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Hamiltonian Mechanics

Marcello Seri
Bernoulli Institute
University of Groningen
m.seri (at) rug.nl

Version 1.20 alpha
August 1, 2024

Contents

Preface

1 1: Rewrite

The literature on classical (or analytical, as it was called by Lagrange) mechanics is full of good material. Different sources present the topic from different perspectives and with different points of view, and may suite different people differently. In these lecture notes I am iterating on my perspective. The approach I am taking has been heavily influenced by [Arn89; Kna18; Dub10; Low12; MR99; Ton05; Lan+76].

The book [Kna18] will be a good extra reference for the course. In addition to being a good introduction to many topics in classical mechanics that we will not have time to discuss during the lectures, it covers almost all the material that we will treat in chapters 1, 6, 8, 10, 11, 13, 15. Furthermore, the book can be freely and legally accessed via the University proxy using the SpringerLink service.

I had lots of fun reading [Sch19], if none of the references above satisfies you, consider having a look at that. And for an unusual approach to the topic, centered around the idea that everything should be explicit enough to be directly computable by a computer, you can have a look at the marvellous [SW15]. Finally, a good and enjoyable classical reference is [GPS13].

These notes are not (yet?) exhaustive. They will be updated throughout the course, and it will likely take a few years of course iterations before they fully stabilize. Some topics, examples and exercises discussed in class will not be in these notes, but may appear as examples, exercises or problems in the literature presented above or on the material posted in the course webpage.

Throughout the course, we will discuss the main ideas in Newtonian, Lagrangian and Hamiltonian mechanics and the relations between them. This will include symmetries and elementary phase space reduction, normal modes and small oscillations. We will then move to study action–angle coordinates and integrability, and finally we will discuss elementary perturbation theory. On the latter, we will only briefly discuss resonances and mostly focus on the meaning of the Kolmogorov-Arnold-Moser theorem and the Nekhoroshev theorem.

Topics that I intend to add over time include: contact mechanics, non-holonomic mechanics, sub-riemannian geometry, numerical methods, rigid bodies, mathematical billiards, integrability via Lax pairs... I don’t know if this is just whishful thinking or if it will happen at some point, but I welcome any contribution in these directions.

A number of extra references covering some of the topics mentioned above is collected on my blog. I will make sure to keep the post updated and the links working.

Please don’t be afraid to send me comments to improve the course or the text and to fix the many typos that will surely be in this first draft. They will be very appreciated.

I am extremely grateful to Nithesh Balasubramanian, Senan Bird, Riccardo Bonetto, Josephine van Driel, Ramsay Duff, Mollie Jagoe-Brown, Jesse Mulder, Dijs de Neeling, Anouk Pelzer, Iisekki Rotko, Danique de Ruiter, Robbert Scholtens, Felix Semler, Albert Šilvans, Marit van Straaten and Jermain Wallé for their careful reading of the notes and their useful comments and corrections.

Last but not least, I owe a huge debt of gratitude to Constanza Rojas-Molina, who kindly allowed me to use some of her beautiful artworks in these notes.