Effective bounds in E.Hopf rigidity for billiards and geodesic flows

In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the E.Hopf rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bounds for Burago-Ivanov theorem.Comment: 11 page

A remark on the number of invisible directions for a smooth Riemannian metric

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set K in these directions remain the same straight lines on exit. For example in the plane our construction gives three invisible directions. This is in contrast with billiard type obstacles where a very sophisticated example due to A.Plakhov and V.Roshchina gives 2 invisible directions in the plane and 3 in the space. We use reflection group of the root system An in order to make the directions of the roots invisible.Comment: 4

Shock formation for forced Burgers equation and application

We study the inviscid Burgers equation in the presence of spatially periodic potential force. We prove that for foliated initial value problem there are always solutions developing shocks in a finite time. We give an application of this result to a quasi-linear system of conservation laws which appeared in the study of integrable Hamiltonian systems with 1.5 degrees of freedom.Comment: 7 pages, no figure

Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.Comment: 12 p, revised version, corrected typo

Gutkin billiard tables in higher dimensions and rigidity

E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be either a sphere or to have a very special geometric properties. In 2-dimensional case we prove a rigidity result for Gutkin billiard tables. This is done with the help of a new generating function introduced recently for billiards in our joint paper with A.E. Mironov. A formula for this generating function in higher dimensions is found.Comment: 11

On Newton equations which are totally integrable at infinity

In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions $3$ and higher the following rigidity results holds true: If all the orbits in a neighborhood of infinity are action minimizing then the potential must be constant. This gives a generalization of the previous result \cite{B3}, where it was required all the orbits to be minimal. As a result we have the following application: Suppose that for the time-1 map of the Hamiltonian flow there exists a neighborhood of infinity which is filled by invariant Lagrangian tori homologous to the zero section. Then the potential must be constant. Remarkably, the statement is false for $n=1$ case and remains unknown to the author for $n=2$.Comment: 8

Rigidity for periodic magnetic fields

We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ``twisted'' version of the recent generalisation of Hopf's rigidity result obtained by Burago and Ivanov.Comment: 8 pages, no figure

CO/H2, C/CO, OH/CO, and OH/O2 in Dense Interstellar Gas: From High Ionization to Low Metallicity

We present numerical computations and analytic scaling relations for interstellar ion-molecule gas phase chemistry down to very low metallicities ($10^{-3} \times$ solar), and/or up to high driving ionization rates. Relevant environments include the cool interstellar medium (ISM) in low-metallicity dwarf galaxies, early enriched clouds at the reionization and Pop-II star formation era, and in dense cold gas exposed to intense X-ray or cosmic-ray sources. We focus on the behavior for H$_2$, CO, CH, OH, H$_2$O and O$_2$, at gas temperatures $\sim 100$ K, characteristic of a cooled ISM at low metallicities. We consider shielded or partially shielded one-zone gas parcels, and solve the gas phase chemical rate equations for the steady-state "metal-molecule" abundances for a wide range of ionization parameters, $\zeta/n$, and metallicties, $Z'$. We find that the OH abundances are always maximal near the H-to-H$_2$ conversion points, and that large OH abundances persist at very low metallicities even when the hydrogen is predominantly atomic. We study the OH/O$_2$, C/CO and OH/CO abundance ratios, from large to small, as functions of $\zeta/n$ and $Z'$. Much of the cold dense ISM for the Pop-II generation may have been OH-dominated and atomic rather than CO-dominated and molecular.Comment: Accepted for publication in MNRAS (with some improvements following referee report). 22 pages, 17 figure

Could Solar Radiation Pressure Explain 'Oumuamua's Peculiar Acceleration?

`Oumuamua (1I/2017 U1) is the first object of interstellar origin observed in the Solar System. Recently, \citet{Micheli2018} reported that `Oumuamua showed deviations from a Keplerian orbit at a high statistical significance. The observed trajectory is best explained by an excess radial acceleration $\Delta a \propto r^{-2}$, where $r$ is the distance of `Oumuamua from the Sun. Such an acceleration is naturally expected for comets, driven by the evaporating material. However, recent observational and theoretical studies imply that `Oumuamua is not an active comet. We explore the possibility that the excess acceleration results from Solar radiation pressure. The required mass-to-area ratio is $(m/A)\approx 0.1$ g cm$^{-2}$. For a thin sheet this requires a thickness of $\approx 0.3-0.9$ mm. We find that although extremely thin, such an object would survive an interstellar travel over Galactic distances of $\sim 5$ kpc, withstanding collisions with gas and dust-grains as well as stresses from rotation and tidal forces. We discuss the possible origins of such an object. Our general results apply to any light probes designed for interstellar travel.Comment: To be published in "The Astrophysical Journal Letters" on November 12, 201

New Semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form