On some important problems in analytical dynamics of non-holonomic systems

By using deductive method,Chetaev condition is derived in this paper.We pointout that the processes of variation and differentiation are not permutable innon-holonomic dynamics is a misunderstanding.The paper gives two classical relationsof non-holonomic systems and points out integral variational principles can be appliedin non-holonomic systems.     

The planning of optimal motions of non-holonomic systems

Nonlinear Dynamics - A new method to the planning of optimal motions of the non-holonomic systems is presented. It is based on a non-classical formulation of the Pontryagin Maximum Principle given...     

On the elastic stability of static non-holonomic systems

This paper presents a new formulation for the elastic stability of static non-holonomic structural systems. The theory is developed within the tradition of discrete (or discretized) systems written in terms of a set of generalized coordinates and control parameters. The non-holonomic conditions are written as constraint functions. The formulation employs a Lagrangian functional in terms of the total potential energy, the constraint functions and multipliers. Critical states are identified and the solution is next expanded by regular perturbations. This allows to establish a classification of critical states and identify the initial postcritical behavior. This solution is valid provided that there is no change in the active constraints of the system. The paper presents a mathematical analysis of the critical condition, and concludes with simple examples of two degree-of-freedom systems previously investigated by other authors.     

Variational problem of general displacements in non-holonomic systems

In this paper, the difference of definitional regions of two kinds of q is discussed in analytical dynamics of non-holonomic systems. Its mechanical reason is given. Finally, some difficult problems which have remained unanswered up to now are solved.     

Extension of Whittaker equations to non-holonomic mechanical systems

In 1904, using the energy integral Whittaker studied the reduction of a dynamical problem to a problem with fewer degrees of freedom for the holonomic conservative systems and obtained the Whittaker equation^[1].In this article, Whittaker equations are extended to non-holonomic systems and the generalized Whittaker equations are obtained. And then these equations are transformed into Kiel-sen’s form.Finally an example is given.     

Constructing chaotic systems with conditional symmetry

Nonlinear Dynamics - Asymmetric dynamical systems sometimes admit a symmetric pair of coexisting attractors for reasons that are not readily apparent. This phenomenon is called conditional symmetry...     

THE STATIONARY VALUE PROPERTY OF HAMILTON'S PRINCIPLE IN NON-HOLONOMIC SYSTEMS

I.IntroductionHertzintroducedtheconceptofnon-holonomics}stems(NHS)in1894ti].HeconsideredthatHamiltoll'sprinciple\"asnotsatisfiedforNHSandinparticulardidnotyieldLagrangeequationswithundeterminedmultipliers.Forthisfeason.Hblderl=],Suslovi3]andVoronetzl']dev…     

Calculation of some magnetic and electric fields with cylindrical symmetry

Summary It is assumed that the magnetic and electric fields have cylindrical symmetry and at the same time mirror symmetry with respect to a median plane. The field strength in the median plane is given. The problem is to calculate the field at an arbitrary point in space, the equipotential surfaces, and the form of the polepieces of magnets. Solutions are given in the form of power expansions. Analytical solutions are given when the field strength in the median plane as a function of the radius is a polynomial.     

Forces associated with non-linear non-holonomic constraint equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by non-holonomic equations that are inherently non-linear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known non-holonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as non-holonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor first to form the former, larger set and subsequently perform matrix multiplications.     

On the Noether symmetry and Lie symmetry of mechanical systems

The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates. The Lie symmetry is an invariance of the differential equations of motion under the transformations. In this paper, the relation between these two symmetries is proved definitely and firstly for mechanical systems. The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold. The project supported by the National Natural Science Foundation of China (19972010)     

A unified symmetry of lagrangian systems

In this paper, the definition and the criterion of a unified symmetry are presented. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, are obtained. Some examples are given to illustrate the application of the results. The project supported by the National Natural Science Foundation of China (10272021)     

The solution of Navier equations of classical elasticity using Lie symmetry groups

The analytical solutions of axially-symmetric Navier equations in classical elasticity are found by applying Lie group theory. We investigate two different systems of partial differential equations corresponding elastostatics and elastodynamics problems, and find similarity solutions of both cases by solving the reduced system of ordinary differential equations which have fewer independent variables. As an example of the elastostatics case, the displacements and stress components are obtained for porous, polymeric foam material by using similarity solutions.     

New planforms in systems of partial differential equations with Euclidean symmetry

Dionne & Golubitsky [10] consider the classification of planforms bifurcating (simultaneously) in scalar partial differential equations that are equivariant with respect to the Euclidean group in the plane. In particular, those planforms corresponding to isotropy subgroups with one-dimensional fixed-point space are classified.Many important Euclidean-equivariant systems of partial differential equations essentially reduce to a scalar partial differential equation, but this is not always true for general systems. We extend the classification of [10] obtaining precisely three planforms that can arise for general systems and do not exist for scalar partial differential equations. In particular, there is a class of one-dimensional pseudoscalar partial differential equations for which the new planforms bifurcate in place of three of the standard planforms from scalar partial differential equations. For example, the usual rolls solutions are replaced by a nonstandard planform called anti-rolls. Scalar and pseudoscalar partial differential equations are distinguished by the representation of the Euclidean group.     

Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry

Imperfection sensitivity of a hilltop branching point occurring as a coincidence of a limit point and a double bifurcation point of a finite-dimensional, elastic, conservative system equivariant to the dihedral group is investigated. In the neighborhood of this point, the potential is expanded into a power series of independent state variables, loading parameter and imperfection magnitude. The form of the expansion is determined through exploitation of dihedral-group symmetry. For the perfect system, the hilltop branching point and bifurcated paths are shown to be all unstable. For an imperfect system, equilibrium paths in general break into a series of paths: including fundamental, complementary and aloof paths. The imperfection sensitivity laws for maximum (critical) points of loading on these paths are obtained as a novel finding of this paper. Critical points on the fundamental and complementary paths enjoy a piecewise linear law, which is less severe than a one-half or two-thirds power law for the double bifurcation point. By contrast, maximum points on aloof paths suffer more severe sensitivity. The hilltop branching point thus displays complex system of imperfection sensitivities. As numerical examples, imperfection sensitivity of simple structural models with the hilltop point is investigated to ensure the validity of the present formulation.     

Form invariance, Noether symmetry and Lie symmetry of Hamiltonian systems in phase space

For the holonomic and non-holonomic Hamiltonian systems in phase space, the definitions and criterions of the form invariance of both Hamilton and generalized Hamilton canonical equations are given. The relations among the form invariance, Noether symmetry and Lie symmetry are studied. Two examples are given to illustrate these results.     

Distributed finite-time control for coordinated circumnavigation with multiple non-holonomic robots

Nonlinear Dynamics - The finite-time coordinated circumnavigation problem for multiple non-holonomic robots is investigated in this paper. All robots are required to be evenly circled a target, and...     

Dirac reduction of binary asteroid systems with symmetry and stability of relative equilibria

In this paper, we study symmetry reduction for a binary asteroid system modeled by a rigid body and a particle. In particular, we demonstrate how translational and rotational symmetry reduction appeared in the binary asteroid system can be carried out in the context of Dirac reduction by stages and with the associated reduction of implicit Hamiltonian systems. Then we investigate stability of relative equilibria of the asteroid pair and show stability regions by using the energy-momentum method. Lastly, we illustrate some numerical simulations for stable and unstable orbits near from relative equilibria of the Collinear and T configurations.