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Symmetry in Geometry and Mechanics

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Symmetry in Geometry and Mechanics Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona Symmetry in Geometry and Mechanics Autor: Pablo Ruiz Director: Dr. Ignasi Mundet Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, June 29, 2017 ABSTRACT In this work we have tried to give a theoretical description, as well as a his- torical review, of the differential geometry behind different areas of physics (Classical Mechanics, Electromagnetism, Yang-Mills...) trying to capture the deep geometric nature of modern theoretical physics. Firstly we start by in- troducing classical differential-geometric notions from the modern viewpoint (emphasizing the role played by Bundles and Sections). Then we will devote to the study of Hamiltonian Systems both in the Symplectic and in the Pois- son formalisms, in here we will see two results that we believe are specially important, geodesic flow from the dynamics point of view (where we will analyze the notion of geodesics through the lenses of physical theories, and a proof of Clairaut’s formula in classical differential geometry by the use of a fundamental concept in modern geometric mechanics, the moment map. Contents 0.1 Review of classical differential geometry . .1 0.1.1 Connections in TM tensor bundles and abstract vector bundles . 10 0.1.2 The importance of the Levi-Civita connection in classical differential geometry . 15 0.2 Élie Cartan and the revolutionarization of classical differential geometry . 18 0.2.1 Differential forms and exterior derivative . 18 0.2.2 Lie derivative in smooth manifolds . 25 0.3 Hamiltonian mechanics and Symplectic geometry . 27 0.3.1 Cotangent Bundle of a smooth manifold . 28 0.3.2 Symplectic and Hamiltonian vector fields . 29 0.3.3 Poincare’s lemma . 32 0.3.4 Darboux’s theorem . 32 0.3.5 Poisson Brackets on Symplectic manifolds . 35 0.4 Poisson Manifolds . 37 0.4.1 Hamiltonian vector fields in Poisson Manifolds . 38 0.4.2 Poisson maps/canonical transformations . 41 0.5 Moment Maps . 47 0.5.1 Theory . 47 0.5.2 Computation and Properties of Momentum Maps . 53 0.6 More differential geometry, Principal and Associated Bundles (Yang-Mills theory). 55 0.6.1 Geometrization of the Kaluza-Klein theory of electro- magnetism . 57 0.6.2 Principal Fiber Bundles and their associated Ehresmann connections . 61 0.6.3 The notion of Ehresmann connection . 64 0.6.4 Linear connections in Associated Vector Bundles com- ming from Ehresmann connections in Principal Bundles 64 0.6.5 The electromagnetic field as a U(1)-connection . 66 0.6.6 Line Bundle defined by a closed 2-form on a manifold . 68 0.6.7 Quantization using Complex Linear Connections in Com- plex Vector Bundles . 71 0.6.8 Hermitian Vector Bundles and Connections . 72 1 .1 Tensors, a brief description . 74 Bibliography 83 During the work we will be mainly using Einstein’s summation convention when we operate on tensorial objects, this just means that we omit the ∑ where we sum, and we know that we have to sum those indices that are repeated (usually one up/one down). 0.1 Review of classical differential geometry This section pretends to be just a brief discussion of classical differential geometry, so that the reader knows and recalls some of the concepts that will be essential later, and to develop the language that we will be using in the rest of the work. We will recast classical differential geometry via the fundamen- tal concept of fiber bundles, since they are the abstract framework that allows us to better understand how differential geometry works. We start with the standard definition of a manifold, that is mainly a result of Riemman’s revo- lutionarization of geometry two centuries ago that deeply changed the notion of geometry and physics. The abstract definition of manifold just tries to cap- ture a simple idea, the fact that one needs n magnitudes to determine position in an n-dimensional manifold. How this quantities get represented numer- ically, or what do they represent is just a matter of choice, what we would call a choice of local patch in modern language. What we also require is that the multiple ways to look at this magnitudes coincide (that we have the same geometry/physics). Thus we will now give the formal definition that can be found in any standard book on differential geometry as a starting point. Definition 0.1.0.1. A manifold of dimension n is a Haussdorff second-countable topological space M together with certain structure ,that we usually call atlas (in the case of a smooth manifold we usually call it a differentiable structure): 1. An open cover fUigi2I of M. n 2. A collection of continuous injective maps Fi : Ui ! R called coordinate charts/local coordinates such that Fi(Ui) is an open set. We also require the existence of smooth transition maps, what this means is that if Ui \ Uj 6= f, then the following map transition map is smooth: −1 n n Fj ◦ Fi : Fi(Ui \ Uj) ⊂ R ! Fj(Ui \ Uj) ⊂ R , we say the charts are (smoothly) compatible. The beauty of this defi- nition is that it can be restricted to the class of functions we are spe- cially interested (smooth functions usually in differential geometry), but we could consider analytical functions, meromorphic functions, class Ck functions... Riemann himself used this to revolutionize classical complex analysis with the introduction of Riemann surfaces. 