Chaplygin ball over a fixed sphere: an explicit integration

Numerical Integration Over a Sphere*

An explicit class of min-max polynomials on the ball and on the sphere

Let Π d n+m−1 denote the set of polynomials in d variables of total degree less than or equal to n + m − 1 with real coefficients and let P(x), x = (x1, . . . , xd ), be a given homogeneous polynomial of degree n + m in d variables with real coefficients. We look for a polynomial p ∈ Π d n+m−1 such that P − p has least max norm on the unit ball and the unit sphere in dimension d, d ≥ 2, and cal...

Integration by Rbf over the Sphere

In this paper we consider numerical integration over the sphere by radial basis functions (RBF). After a brief introduction on RBF and spherical radial basis functions (SRBF), we show how to compute integrals of functions whose values are known at scattered data points. Numerical examples are given.

A Discretization of the Nonholonomic Chaplygin Sphere Problem

The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin s...

Explicit, parallel Poisson integration of point vortices on the sphere

Solutions to ideal fluid flow where the vorticity field is assumed as a sum of singular point vortices result in a Poisson system describing the motion of the vortex centres. We construct Poisson integration methods for these dynamics by splitting the Hamiltonian into its constituent vortex pair terms. From backward error analysis, the method is formally known to provide solutions to a modified...