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Register a new userRolling systems and their billiard limits
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Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip billiards resemble non-holonomic mechanical systems in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of non-holonomic (rolling) systems, in a way that is akin to how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian manifold.
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Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the body cone, along another, the space cone, with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean $mathbb{R}^3$, thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular $ 2 times 2 $ matrices in terms of rolling cones in 3-dimensional Minkowski space $mathbb{R}^{2,1}$ and the associated pseudo spherical curves, in either the hyperbolic plane $H^2$ or its Lorentzian analog $H^{1,1}$. In particular, this yields an apparently new geometric interpretation of Schrodingers (or Hills) equation $ ddot x + q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolic plane.
We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.
The dancing metric is a pseudo-riemannian metric $pmb{g}$ of signature $(2,2)$ on the space $M^4$ of non-incident point-line pairs in the real projective plane $mathbb{RP}^2$. The null-curves of $(M^4,pmb{g})$ are given by the dancing condition: the point is moving towards a point on the line, about which the line is turning. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves...) and pseudo-riemannian 4-dimensional conformal geometry (null-curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature,...). There is also an unexpected bonus: by applying a twistor construction to $(M^4,pmb{g})$, a $mathrm G_2$-symmetry emerges, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the dancing condition by a higher-order condition, expressed in terms of the osculating conic along a plane curve. The outcome is a correspondence between curves in the projective plane and its dual, a projective geometry analog of the more familiar rolling without slipping and twisting for a pair of riemannian surfaces.
We report an unexpected reverse spiral turn in the final stage of the motion of rolling rings. It is well known that spinning disks rotate in the same direction of their initial spin until they stop. While a spinning ring starts its motion with a kinematics similar to disks, i.e. moving along a cycloidal path prograde with the direction of its rigid body rotation, the mean trajectory of its center of mass later develops an inflection point so that the ring makes a spiral turn and revolves in a retrograde direction around a new center. Using high speed imaging and numerical simulations of models featuring a rolling rigid body, we show that the hollow geometry of a ring tunes the rotational air drag resistance so that the frictional force at the contact point with the ground changes its direction at the inflection point and puts the ring on a retrograde spiral trajectory. Our findings have potential applications in designing topologically new surface-effect flying objects capable of performing complex reorientation and translational maneuvers.
In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.
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