31 Lectures on Geometric Mechanics

Abstract

These lecture notes in geometric mechanics are meant to convey insight through clear definitions and workable examples. The lecture format adopted here is intended to convey the immediacy of the taught course and to be useful as a basis for other courses. The lecture notes comprise: AP = Applications of Pure maths, e.g., Noether's theorem: Lie group symmetry of Hamilton's variational principle implies conservation laws for its equations of motion.\smallskip PA = Purifications of Applied maths, e.g., Euler fluid dynamics describes geodesic flow on the manifold of smooth invertible maps acting on the domain of flow. \smallskip Both AP and PA appear here, though the difference is not mentioned. It is left to the reader to decide whether it was AP or PA in each of the lectures containing well over sixty solved exercises. An aspect of modern applications emphasised here is the use of the composition of evolutionary maps for multi-physics, multi-timescale interactions including waves interacting with flows in the Euler--Poincar\'e framework in geophysical fluid dynamics (GFD) for ocean and atmosphere dynamics, and in magnetohydrodynamics (MHD) for applications in plasma physics such as magnetic confinement fusion (MFC) and astrophysical processes such as Alfv\'en waves and gravity waves propagating on the Solar tachocline. The topics covered in each lecture can also be gleaned from its table of contents listed at the onset of each lecture.