On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D’Alembert–Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results.     

The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler — Jacobi — Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.     

Hirota-Kimura type discretization of the classical nonholonomic Suslov problem

We constructed Hirota-Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In n-dimensional case we give discrete equations.      

A Variational Integrator for the Chaplygin–Timoshenko Sleigh

The paper introduces a mechanically inspired nonholonomic integrator for numerical simulation of the dynamics of a constrained geometrically exact beam that is a field-theoretic analogue of the Chaplygin sleigh. The integrator features an exact constraint preservation, an excellent numerical energy conservation throughout a large number of iterations, while avoiding the use of unnecessary Lagrange multipliers. Simulations of the dynamics of the constrained beam reveal typical for nonholonomic system’s behavior, such as motion reversals and locomotion generation.

     

刚体动力学方程的一个辛积分方法

针对四元数和对应广义动量表示的刚体定点动力学方程,利用一种位移格式的微分—代数方程积分方案,实现了非独立广义动量表示的拉格朗日方程的辛积分算法．数值实验显示该算法具有精度高和保持系统守恒量的特点．更为重要的是,广义动量表示的拉格朗日方程较之传统形式的拉格朗日方程在辛积分中表现出独特的优越性．     

El concepto de hiperviga como generalización de los modelos unidimensionales en la estática de piezas alargadas

This paper is concerned with the general formulation of linear problems in rod elastostatics, and with the identification of their common formal and structural features, valid for every kinematical hypothesis. The generalized variables (section forces and generalized deformations) defining the 1-dimensional model are introduced in a consistent and natural way, through a convenient factorization of the density of complementary potential energy, for every kinematical constraint which can be expressed as a linear combination of the generalized displacements. The identification of this complementary energy function with the Hamiltonian functional of analytical mechanics allows a systematic procedure to construct the equations which rule this class of problems. In this frame, the main target is to establish the required conditions to write the rod equilibrium equations in a purely statical form (with no interplay of kinematical variables). We primarily conclude that this is possible when the kinematics of the cross-sections is constrained to a rigid body movement. As a consequence, the concept of hyperbeam is proposed in order to define those models with deformation modes beyond rigid body movements of the cross section: in them, section forces and generalized displacements are coupled in the equilibrium equations. This is related to the idea of local static indeterminacy (hyperstaticity) and justifies the new name.     

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.     

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

The heat and moisture transfer balance theory of garment simulation

To establish the human body model to analyze the heat and moisture transfer on body surface, a new explicit definition of rational L-recursion surface is given and the L-recursion surfaces, in Grassmann spaces, are constructed by using blossom method of the homogeneous normal pyramid form. Based on our human body model, the balance theory of garment simulation, the heat and moisture transfer balance equations, called ICAD-balance equations are obtained. The balance theory of garment simulation integrally studies the complex system of human body–fabric–environment. At the same time, the method of obtaining the heat and moisture transfer balance equations is also based on the mass conservation law, the energy conservation law and the Fish law of capillarity. A finite volume method is employed to solve the ICAD-balance equations.     

Analysis of a coupled system of kinetic equations and conservation laws: Rigorous derivation and existence theory via defect measures

In this paper we introduce a coupled system of kinetic equations of B.G.K. type and then study its hydrodynamic limit. We obtain as a consequence the rigorous derivation and existence theory for a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit. The latter is a particular case of the coupled system of Boltzmann and Euler equations. A fundamental element in this study is the rigorous derivation and justification of the interface conditions between the kinetic model and its hydrodynamic conservation laws limit, which is obtained using a new regularity theory introduced herein.

     

Constraint Control of Nonholonomic Mechanical Systems

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov’s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction \(\varvec{\xi }\). We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov’s problem for the rotation group \(\textit{SO}(3)\). We show that it is possible to control the system using the constraint \(\varvec{\xi }(t)\) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.     

Rigid Body Dynamics Using Clifford Algebra

In this paper the dynamics of rigid bodies is recast into a Clifford algebra formalism. Specifically, the algebra Cℓ(0, 6, 2), is used and it is shown how velocities, momenta and inertias can be represented by elements of this algebra. The equations of motion for a rigid body are simply derived by differentiating the momentum of the body.     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

大气环流闭合方程组的总能量守恒格式及全球预报误差的严格估计

本文讨论大气环流闭合方程组,由于同时考虑了热传导效应,内摩擦效应及表达动能向内能转化的耗散项,因此符合总能量守恒律,文中对这一方程组建立了加权平均守恒型差分格式,并证明当选择最优参数时,它满足离散形式的总能量守恒律,通常的二次守恒格式是其次优的情况,文中还综合应用了Jessen不等式,Hardy不等式等等,从而严格证明了在一定条件下,存在t₀>0,当t     

New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.     

New Families of Conservative Systems on S2 Possessing an Integral of Fourth Degree in Momenta

There is a well-known example of an integrable conservative system on S2, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. In this paper we propose new families of examples of conservative systems on S2 possessing an integral of fourth degree in momenta.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

Energy-Preserving Integrators Applied to Nonholonomic Systems

We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple \(({{\mathcal {D}}}^*, \varPi , \mathcal {H})\), where \({{\mathcal {D}}}^*\) is the dual of the vector bundle determined by the nonholonomic constraints, \(\varPi \) is an almost-Poisson bracket (the nonholonomic bracket) and \( \mathcal {H}: {{\mathcal {D}}}^*\rightarrow \mathbb {R}\) is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.

     

Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems

This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.