A class of nonholonomic kinematic constraints in elasticity

We propose a first example of a simple classical field theory with nonholonomic constraints. Our model is a straightforward modification of a Cosserat rod. Based on a mechanical analogy, we argue that the constraint forces should be modeled in a special way, and we show how such a procedure can be naturally implemented in the framework of geometric field theory. Finally, we derive the equations of motion and we propose a geometric integration scheme for the dynamics of a simplified model.Comment: 28 pages, 7 figures, uses IOPP LaTeX style (included) (v3: section 2 entirely rewritten

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.Comment: 22 pages, 1 figure; final version as published with minor typos correcte

The momentum map for nonholonomic field theories with symmetry

In this note, we introduce a suitable generalization of the momentum map for nonholonomic field theories and prove a covariant form of the nonholonomic momentum equation. We show that these covariant objects coincide with their counterparts in mechanics by making the transition to the Cauchy formalism

Continuous and discrete aspects of Lagrangian field theories with nonholonomic constraints

This dissertation is a contribution to the differential-geometric treatment of classical field theories. In particular, I study both discrete and continuous aspects of classical field theories, in particular those with nonholonomic constraints. After some introductory chapters dealing with the geometric structures inherent in field theories and the discretization of field theories, the first part of the thesis is concerned with discrete field theories taking values in Lie groupoids. It is shown that many previously known discrete field theories are particular instances of Lie groupoid field theories, and the geometry of Lie groupoids is used to construct a unifying framework for this class. In two further chapters, the effect of symmetry upon this setup is described, with particular attention to the case of Euler-Poincaré reduction, which can be rephrased using concepts of discrete differential geometry. In the second part of the thesis, nonholonomic constraints for field theories are described. A number of differential-geometric results that characterize the nature of nonholonomic constraints are derived: in particular, a version of the De Donder-Weyl equation suitable for constrained field theories is discussed and a so-called momentum lemma is derived (describing the influence of symmetry upon the nonholonomic framework). In the last chapter, a physical example of a nonholonomic field theory is given, based on the theory of Cosserat media. This example is treated using the theory of the preceding chapters. Furthermore, a geometric numerical integration scheme is derived and used to give a quantitative insight into the dynamics

Routh Reduction by Stages

This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group

The motion of solid bodies in potential flow with circulation: a geometric outlook

The motion of a circular body in 2D potential flow is studied using symplectic reduction. The equations of motion are obtained starting front a kinetic-energy type system on a space of embeddings and reducing by the particle relabelling symmetry group and the special Euclidian group. In the process, we give a geometric interpretation for the Kutta-Joukowski lift force in terms of the curvature of a connection on the original phase space