This paper introduces dual field-theoretic Hamel’s integrator in the form of discrete mechanical systems that utilize dynamic variables measured relative to moving frames. As in the continuous setting, the use of moving frames results in a flexible and overall more efficient formalism. The integrator is derived in a systematic way using the formalism of composition maps. It is then utilized for numerical integration of geometrically exact beams.

Nonholonomic systems are mechanical systems with ideal linear velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper surveys discrete-time nonholonomic mechanics with applications to numerical integration of nonholonomic systems. This includes an exposition of key elements of discrete mechanics, discrete Lagrange – d’Alembert principle, and exact nonholonomic integrators on vector spaces. Exact variational integrators were introduced and exposed in the context of Lagrangian mechanics by Marsden and West. These integrators sample the trajectories of mechanical systems and are useful for developing practical mechanical integrators.

Nonholonomic systems are mechanical systems with ideal velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper reviews infinite-dimensional and field-theoretic nonholonomic systems as well as Hamel’s formalism for these settings.

Exact variational integrators were exposed in the context of Lagrangian mechanics in Marsden and West (2001). These integrators sample the trajectories of holonomic mechanical systems and are useful for developing practical mechanical integrators. This paper introduces an exact variational integrator for Hamel’s equations, which are interpreted as a noncanonical form of Hamilton’s equations. This exact Hamel integrator is then adopted for a systematic construction of low-order constraint-preserving integrators for nonholonomic mechanical systems.

This paper introduces Hamel’s formalism for classical field theories with the goal of analyzing the dynamics of continuum mechanical systems with velocity constraints. The developed formalism is utilized to prove the existence and uniqueness of motions of an infinite-dimensional generalization of the Chaplygin sleigh, a canonical example of nonholonomic dynamics. The formalism is very flexible and, for mechanical field theories, includes the Eulerian and Lagrangian representations of continuum mechanics as special cases. It also provides a useful approach to analyzing symmetry reduction.

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.

In this paper, we introduce Hamel’s formalism for infinite-dimensional mechanical systems and in particular consider its applications to the dynamics of nonholonomically constrained systems. This development is a nontrivial extension of its finite-dimensional counterpart. The analysis is applied to several continuum mechanical systems of interest, including coupled systems and systems with infinitely many constraints.

This paper reviews recent results on the extension of Hame?s formalism to infinite-dimensional mechanical systems and to variational integrators. Of a particular interest are applications to the dynamics and numerical integration of systems with velocity constraints.

In this chapter we consider the relationship between the classical inverse problem of the calculus of variations and the method of controlled Lagrangians. The latter is a technique for deriving stabilizing feedback controls for nonlinear controlled mechanical systems. It relies on deriving a Lagrangian which describes the feedback controlled dynamics. This is a nontrivial extension to the theory of the inverse problem as it involves controls. We discuss various aspects of both subjects and illustrate the theory with examples.

Hamel's formalism is a representation of Lagrangian mechanics obtained by measuring the velocity components relative to a frame that generically is not induced by configuration coordinates. The use of this formalism often leads to a simpler representation of dynamics. Utilizing the variational discretization approach, this paper develops a discrete Hamel's formalism with applications to nonholonomic integrators.

In this paper, we discuss the integrability of a nonholonomic mechanical system—a generalized Klebsh–Tisserand case of the Suslov problem. Using the theory of Hamiltonization and the Poincaré–Hopf theorem we analyze the topology of the invariant manifolds and in particular describe their genus. We contrast the results with those for Hamiltonian systems.

This paper discusses Hamel’s formalism and its applications to structure-preserving integration of the ndimensional spherical pendulum. It utilizes redundant coordinates in order to eliminate multiple charts on the configuration space of the pendulum as well as nonphysical artificial singularities induced by local coordinates, while keeping the minimal possible degree of redundancy and avoiding integration of differential-algebraic equations. We show that by a suitable choice of reconstruction equation, this approach leads to an energy-momentum integrator for the n-dimensional spherical pendulum. Long-time numerical simulations are performed that compare the numerical performance of the proposed Hamel integrator with Störmer–Verlet and RATTLE.

This paper discusses Hamel's formalism and its applications to structure-preserving integration of mechanical systems. It utilizes redundant coordinates in order to eliminate multiple charts on the configuration space as well as nonphysical artificial singularities induced by local coordinates, while keeping the minimal possible degree of redundancy and avoiding integration of differential-algebraic equations.

The method of controlled Lagrangians for discrete mechanical systems is extended to the problem of stabilization of the rotations of a spacecraft with a symmetric rotor. In particular, stabilization about its intermediate axis of inertia is considered. The Moser-Veselov discretization is used to obtain the discrete dynamics of the system. Stabilization conditions for the continuous model and its discretization are compared. It is shown that stability of the discrete system is sufficient for stability of its continuous counterpart but not vice versa.

Hamel's equations are an analogue of the Euler–Lagrange equations of Lagrangian mechanics when the velocity is measured relative to a frame which is not related to system's local configuration coordinates. The use of this formalism often leads to a simpler representation of dynamics but introduces additional terms in the equations of motion. The paper elucidates the variational nature of Hamel's equations and discusses their utility in control and stabilization. The latter is illustrated with the problem of stabilization of a falling disk.

The equations of nonholonomic mechanics may be derived using a number of variational principles. This paper studies some of these principles from the contemporary geometric point of view, taking into account various bundle structures that are intrinsically present in the nonholonomic setting.

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.

Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria of discrete mechanical systems with symmetry and equilibria of discrete mechanical systems with broken symmetry. Unexpected phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic shaping procedure. New terms in the controlled shape equation that are necessary for potential shaping in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline, and the application of the theory to the construction of digital feedback controllers is also discussed.