakes or seas could possibly be interconnected with subsurface liquid reservoirs of alkanes. In this paper, we investigate the interplay that would happen between a reservoir of liquid hydrocarbons located in Titan's subsurface and a hypothetical clathrate reservoir that progressively forms if the liquid mixture diffuses throughout a preexisting porous icy layer. To do so, we use a statistical-thermodynamic model in order to compute the composition of the clathrate reservoir that forms as a result of the progressive entrapping of the liquid mixture. This study shows that clathrate formation strongly fractionates the molecules between the liquid and the solid phases. Depending on whether the structure I or structure II clathrate forms, the present model predicts that the liquid reservoirs would be mainly composed of either propane or ethane, respectively. The other molecules present in the liquid are trapped in clathrates. Any river or lake emanating from subsurface liquid reservoirs that significantly interacted with clathrate reservoirs should present such composition. On the other hand, lakes and rivers sourced by precipitation should contain higher fractions of methane and nitrogen, as well as minor traces of argon and carbon monoxide.Comment: Accepted for publication in Icaru

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.Comment: v2: references added, a few typos corrected; v3: substantial revision, added three sections: (3) relation to parahoric bundles, (7) moduli spaces, (8) examples, 60 pages; v4: several corrections, more details included on the Hitchin-Kobayashi correspondence, some proofs shortened, 57 page

On the volatile enrichments and composition of Jupiter

Using the clathrate hydrates trapping theory, we discuss the enrichments in volatiles in the atmosphere of Jupiter measured by the \textit{Galileo} probe in the framework of new extended core-accretion planet formation models including migration and disk evolution. We construct a self-consistent model in which the volatile content of planetesimals accreted during the formation of Jupiter is calculated from the thermodynamical evolution of the disk. Assuming CO2:CO:CH4 = 30:10:1 (ratios compatible with ISM measurements), we show that we can explain the enrichments in volatiles in a way compatible with the recent constraints set from internal structure modeling on the total amount of heavy elements present in the planet.Comment: Accepted in ApJLetter