Triangular Orbits Research Articles

New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in “Can the elliptic billiard still surprise us?” (2020) Math. Intelligencer 42(1): 6–17, some of which were generalized to N > 3. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.

On the Circumcenters of Triangular Orbits in Elliptic Billiard

On an elliptic billiard, we study the set of the circumcenters of all triangular orbits and we show that this is an ellipse. This article follows Romaskevich (L’Enseig Math 60:247–255, 2014), which proves the same result with the incenters, and Glutsyuk (Moscow Math J 14:239–289, 2014), which among others, introduces the theory of complex reflection in the complex projective plane. The result we present was found at the same time in Garcia (Amer Math Monthly 126:491–504, 2019). His proof uses completely different methods of real differential calculus.

Complex Caustics of the Elliptic Billiard

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. In the case of 4-periodic orbits, we get generically three caustics. We also give an upper bound on the number of caustics for orbits with a fixed number of sides, and explain how to compute its exact value.

On the incenters of triangular orbits on elliptic billiards

We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have suggested that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection.

On Odd-periodic Orbits in Complex Planar Billiards

The famous conjecture of V. Ya. Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper, we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: (1) triangular orbits; (2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invisibility conjecture.

Dynamics of a paramagnetic colloidal particle driven on a magnetic-bubble lattice

We present a theoretical study of the recently observed dynamical regimes of paramagnetic colloidal particles externally driven above a regular lattice of magnetic bubbles [P. Tierno, T. H. Johansen, and T. M. Fischer, Phys. Rev. Lett. 99, 038303 (2007)]. An external precessing magnetic field alters the potential generated by the surface of the film in such a way to either drive the particle circularly around one bubble, ballistically through the array, or in triangular orbits on the interstitial regions between the bubbles. In the ballistic regime, we observe different trajectories performed by the particles phase locked with the external driving. Superdiffusive motion, which was experimentally found bridging the localized and delocalized dynamics, emerge only by introducing a certain degree of randomness into the bubbles size distribution.

Low-temperature features of electron scattering in cadmium

The scattering rates nu (T) of various groups of electrons in cadmium have been measured down to 0.25 K using the radio-frequency size effect. For electrons on the third band lens of the fermi Surface nu (T) can be analyzed below about 2.5 K as the superposition of two terms, alpha T2 and beta T3. While the latter is associated with electron-phonon scattering the former component can be attributed to electron-electron scattering. On two triangular orbits in the first and second bands an apparently large T2 term is observed between 5 and 1.5 K. At lower temperatures nu (T) drops rapidly suggesting that this behaviour is due to electron-phonon scattering. An explanation for large T2 terms has been proposed by Lawrence et al. (1986) on the basis of Umklapp electron-phonon scattering together with the particular shape of the Fermi surface of cadmium. Simple calculations on these lines are shown to reproduce semi-quantitatively the observed variation of nu (T) and to aid in the interpretation of another second band orbit.

Influence of breakdown and spin - orbit coupling on the magnetoresistance of metals

A further exploration is made of the theory of Falicov, Pippard and Sievert explaining Stark's magnetoresistance data for zinc with a magnetic field aligned along the hexad axis, where the effects of spin splitting on the triangular orbits were observed. An alternative model for the spin splitting is suggested in terms of a spin-dependent phase shift upon Bragg reflection. Because the triangles are very small it is checked that the Pippard network model is in good agreement with a calculation using the effective Hamiltonian. The formal theory of spin precession along any arc of an orbit is also reconsidered along lines suggested by Roth. It seems that spin flip caused by spin-orbit coupling is unlikely in zinc, but a calculation of the effects of such a hypothetical spin flip on the magnetoresistance is made and the agreement with experiment is substantially improved.