1 n We can interpret each chart Fi as a collection of n functions (x , ..., x ) on Ui, and we also interpret transition maps as a coordinate transformation in the sense that they can be interpreted as a family of maps: (x1, ..., xn) 7! (y1(x1, ..., xm), ..., yn(x1, ..., xm)) One important remark to make here is that one can also define the no- tion of manifold in the infinite-dimensional case, and there is a whole theory built on them (by modeling them on Banach Spaces, rather than on the finite- dimensional Rn. Since the major focus of this work is on finite-dimensional manifolds we will stick to the definition we gave and restrict our attention to the finite-dimensional case. In sections of the work (mainly the last ones) some of the spaces that will pop up in the description will be infinite-dimensional, and thus we will have little to say about them, what we can get is an intuitive idea on the object, rather than a deep comprehension, that would obviously require us to develop the highly non-trivial theory of infinite-dimensional man- ifolds. Now that we have the fundamental notion of a manifold, that is essential to geometry and physics, we connect with standard calculus on euclidean space just by noting that trivially Rn is a smooth manifold with a natural smooth n n structure that has a unique chart, the identity mapping Idn : R ! R . The formal notion of a manifold as we know it today was the result of a deep historical development, this definition is not essentially fundamental, what we want to emphasize is that this definition is nothing more than the formal definition that comes in hand with Riemman’s ideas. Since it is some- thing that in mathematics usually gets forgotten, but definitions are not the start of anything, but a guide we allow ourselves to be clear about the ideas we are talking about, and definitions get superseded by others that are more advanced and that represent better the nature of the object (the case of con- nections, the notion of derivative, even the very nature of what does geometry mean). We could argue, in a similar fashion to what the german Philosopher Hegel had in mind with his criticism of Kant’s phylosophy, that the concrete definition/result has to be understood as a realization of the whole Historical Devel- opment. (In our case this just means that to better understand mathematical concepts one is almost obligated to analyze the History that led to them). The main contribution to this revolutionarization of geometry comes from Riemann, in his doctoral dissertation on the nature of geometry [11]. And this same notion turned out to be indispensable for Einstein while seeking for a geometrical description of the physical universe. In fact, due to the fact that differential geometry was not as well understood at the beginning of last century, Einstein had to develop a lot of intuition on how it worked, and needed a lot of mathematically-minded people to develop the mathematics needed to describe the universe at a cosmological level and to elaborate the best theory of gravity we still have. To continue we now introduce the notion of a smooth map between smooth manifolds, that is just a natural generalization of the notion on euclidean space (we just require that it works locally and for every chart). Definition 0.1.0.2. We say that a map between two smooth manifolds f : M ! N (of dimensions m and n respectively) is smooth if for every local charts f on M and y on N the composition y ◦ f ◦ j−1 : Rm ! Rn is a smooth map (in the standard calculus sense). And further than that, we introduce the concept of "equality" inside the category of smooth manifolds, the diffeomorphism. Definition 0.1.0.3. Let f : M ! N be a smooth map, we say f is a diffeomor- phism if it is an invertible function and the inverse function is also smooth. If we have a diffeomorphism between to manifolds M, N we say that the manifolds are diffeomorphic. In a certain smooth manifold M we will denote the space of all smooth functions (what physicists call observables) in the following way: C¥(M) = f f : M ! R, f smoothg.
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