Chaplygin Systems Research Articles

Noether theorem and its inverse for nonstandard generalized Chaplygin systems

Noether theorem and its inverse for nonstandard generalized Chaplygin systems

Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres

We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe an invariant measure and provide examples of $SO(n-2)$--symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional $SO(2)$--symmetry we prove that the obtained magnetic geodesic flows on the sphere $S^{n-1}$ are integrable. In particular, we introduce the generalized Demchenko case in $\mathbb R^n$, where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres $S^{n-1}$ endowed with the round-sphere metric, under the influence of a homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for $n>3$ was not studied before. We perform explicit integrations in elliptic functions of the systems for $n=3$ and $n=4$, and provide the case study of the solutions in both situations.

Shortest and straightest geodesics in sub-Riemannian geometry

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and generalization of sub-Riemannian geodesics and discuss interrelations between different definitions. For Chaplygin transversally homogeneous sub-Riemannian manifold Q, we prove that straightest geodesics (defined as geodesics of the Schouten partial connection) coincide with shortest geodesics (defined as the projection to Q of integral curves (with trivial initial covector) of the sub-Riemannian Hamiltonian system). This gives a Hamiltonization of Chaplygin systems in non-holonomic mechanics.We consider a class of homogeneous sub-Riemannian manifolds, where straightest geodesics coincide with shortest geodesics, and give a description of all sub-Riemannian symmetric spaces in terms of affine symmetric spaces.

The geometry of nonholonomic Chaplygin systems revisited

We consider nonholonomic Chaplygin systems and associate to them a tensor field on the shape space, that we term the gyroscopic tensor, and that measures the interplay between the non-integrability of the constraint distribution and the kinetic energy metric. We show how this tensor may be naturally used to derive an almost symplectic description of the reduced dynamics. Moreover, we express sufficient conditions for measure preservation and Hamiltonisation via Chaplygin’s reducing multiplier method in terms of the properties of this tensor. The theory is used to give a new proof of the remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by Fedorov and Jovanović in Fedorov and Jovanović (2004 J. Nonlinear Sci. 14 341–81); Fedorov and Jovanović (2009 Regul. Chaotic Dyn. 14 495–505).

A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.

Cartan meets Chaplygin

In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan?s example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin?s marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal. 1.

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.

Noether theorem for generalized Chaplygin system on time scales

In this paper, the Noether theorem for generalized Chaplygin system on time scales is proposed and studied. The generalized Chaplygin formula for nonholonomic system on time scales is derived. The Noether theorems for generalized Chaplygin system on time scales are established, and two special cases of the Noether theorems for continuous and discrete generalized Chaplygin systems are given. Finally, two examples are given to illustrate the applications of the results.

The initial motions for holonomic and nonholonomic mechanical systems

Whittaker first put forward a new approach, called the initial motions, to solve the differential equations of motion aimed at holonomic systems. Since most of the differential equations of motion for mechanical systems are nonlinear ordinary ones, which are difficult to find the analytic solutions. Fortunately, the concept of initial motions can manage these situations and study its subsequent motions. This work is devoted to discuss and investigate the initial motions for mechanical systems, particularly for nonholonomic systems. First, the differential equations for holonomic systems are formulated, and the formulation and solution of initial motions of the systems are proposed. Second, the differential equations of motion for nonholonomic systems are established, based on the new method of initial motions to obtain the initial values of high-order derivatives of generalized velocities, the formulation and solution of initial motions are introduced in the general nonholonomic systems and Chaplygin systems. The methods and results obtained are illustrated by a number of classical examples, both for holonomic and nonholonomic systems.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Bäcklund transformations relating different Hamilton-Jacobi equations

We discuss one of the possible finite-dimensional analogues of the general Backlund transformation relating different partial differential equations. We show that different Hamilton-Jacobi equations can be obtained from the same Lax matrix. We consider Henon-Heiles systems on the plane, Neumann and Chaplygin systems on the sphere, and two integrable systems with velocity-dependent potentials as examples.

Geometry of non-holonomic diffusion

We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For G -Chaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard.

Simultaneous separation for the Neumann and Chaplygin systems

The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separations for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. It allows us to say about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.

A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid

The paper is concerned with the topological analysis of the Chaplygin integrable case in the dynamics of a rigid body in a fluid. A full list of the topological types of Chaplygin systems in their dependence on the energy level is compiled on the basis of the Fomenko-Zieschang theory. An effective description of the topology of the Liouville foliation in terms of natural coordinate variables is also presented, which opens a direct way to calculating topological invariants. It turns out that on all nonsingular energy levels Chaplygin systems are Liouville equivalent to the well-known Euler case in the dynamics of a rigid body with fixed point.Bibliography: 23 titles.

Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton–Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.

О неголономных системах Веселовой и Чаплыгина

О неголономных системах Веселовой и Чаплыгина

A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (II)

We show that the sub-Riemannian connection ∇, constructed in the first part of our study [2], is the one which enables us to express the Lagrange–d'Alembert equations for a nonholonomic mechanical system in a form that is similar to the Newton's equations from classical mechanics (cf. (2.21)). Also, we define the generalized Chaplygin systems and show that the Lagrange–d'Alembert equations for these systems depend only on some horizontal variables. Two examples are given to illustrate our theory in comparison with other studies.

Nonholonomic Hamilton–Jacobi theory via Chaplygin Hamiltonization

We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton–Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton–Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.

The Weitzenböck connection and time reparameterization in nonholonomic mechanics

We show that the torsion of the Weitzenböck connection is responsible for the fictitious pseudogyroscopic force experienced by a general mechanical system in a noncoordinate moving frame. In particular, we show that for the class of mechanical systems subjected to nonintegrable constraints known as non-abelian nonholonomic Chaplygin systems, the constraint reaction force directly depends on this torsion. For these Chaplygin systems, we show how this torsional force can in some cases be removed by an appropriate choice of frame depending on a multiplier f(q), linking these results to the process of Chaplygin Hamiltonization through time reparameterization. Lastly, we show that the cyclic symmetries of f in some cases lead to the existence of momentum conservation laws for the original nonholonomic system and illustrate the results through several examples.

Controlling nonholonomic Chaplygin systems

In this paper we deal with the problem of controlling some Chaplygin systems in the framework of the vakonomic approach for nonholonomic systems. Equations of motion for these systems are obtained which contain a free parameter that permits to control the system. It is show that given a prescribed path it is possible to determine the parameter of control which inserted in the equations of motion compel the trajectory of the system to follow the input function.