# Discrete Mechanics, Geometric Integration and Lie-Butcher Series

Instituto de Ciencias Matemáticas
C/ Nicolás Cabrera, 13-15
Campus Cantoblanco - UAM
www.icmat.es
Telephone: +34 91 2999700

### Activities

Here all activities such as mini-courses, seminars, etc will be announced.

### Workshop:

ICMAT(CSIC-UAM-UC3M-UCM), Campus Cantoblanco, Madrid May 25-28, 2015

Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series

### Seminars:

Tuesday, April 14, 2015, 12:30, NEW lunchroom. University of Bergen, Norway.

Speaker: Kurusch Ebrahimi-Fard

Continue with a different application of post-Lie algebras in the theory of integrable differential equations (Lax-pairs, R-matrices, matrix factorisations …). This is the same algebraic structure arising in a different context, and provides another nice example of this IMPORTANT(!) algebraic structure.

Friday, April 10, 2015, 12:30, Old Lunchroom, University of Bergen, Norway.

Topic: More on Post Lie algebras

Literature for Friday seminar:

Kurusch Ebrahimi-Fard, Alexander Lundervold, Hans Munthe-Kaas On the Lie enveloping algebra of a post-Lie algebra.
http://arxiv.org/abs/1410.6350

### Past Seminars:

Monday, February 16, 2015, 11:00-12:00, q-math seminar, Room 2.2.D.08, UC3M, Leganés.

Speaker: Hans Munthe-Kaas (University of Bergen-Norway)

Abstract: Symmetric spaces (e.g. spheres and Grassman manifolds) are Riemannian manifolds with constant curvature tensor.
Algebraically they are characterised as spaces with a symmetric product, inducing a Lie triple structure on the tangent space.
On the group level they can be realized as quotients of Lie groups, which is closely related to Cartan decomposition of the Lie algebra; it splits in a sum of a sub algebra and a Lie triple system.
We will discuss symmetric spaces and their relationship to post-Lie algebras, and we will discuss various numerical algorithms based on Cartan decompositions and triple systems.

Tuesday, January 27, 2015, 12:30-13:30, University of Bergen.

Speaker: Kurusch Ebrahimi-Fard (CSIC, Spain)

Friday, January 23, 2015, 10:15-12:00, University of Bergen.

Speaker: Kurusch Ebrahimi-Fard (CSIC, Spain)

Wednesday, November 12, 2014, 14:15 - 15:00, University of Bergen.

Speaker: María Barbero (CSIC, Spain)

Abstract: Mechanical systems are governed by differential equations which are usually difficult to solve or it is impossible to solve them analytically . Numerical integrators are needed to handle these problems. The crucial point is to find "good" numerical algorithms and to decide what "good" means in these cases.

The characterization of mechanics from a variational viewpoint allows the construction of variational integrators which preserve geometric properties of the system such as symplecticity, momentum, good energy behaviour, etc. From the discrete Euler-Lagrange equations is possible to construct variational integrators that have been proved to perform well in numerical simulations and they admit interesting extensions (nonholonomic systems, systems with external forces, classical field theories...).

In this talk we provide introductory notions on this topic.

References:

Hairer, Lubich, Wanner, Geometric Numerical Integrators, Springer, Berlin-Heidelberg-New York, 2001.

Marsden, West, Discrete mechanics and variational integrators, Acta Numerica 10, 357-514, 2001.

Tuesday, October 7, 2014, 10:00 - 11:30, ICMAT, Aula Gris 1

Speaker: Alexander Lundervold (Bergen University College-Norway)

Abstract: I will talk about some relatively recent developments in the theory of Butcher series. Butcher series belong to the field of geometric numerical integration, whose aim it is to construct and study numerical integration methods for differential equations preserving some geometric structure of the underlying system. I will focus on some of the algebraic underpinnings of Butcher series, and the generalization of Butcher series to numerical integration methods for differential equations on manifolds, the so-called Lie--Butcher series. The talk aims to serve as one of the starting points for our discussions during the NILS project

Lie–Butcher series

Tuesday, October 14, 2014, 10:00 - 11:30, ICMAT, Aula Gris 1

Speaker: Mauricio Godoy (University of Bergen-Norway)

Abstract: The Baker-Campbell-Hausdorff formula is a powerful tool that relates explicitly the Lie bracket on a Lie algebra and the product on an associated Lie group via an infinite series of brackets of increasing length. The formula can be seen as the correction that one needs to do when the identity $$e^{A+B}= e^{A} e^{B}$$ does not hold.

In this talk I will explain a combinatorial approach to find the coefficients of this series proposed by Strichartz, which is obtained by cleverly solving a differential equation. Similar formulas were derived in different manner by other authors, and I will try to comment on these other approaches. One of the important observations of Strichartz that I will stress is that this formula can be applied to find good aproximations of problems in sub-Riemannian geometry and control.

Project coordinator:

María Barbero Liñán
(ICMAT-UC3M-Spain)

Members:

Kurusch Ebrahimi Fard
(ICMAT-CSIC-Spain)
Marta Farré Puiggalí
(ICMAT-CSIC-Spain)
David Martín de Diego
(ICMAT-CSIC-Spain)

Mauricio Godoy
(University of Bergen-Norway)
Kristoffer Foellesdal
(University of Bergen-Norway)
Alexander Lundervold
(Bergen University College-Norway)
Hans Munthe-Kaas
(University of Bergen-Norway)
Antonella Zanna
(University of Bergen-Norway)

Contact:

Supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism. Operated by Universidad Complutense de Madrid.

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