Find results with:

Add another field

Search by date

to

Searching collections:

Dartmouth Dissertations

Add or remove collections

Home Dartmouth Dissertations epstein2016 1

Reference URL

To link to the entire object, paste this link in email, IM or document

To embed the entire object, paste this HTML in website

To link to this page, paste this link in email, IM or document

To embed this page, paste this HTML in website

Dynamics of magnetic flows on nilmanifolds / by Jonathan Epstein

you wish to report:

Your comment:

Your Name:

Submit

Cancel

...

Back to top

Powered by CONTENTdm®

Author	Epstein, Jonathan M.
Title	Dynamics of magnetic flows on nilmanifolds / by Jonathan Epstein
Date	2016
Abstract	In this thesis, we study the dynamics of magnetic flows on compact nilmanifolds. Magnetic flows are generalizations of geodesic flows. They model the motion of a particle of unit mass and unit charge in a smooth manifold M in the presence of a magnetic field. As such, their dynamical properties are in influenced by both the underlying Riemannian geometry and the closed 2-form on M which plays the role of the magnetic field. At the same time, nilmanifolds are a rich and varied class of examples, as well as a source of conjectures and counterexamples in Riemannian geometry. More precisely, we consider nilmanifolds of the form M = Γ∕G, where G is a simply connected 2-step nilpotent Lie group and Γ < G is a cocompact discrete subgroup. The manifold M is endowed with a Riemannian metric g and closed 2- form, or magnetic field, σ, each of which pulls back to a left-invariant tensor field on G. First, we focus on the case when G is the 2n+1 dimensional Heisenberg group, and σ is exact. We calculate the Mañé critical value and the lengths of closed magnetic geodesics in nontrivial free homotopy classes. Next we consider the topological entropy of magnetic flows on arbitrary 2-step compact nilmanifolds. When σ represents a rational cohomology class and its restriction to g = TₑG vanishes on the derived algebra, we prove that the associated magnetic ow has zero topological entropy on a dense set of energy levels. In particular, this is the case when σ represents a rational cohomology class and is exact. Lastly, we provide an example of a magnetic field on a 2-step compact nilmanifold that has positive topological entropy for arbitrarily high energy levels. The salient difference in this case is that σ is not exact. We discuss the relationship to Mañé's critical value. The main tool is a symplectic reduction of the cotangent bundle of a nilmanifold of one more dimension than M.
Description	1 online resource (vi, 82 leaves) : illustrations
Format	application/pdf
College	Dartmouth College -- Department of Mathematics.
Issuing Body	Dartmouth Dissertations are made available through the Dartmouth Digital Library.
Degree	Thesis (Ph.D)--Dartmouth College.
Copyright	Made available by the author under a Creative Commons Attribution-NonCommercial License.; Made available by the Trustees of Dartmouth College under a Creative Commons Attribution-NonCommercial License. https://creativecommons.org/licenses/by-nc/4.0/
DOI	https://doi.org/10.1349/ddlp.2168

Title	epstein2016 1
Full Text	DYNAMICS OF MAGNETIC FLOWS ON NILMANIFOLDS A Thesis Submitted to the Faculty in partial ful llment of the requirements for the degree of Doctor of Philosophy in Mathematics by Jonathan Epstein DARTMOUTH COLLEGE Hanover, New Hampshire April 20, 2016 Examining Committee: Carolyn Gordon, Chair David Webb Craig Sutton Tracy Payne F. Jon Kull, Ph.D. Dean of Graduate Studies

Find results with:

Searching collections:

Home Dartmouth Dissertations epstein2016 1

Dynamics of magnetic flows on nilmanifolds / by Jonathan Epstein

Object Description

Description

Dynamics of magnetic flows on nilmanifolds / by Jonathan...

Dynamics of magnetic flows on nilmanifolds / by Jonathan...

	All
	Pages Enter page numbers or page ranges separated by commas. For example: 1,3,5-12