Academics

Introduction to Geometries and Mechanics

Time:2023-10-10 ~ 2023-12-28 Tue,Thu 09:50-11:25

Venue:Venue: A3-4-101 Zoom: 361 038 6975 (PW: BIMSA)

Speaker:Andrey Tsiganov (Visiting Professor)

Introduction

In classical mechanics, we study the evolution of a given system in configuration space. The properties of this space are described by Euclidean, projective, Riemannian, pseudo and sub-Riemannian geometries. In Hamiltonian mechanics, we have a phase space and use symplectic, Poisson, and contact geometries. To solve the equations of motion, we also use algebraic, differential, and so on other geometries. The most important and interesting mathematical models lie at the intersection of these geometric theories.

In this course we will start with Euclidean geometry and discuss general concepts such as metrics, geodesic motion, isometries, symmetries, Noether's theorem, Killing vector fields, hidden symmetries, equivalent metrics, Killing tensors, flatness and reduction of bilinear forms, orthogonal curvilinear coordinates, and so on. Then we consider some examples of constrained Hamiltonian mechanics imposing holonomic and nonholomic constraints. We then discuss the Erlangen program proposed by Klein, which is a method of characterizing geometries based on group theory and projective geometry. Some applications of Riemannian and pseudo-Riemannian geometry in mechanics have also been considered. Finally, the role of sub-Riemannian geometry in control theory is briefly discussed.

The main aim of this course is to give an overview of modern geometric methods used in various branches of classical mechanics to solve open problems. Students can discuss possible approaches to solving these open problems separately with the instructor.


Lecturer Intro

Andrey Tsiganov currently works at the Department of Computational Physics, Saint Petersburg State University, Russia. His main research interests are integrable and superintegrable systems in classical and quantum mechanics, nonholonomic and vakonomic mechanics, geometry and topology of dynamical systems, see profile at https://www.researchgate.net/profile/Andrey-Tsiganov. He is one of the organizers of the BIMSA Integrable System Seminar, see https://researchseminars.org/seminar/BIMSA-ISS and https://sites.google.com/view/bimsa-iss.


DATEOctober 10, 2023
SHARE
Related News
    • 0

      Exactly Solved Models in Statistical Mechanics

      IntroductionThe course intends to provide an introduction to the theory of integrable lattice models. Basic examples are the two-dimensional Ising model in a zero magnetic field, the six-vertex model, as well as related two-dimensional models and spin chains.It is planned to explain with simple model examples the concept of matrix transfer, duality between high and low temperatures, the concept...

    • 1

      Introduction to Spin(7) geometry

      IntroductionThe classification of Riemannian manifolds with special holonomy contains two “exceptional” cases: G2 and Spin(7). Manifolds with holonomy contained in G2 or Spin(7) are called G2-manifolds or Spin(7)-manifolds, respectively. In this course, I will introduce various topics of G2 and Spin(7) geometry, mainly focusing on the Spin(7) case. We start from the linear algebra in Spin(7) ...