# Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

@article{Ponte2008DiscreteNL, title={Discrete Nonholonomic Lagrangian Systems on Lie Groupoids}, author={David Iglesias Ponte and Juan Carlos Marrero and David Mart{\'i}n de Diego and Eduardo Mart{\'i}nez}, journal={Journal of Nonlinear Science}, year={2008}, volume={18}, pages={221-276} }

Abstract
This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie…

## 49 Citations

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

- Mathematics
- 2014

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…

### New developments on the geometric nonholonomic integrator

- Mathematics
- 2013

In this paper, we will discuss new developments regarding the geometric nonholonomic integrator (GNI) (Ferraro et al 2008 Nonlinearity 21 1911–28; Ferraro et al 2009 Discrete Contin. Dyn. Syst.…

### Momentum and energy preserving integrators for nonholonomic dynamics

- Mathematics
- 2007

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian , where Q is the…

### Geometric discretization of nonholonomic systems with symmetries

- Computer Science
- 2009

A family of nonholonomic integrators that are general, yet simple and easy to implement, are obtained and applied to two examples-the steered robotic car and the snakeboard.

### Lagrangian reduction of nonholonomic discrete mechanical systems by stages

- MathematicsJournal of Geometric Mechanics
- 2020

In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincare systems, and study some of its elementary properties. Examples of…

### ON THE CONSTRUCTION OF VARIATIONAL INTEGRATORS FOR OPTIMAL CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS

- Mathematics
- 2014

In this paper we derive variational integrators for optimal control problems of nonholonomic mechanical systems. We rewrite the system as a constrained second-order variational problem, that is, as a…

### Lagrangian reduction of nonholonomic discrete mechanical systems

- Mathematics, Computer Science
- 2010

This paper proposes a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle and produces a discrete dynamical system that is called the discrete reduced system.

### The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators

- Mathematics
- 2016

Geometric integrators for nonholonomic systems were introduced by Cortes and Martinez in [ 4 ] by proposing a discrete Lagrange-D'Alembert principle. Their approach is based on the definition of a…

### Exact discrete Lagrangian mechanics for nonholonomic mechanics

- MathematicsNumerische Mathematik
- 2022

The exponential map associated to a nonholonomic system is constructed that allows this discrete constraint submanifold to be defined and an exact discrete version of the non holonomic equations is derived.

### Feedback Integrators for Nonholonomic Mechanical Systems

- MathematicsJ. Nonlinear Sci.
- 2019

The theory of feedback integrators is extended to handle mechanical systems with nonholonomic constraints with or without symmetry, so as to produce numerical integrators that preserve the…

## References

SHOWING 1-10 OF 60 REFERENCES

### Discrete nonholonomic LL systems on Lie groups

- Mathematics
- 2005

This paper studies discrete nonholonomic mechanical systems whose configuration space is a Lie group G. Assuming that the discrete Lagrangian and constraints are left-invariant, the discrete…

### Nonholonomic Lagrangian systems on Lie algebroids

- Mathematics
- 2005

This paper presents a geometric description on Lie algebroids of
Lagrangian systems subject to nonholonomic constraints. The Lie
algebroid framework provides a natural generalization of classical…

### Nonholonomic mechanical systems with symmetry

- Mathematics
- 1996

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical…

### Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

- MathematicsJ. Nonlinear Sci.
- 2004

It is proved that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere Sn-1, the first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension.

### On the geometry of non‐holonomic Lagrangian systems

- Mathematics
- 1996

We present a geometric framework for non‐holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the…

### Lagrangian reduction by stages for non-holonomic systems in a Lie algebroid framework

- Mathematics
- 2005

The Lagrange-d'Alembert equations of a non-holonomic system with symmetry can be reduced to the Lagrange-d'Alembert-Poincarequations. In a previous contribution we have shown that both sets of…

### Integrators for Nonholonomic Mechanical Systems

- MathematicsJ. Nonlinear Sci.
- 2006

A discrete analog of the Lagrange-d'Alembert principle of nonhonolomic mechanics is studied and given conditions for it to define a map and to be reversible and can generate linearly implicit, semi-implicit, or implicit numerical integrators for nonholonomic systems which exhibit superior preservation of the dynamics.

### Discrete Time Lagrangian Mechanics on Lie Groups,¶with an Application to the Lagrange Top

- Mathematics
- 1999

Abstract:We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting.…

### A discretization of the nonholonomic Chaplygin sphere problem.

- Mathematics
- 2007

The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational…

### Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids

- Mathematics
- 2005

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. From a variational principle we derive the discrete Euler–Lagrange equations and…