Routh Sphere Research Articles

On the Nonholonomic Routh Sphere in a Magnetic Field

This paper is concerned with the motion of an unbalanced dynamically symmetric sphere rolling without slipping on a plane in the presence of an external magnetic field. It is assumed that the sphere can consist completely or partially of dielectric, ferromagnetic, superconducting and crystalline materials. According to the existing phenomenological theory, the analysis of the sphere’s dynamics requires in this case taking into account the Lorentz torque, the Barnett-London effect and the Einstein-de Haas effect. Using this mathematical model, we have obtained conditions for the existence of integrals of motion which allow one to reduce integration of the equations of motion to a quadrature similar to the Lagrange quadrature for a heavy rigid body.

Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball

The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th birthday we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has a centre of mass offset from the geometric centre, but it has an axis of symmetry through both these points, and equal moments of inertia about all axes orthogonal to the symmetry axis. There are three constants of motion: the total energy and two quantities involving the angular momenta. It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether’s invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation. As a further demonstration of the power and utility of this method, we find the Noetherian symmetries and corresponding integrals for a system introduced recently, the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether’s theorem.

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from Garc?a-Naranjo [21] and Garc?a-Naranjo and Marrero [22], we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin?s reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.

On Noisy Extensions of Nonholonomic Constraints

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange–d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with “rolling ball type” constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.

On the Routh sphere problem

We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra e*(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

О сфере Рауса

О сфере Рауса

Poisson structures for reduced non-holonomic systems

Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk, the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.

What is a completely integrable nonholonomic dynamical system?

We compare the geometry of a toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. We illustrate their differences using the following examples: the nonholonomic oscillator, Chaplygin's skate, Routh's sphere and the rolling oblate ellipsoid of revolution.

Routh's sphere

In this paper we show that the integral map of Routh's sphere has monodromy when the sphere becomes gyroscopically unstable. This uses the non-Hamiltonian monodromy theorem.