Computing planar and spherical choreographies

An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix. New choreographies on the sphere are presented

Computer-assisted proofs in geometry and physics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references.In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies.by Gregory T. Minton.Ph.D

On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic

Paper mechanisms for sonic interaction

ABSTRACT Introducing continuous sonic interaction in augmented popup books enhances the expressive and performative qualities of movables, making the whole narrative experience more engaging and personal. The SaMPL Spring School on Sounding Popables explored the specific topic of paper-driven sonic narratives. Working groups produced several sketches of sonic interactions with movables. The most significant sketches of sounding popables are presented and analyzed

Basic gestures as spatiotemporal reference frames for repetitive dance/music patterns in samba and charleston

THE GOAL OF THE PRESENT STUDY IS TO GAIN BETTER insight into how dancers establish, through dancing, a spatiotemporal reference frame in synchrony with musical cues. With the aim of achieving this, repetitive dance patterns of samba and Charleston were recorded using a three-dimensional motion capture system. Geometric patterns then were extracted from each joint of the dancer's body. The method uses a body-centered reference frame and decomposes the movement into non-orthogonal periodicities that match periods of the musical meter. Musical cues (such as meter and loudness) as well as action-based cues (such as velocity) can be projected onto the patterns, thus providing spatiotemporal reference frames, or 'basic gestures,' for action-perception couplings. Conceptually speaking, the spatiotemporal reference frames control minimum effort points in action-perception couplings. They reside as memory patterns in the mental and/or motor domains, ready to be dynamically transformed in dance movements. The present study raises a number of hypotheses related to spatial cognition that may serve as guiding principles for future dance/music studies

Classification of positive elliptic-elliptic rotopulsators on Clifford tori

We prove that positive elliptic-elliptic rotopulsator solutions of the n-body problem in spaces of constant Gaussian curvature that move on Clifford tori of nonconstant size either lie on great circles, or project onto regular polygons. We additionally prove for the case that the configurations project onto regular polygons that all masses are equal and show that all these different types of positive elliptic-elliptic rotopulsator exist.</p

Determination of the doubly-symmetric periodic orbits in the restricted three-body problem and Hill's lunar problem

We review some recent progress on the research of the periodic orbits of the N-body problem,and propose a numerical scheme to determine the spatial doubly-symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill's lunar problem. Doubly symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden's method with a line-search, and the algorithm is reviewed. Some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of "Copenhagen problem" and the Sun-Jupiter-asteroid model are considered. New SDSPs are also numerically found in Hill's lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and most of the searched SDSPs are linearly stable.Comment: 29 pages, 11 figure