Sub-Riemannian integrators: towards a new class of geometric integrators for non-holonomic Hamiltonian systems

Samenvatting

The last few years have seen the rise of importance of the fields of Contact Geometry and sub-Riemannian Geometry, also representable in terms of non-holonomic Hamiltonian mechanics. Research ranges from image detection on new models of the visual cortex to new approaches to thermodynamics. As the theory gets closer to potential physical applications, the lack of numerical integrators capable of preserving the Contact form or the sub-Riemannian non-holonomic constraints is becoming a more and more central issue.

The objective of this project is to develop a new class of numerical integrators suitable to give good qualitative and quantitative solutions for non-holonomic Hamiltonian systems. As this is a very general and hard objective, the work will proceed in three steps:

1. October 2018 - January 2018
We focus on contact Hamiltonian systems, as they provide a class of constraints that have the very specific shape of the kernel of a differential form and a better understood structure. This is very interesting on its own as contact Hamiltonian dynamics has applications in the classical mechanics of dissipative systems, fluid dynamics, statistical physics, statistics, quantum mechanics, information geometry, shape dynamics, biology, and integrable systems. So far, in collaboration with Dr. Alessandro Bravetti and Dr. Mats Vermeeren, we have already exploited Herglotz’ variational principle to obtain a geometric integrator for time independent contact systems that naturally preserves the energy-like function and the contact form. We expect to send this work for publication soon, after completing a finer analysis of the errors and adding further comparisons and benchmark examples.

2. February 2018 - June 2019
Before moving on to more general non-holonomic systems, we will extend the current integrator to time dependent contact systems, so that the integrator is actually suitable to be used in all the applications that we are currently aware of. A recently published generalization of Herglotz’ principle should allow us to obtain both time-dependent and higher order integrators. Preliminary results show that our current integrator can be generalized to this setting and therefore we expect to be able to complete this second work in a reasonable amount of time, also including examples that are of interest in celestial mechanics and in thermodynamics.

3. July 2019 - June 2020
The Herglotz principle does not hold for general non-holonomic systems, but the structure of the geometric integrators that we would have developed up to this stage, should be enough to suggest an ansatz on what kind of restrictions or relaxations we need to be able to address more general systems. Here we will likely have to deal also with regularization and preconditions to try and reduce the instability caused by singularities present in the more general sub-Riemannian metrics. This generalization will proceed in multiple sub-steps in which we will remove further and further restrictions from the structure of the phase space constraints, trying to finally arrive to the bracket-generating distributions that define the most general meaningful sub-Riemannian system and that would have very interesting applications to control theory.

Kenmerken

Projectnummer

040.11.698

Hoofdaanvrager

M. Seri

Verbonden aan

Rijksuniversiteit Groningen, Faculty of Science and Engineering (FSE), Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence

Looptijd

07/10/2019 tot 25/10/2019