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This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. A Landscape of Hamiltonian Phase Spaces - On the foundations and generalizations of one of the most powerful ideas of modern science. Carlos Zapata-Carratal´a Doctor of Philosophy University of Edinburgh 2019 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Carlos Zapata-Carratal´a) iii To my parents and teachers. A mis padres y profesores. iv In memory of Michael Atiyah. v vi Lay Summary Modern science and engineering uses mathematical quantities to represent parts of the world. These quantities need to be operated following some careful rules, known as dimensional analysis, if we want to obtain practical results from the theoretical knowledge we have about the world. One of the oldest and, to this day, most useful, bits of knowledge about the world is understanding how things move around and how some object's motion affects the motion of other objects. This is what we call mechanics. What we know today about how things move is based on a very long history of different ideas and theories that dates back 100s of years. The invention of modern mathematics, at least modern from a historical perspective, has been motivated by the problems of mechanics. For example, the derivatives and integrals that we calculate today are a result of a group of stubborn physicists and mathematicians 300s years ago trying to understand why the planets moved precisely in the way we see them move in the sky. The mathematics that we use to describe the motion of objects nowadays dates back roughly 200 years. But the rules to operate with the quantities we measure in practice were only decided 100 years ago. This difference means that today, when we want to, for example, send a rocket to Mars, we first measure the distance that the rocket needs to travel, let's say 50; 000; 000 km, then the scientists and engineers come up with the right equations to calculate how long it will take to get there and then they introduce the digit 50; 000; 000 without the units `km' for a computer or calculator to give back another digit, let's say 270. If they used all the rules right, it is possible to give a unit to the new digit and conclude that the trip will last for 270 days. This works in practical situations because scientists and engineers are always paying attention to what units their digits carry but when theoretical scientists and mathematicians study equations they don't worry about the units. In this thesis we try to understand two things that can be seen in the example of the rocket: 1. What is the best mathematical way to represent the travel of a rocket to Mars? We need to know where the rocket is at all times but that's not enough, it is also important to know how quickly it is travelling. Imagine that in all our calculations to travel to Mars we only consider our position, then if we arrive on Mars, but we are traveling super fast, we would crash and all our efforts would have been for nothing! 2. Is it possible that my mathematics take care of the units by themselves? I don't want to have to remove them manually and insert them back in every time I want to compute something. It is a hassle and there is more risk of making mistakes! vii viii Abstract In this thesis we aim to revise the fundamental concept of phase space in modern physics and to devise a way to explicitly incorporate physical dimension into geometric mechanics. To this end we begin with a nearly self-contained presentation of local Lie algebras, Lie algebroids, Poisson and symplectic manifolds, line bundles and Jacobi and contact manifolds, that elaborates on the existing literature on the subject by introducing the notion of unit-free manifold. We give a historical account of the evolution of metrology and the notion of phase space in order to illustrate the disconnect between the theoretical models in use today and the formal treatment of physical dimension and units of measurement. A unit-free manifold is, mathematically, simply a generic line bundle over a smooth manifold but our interpretation, that will drive several conjectures and that will allow us to argue in favor of its use for the problem of implementing physical dimension in geometric mechanics, is that of a manifold whose ring of functions no longer has a preferred choice of a unit. We prove a breadth of results for unit-free manifolds in analogy with ordinary manifolds: existence of Cartesian products, derivations as tangent vectors, jets as cotangent vectors, submanifolds and quotients by group actions. This allows to reinterpret the notion of Jacobi manifold found in the literature as the unit-free analogue of Poisson manifolds. With this new language we provide some new proofs for Jacobi maps, coisotropic submanifolds, Jacobi products and Jacobi reduction. We give a precise categorical formulation of the loose term `canonical Hamiltonian mechanics' by defining the precise notions of theory of phase spaces and Hamiltonian functor, which in the case of conventional symplectic Hamiltonian mechanics corresponds to the cotangent functor. Conventional configuration spaces are then replaced by line bundles, what we call unit-free configuration spaces, and, after proving some specific results about the contact geometry of jet bundles, they are shown to fit into a theory of phase spaces with a Hamiltonian functor given by the jet functor. We thus find canonical contact Hamiltonian mechanics. Motivated by the algebraic structure of physical quantities in dimensional analysis, we redevelop the elementary notions of the theory of groups, rings, modules and algebras by implementing an addition operation that is only defined partially. In this formalism, the notions of dimensioned ring and dimensioned Poisson algebra appear naturally and we show that Jacobi manifolds provide a prime example. Furthermore, this correspondence allows to recover an explicit Leibniz rule for the Jacobi bracket, Jacobi maps, coisotropic submanifolds, and Jacobi reduction as their dimensioned Poisson analogues thus completing the picture that Jacobi manifolds are the direct unit-free analogue of ordinary Poisson manifolds. ix x Acknowledgements The first person I would like to thank is Jose Figueroa-O'Farrill, who acted as my supervisor during all my years in Edinburgh and from whom I learnt a great deal. His help in resolving the many issues on which I got constantly stuck and his understanding during difficult times have de facto enabled this PhD thesis to be a reality. I thank Henrique Bursztyn for the opportunity he gave me to visit IMPA, for welcoming me in such a stimulating research environment and for the financial support that covered for my stay in Rio de Janeiro. I am indebted to Luca Vitagliano and Jonas Schnitzer, who very generously read my manuscripts and offered invaluable comments and remarks. I also thank Adolfo de Azcarraga for his continued encouragement and guidance and Jan Waterfield for helping me venture into the world of the baroque harpsichord, something that would provide much needed musical reliefs during my studies. A special mention goes to the late Michael Atiyah, with whom I had the privilege to engage in deep conversations, be present in a circle of academic all-stars, share Tarta de Santiago and weather media storms. Had it not been for the job he offered me as his assistant, the last few months of my PhD would have been financially troublesome. I will always dearly remember Michael's energy and passion about mathematics and science, even to his very final days, and how, in my position of his assistant, I was able to accompany him in the twilight of his life. I would like to thank the many members of the staff at the University of Edinburgh that have been immensely helpful during all this years, oftentimes going the extra mile just for my convenience. In particular Chris, Jill and Iain. I am very grateful to the people that accompanied me during my time in Edinburgh and that I now consider my friends. Xiling for welcoming me into my first office in JCMB for all the great times, from Scottish lakes to Mediterranean beaches. Tim for the very thought-provoking conversations about mathematics, physics, life and everything. Manuel for passing on the baton of PG rep and rightful title of Queen of the Postgraduates and for making our flat into our home. Matt for his precious time spent in my uninformed algebra questions and all the things we have come to enjoy together. Ruth for her musical collaboration. Scott for all the LAN party experiences and being my first gaming-fueled fried. Ross for the camaraderie of sharing supervisor and the many interesting discussions.
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