Katarzyna Grabowska-Geometric Mechanics - Tulczyjew Triples, Algebroids, and Dirac Structures, Part1

This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025.
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 multisymplectic geometry. In this minicourse, we will explore the geometric structures and procedures that comprise Geometric Mechanics. Special attention will be given to mechanical systems facing specific challenges, such as those described by singular Lagrangians or 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.