arXiv subjects

Juergen Scheurle

Publications and source records attributed to Juergen Scheurle.

3 recordsLinked to original sources

arXiv2015

On some aspects of the discretization of the Suslov problem

In this paper we explore the discretization of Euler-Poincar\'e-Suslov equations on $SO(3)$, i.e. of the Suslov problem. We show that the consistency order corresponding to the unreduced and reduced setups, when the discrete reconstruction equation is given by a Cayley retraction map, are related to each other in a nontrivial way. We give precise conditions under which general and variational integrators generate a discrete flow preserving the constraint distribution. We establish general consistency bounds and illustrate the performance of several discretizations by some plots. Moreover, along the lines of [14] we show that any constraints-preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system.

math.NA↗

arXiv2014

On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in local coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$. Moreover, we show that any $D-$preserving discretization may be understood as beeing generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$. Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints. The exploration in this paper lays the ground to analyze the dynamics of appropriate discretizations of nonholonomic mechnical systems in the Lagrangian framework and to relate that dynamics to the dynamics of the original nonholonomic systems. We postpone this analysis to a series of forthcoming papers.

math.NA↗

arXiv2006

Birkhoffian formulation of the dynamics of LC circuits

We present a formulation of general nonlinear LC circuits within the framework of Birkhoffian dynamical systems on manifolds. We develop a systematic procedure which allows, under rather mild non-degeneracy conditions, to write the governing equations for the mathematical description of the dynamics of an LC circuit as a Birkhoffian differential system. In order to illustrate the advantages of this approach compared to known Lagrangian or Hamiltonian approaches we discuss a number of specific examples. In particular, the Birkhoffian approach includes networks which contain closed loops formed by capacitors, as well as inductor cutsets. We also extend our approach to the case of networks which contain independent voltage sources as well as independent current sources. Also, we derive a general balance law for an associated "energy function".

math.DS↗