Melvin Leok’s Professional Blog

I have accepted a tenured associate professorship in Mathematics at the University of California, San Diego, which will commence on July 1, 2009. The computational geometric mechanics group will be relocating to San Diego, and it will be affiliated with the Center for Computational Mathematics, the Program in Computational Science, Mathematics, and Engineering, and the Cymer Center for Control Systems and Dynamics.

As part of Purdue’s annual presidential review of the College of Science, the department of mathematics has commissioned a short video vignette where I describe my research in a broadly accessible fashion. This is available as a streaming video.

I have been selected in an internal competition as one of Purdue University’s two nominees in 2009 for the Packard Fellowship for Science and Engineering. Every year, the Packard Foundation invites the presidents of 50 universities to nominate two professors each from their institutions. An advisory panel of distinguished scientists and engineers then selects 20 Fellows to receive individual awards of $875,000, payable over five consecutive years.

[ PDF | arXiv:0903.0332 [math.NA] ]

This paper presents an analytical model and a geometric numerical integrator for a rigid body connected to an elastic string, acting under a gravitational potential. Since the point where the string is attached to the rigid body is displaced from the center of mass of the rigid body, there exist nonlinear coupling effects between the string deformation and the rigid body dynamics. A geometric numerical integrator, refereed to as a Lie group variational integrator, is developed to numerically preserve the Hamiltonian structure of the presented model and its Lie group configuration manifold. These properties are illustrated by a numerical simulation.

I will be on leave from Purdue University from April 2009 to mid June 2009 to visit the California Institute of Technology as a Visiting Assistant Professor of Control and Dynamical Systems.

Ms. Tatiana Shingel, a graduate student in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, under the research direction of Prof. Arieh Iserles, has accepted a three-year appointment as a research assistant professor of mathematics at Purdue University.

Her appointment will commence in August of 2009, and she will be affiliated with the Computational Geometric Mechanics at Purdue group. Her research expertise is in global constructive approximation and interpolation on Lie groups.

Tomoki Ohsawa, a graduate student from the University of Michigan, will be visiting the Computational Geometric Mechanics at Purdue group from mid February to early March.

His research interests are in Dirac structures and mechanics, as well as nonholonomic Hamilton-Jacobi theory.

I will be visiting the Boston and Baltimore area in the next few weeks, and giving seminars at MIT and Johns Hopkins respectively. These will be colloquium style presentations surveying the field of computational geometric mechanics, as well as highlighting some exciting new research directions.

Massachusetts Institute of Technology, Department of Mathematics
Physical Mathematics Seminar
Tuesday, February 3, 2009
2:30pm-3:30pm, Room 2-105
Website

Johns Hopkins University, Department of Applied Mathematics and Statistics
Applied Mathematics and Statistics Seminar
Thursday, February 12, 2009
4:00pm-5:00pm, 304 Whitehead Hall
Website

Ms. Jingjing Zhang, a PhD candidate at the Academy of Mathematics and Systems Science, Chinese Academy of Science, in Beijing is joining our computational geometric mechanics group starting February 12, 2009, until December 2009.

Her stay at Purdue is funded by a fellowship from the Academy of Mathematics and Systems Science, where she is advised by Prof. Jialin Hong. Her research interests include structure-preserving numerical integrators for multisymplectic field theories, and for stochastic systems.

For candidates interested in joining the Computational Geometric Mechanics @ Purdue group, research positions for postdoctoral scholars and graduate students in the broad area of geometric numerical methods in geometric mechanics and control remain available. Please contact Prof. Melvin Leok for further details.

[ PDF | arXiv:0810.0740 [math.SG] ]

We construct discrete analogues of Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also introduce a discrete Hamilton-Pontryagin variational principle on the discrete Pontryagin bundle, which provides an alternative derivation of the same set of integration algorithms. In so doing, we explicitly characterize the discrete Dirac structures that are preserved by Hamilton-Pontryagin integrators. In addition to providing a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of Dirac mechanics, it provides a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators. Since discrete Lagrangians and discrete Hamiltonians are essentially generating functions of different types, the theoretical framework described in this paper is sufficiently general to encompass all possible Dirac integrators through an appropriate choice of generating functions.