56 research outputs found
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
- …