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Register a new userThe Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids
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Thomas Machon
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We show the Godbillon-Vey invariant arises as a `restricted Casimir invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
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If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.
We study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection. We assume that the three constants of motion are the Hamiltonian, $H$, with the standard form of a kinetic part plus a potential dependent on the position only, the $z$-component of the angular momentum, $L$, and a Hamiltonian-like constant, $widetilde H$, for which the kinetic part is quadratic in the momenta. We find the explicit form of these potentials compatible with complete integrability. The classical equations of motion, written in terms of two arbitrary potential functions, is separated in oblate spheroidal coordinates. The quantization of such systems leads to a set of two differential equations that can be presented in the form of spheroidal wave equations.
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the present context takes the form $0to Jto Ato Bto 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(tau_{GV}^r,sigma_{GV})$ for the pair $Ato B$; $tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, `a la Melrose, of the usual Godbillon-Vey cyclic cocycle $tau_{GV}$; $sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $tau_{GV}$ and a specific 1-cyclic cocycle (Roes 1-cocycle). We call $sigma_{GV}$ the eta cocycle associated to $tau_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class $Ind (D,D^partial)in K_* (A,B)$ and establishing the equality <Ind (D),[tau_{GV}]>=<Ind (D,D^partial), [tau^r_{GV}, sigma_{GV}]>$. The Godbillon-Vey eta invariant $eta_{GV}$ is obtained through the eta cocycle $sigma_{GV}$.
We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model, the Schonmann projection. These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function of the two-sided boundary conditions are not continuous. We prove that they are g-measures, which means that their conditional probabilities have a continuous dependence on one-sided boundary condition.
The recent researches in non equilibrium and far from equilibrium systems have been proved to be useful for their applications in different disciplines and many subjects. A general principle to approach all these phenomena with a unique method of analysis is required in science and engineering: a variational principle would have this fundamental role. Here, the Gouy-Stodola theorem is proposed to be this general variational principle, both proving that it satisfies the above requirements and relating it to a statistical results on entropy production.
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