报告题目：Course on Geometric Mechanics
报 告 人： Katarzyna Grabowska
所在单位：University of Warsaw
报告时间：October 23, 10:30-11:30, 13:30-14:30
October 24, 9:00-10:00, 14:30-15:30
报告地点：吉林大学正新楼209
报告摘要: According to J.L. Lagrange, the variational description of mechanics ``reduces all the laws of motion of bodies to their equilibrium and thus brings dynamics back into statics''. Since statics is conveniently formulated in the language of differential geometry, we can also view variational calculus as a way to integrate differential geometry into the infinite-dimensional space of trajectories in physical systems. Later, the development of symplectic geometry enabled the application of traditional differential geometry in mechanics once again. This approach was similarly extended to classical field theory through the use of multisymplectic geometry. In this mini-course, we will explore the geometric structures and procedures that comprise Geometric Mechanics. Special attention will be given to mechanical systems that face specific challenges, such as those described by singular Lagrangians or those with nonholonomic constraints. We will discuss mechanics on algebroids and introduce the concept of a Dirac algebroid as a tool for deriving phase equations for systems with nonholonomic constraints, both in the Hamiltonian and Lagrangian settings. The course will be divided into four lectures:
1. Introduction. In the first lecture, we shall review the passage from the variational approach to mechanics to a purely geometrical one. As a result, we shall construct the classical Tulczyjew Triple and examine its applications.
2. Crash course on double vector bundles. The classical Tulczyjew triple is composed of certain double vector bundle morphisms. Before we proceed to the more general setting, we need to understand the concept of a double vector bundle and become familiar with several important examples.
3. Tulczyjew triple in an algebroid setting. During the third lecture, we shall introduce the concept of a general algebroid as a particular double vector bundle morphism and build the appropriate version of a Tulczyjew triple. It will then be used to describe mechanical systems with symmetry and constraints.
4. Dirac structures in nonholonomic mechanics. The last lecture will be devoted to the concept of a linear almost Dirac structure called a Dirac algebroid. It can be used in nonholonomic mechanics as a convenient tool for deriving equations with linear nonholonomic constraints with no restrictions on the Hamiltonian or Lagrangian functions. For example, one can include magnetic or gyroscopic forces in the play.
报告人简介：Katarzyna Grabowska is a professor in the Department of Mathematical Methods in Physics at the University of Warsaw. She is interested in differential geometric methods in physics and differential geometry in general.
