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A Landscape of Hamiltonian Phase Spaces: on the foundations and generalizations of one of the most powerful ideas of modern science
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In this thesis we revise the concept of phase space in modern physics and devise a way to explicitly incorporate physical dimension into geometric mechanics. A historical account of metrology and phase space is given to illustrate the disconnect between the theoretical physical models in use today and the formal treatment of units of measurement. Self-contained presentations of local Lie algebras, Lie algebroids, Poisson manifolds, line bundles and Jacobi manifolds are given. A unit-free manifold is defined as a generic line bundle over a smooth manifold that we interpret as a manifold whose ring of functions no longer has a preferred choice of a unit. This point of view allows us to implement physical dimension into geometric mechanics. Unit-free manifolds are shown to share many of the core structure of the category of ordinary smooth manifolds: Cartesian products, derivations as tangent vectors, jets as cotangent vectors, submanifolds and quotients. This allows to reinterpret the notion of Jacobi manifold as the unit-free analogue of Poisson manifolds. With this new language we rediscover known results about Jacobi maps, coisotropic submanifolds, Jacobi products and Jacobi reduction. We give a categorical formulation of the loose term 'canonical Hamiltonian mechanics' by defining the notions of theory of phase spaces and Hamiltonian functor. Conventional configuration spaces are then replaced by line bundles, called unit-free configuration spaces, and, they are shown to fit into a theory of phase spaces with a Hamiltonian functor given by the jet functor. Motivated by the algebraic structure of physical quantities in dimensional analysis, we define dimensioned groups, rings, modules and algebras by implementing an addition operation that is partially defined. Jacobi manfolds are shown to have associated dimensioned Poisson algebras and dimensioned coisotropic calculus.
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The aim of this paper is to prove a normal form Theorem for Dirac–Jacobi bundles using a recent techniques of Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As another application we provide an alternative proof of the splitting theorem of homogeneous Poisson structures.
This paper belongs to a series of works aiming at exploring generalized (complex) geometry in odd dimensions. Holomorphic Jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles. In fact, generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to holomorphic contact groupoids. A crucial tool in our proof is what we call the homogenization scheme, which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids.
The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation. Spacetime structure on the base space induces a closure operator on the idempotent subunits. Restriction is then interpreted as spacetime propagation. This lets us study relativistic quantum information theory using methods entirely internal to monoidal categories. As a proof of concept, we show that quantum teleportation is only successfully supported on the intersection of Alice and Bob's causal future.
In this paper, we apply the geometric Hamilton–Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms.
The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a hamiltonian vector field X H can be projected into a configuration manifold by means of a 1-form , then the integral curves of the projected vector field can be transformed into integral curves of X H provided that W is a solution of the Hamilton–Jacobi equation. In this way, we use the geometric Hamilton–Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton–Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz–Smorodinsky oscillators and for explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.
Galileo Unbound: A Path Across Life, The Universe and Everything traces the journey that brought us from Galileo's law of free fall to today's geneticists measuring evolutionary drift, entangled quantum particles moving among many worlds, and our lives as trajectories traversing a health space with thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets or the collapse of stars into black holes. This book tells the history of spaces of expanding dimension and increasing abstraction and how they continue today to give new insight into the physics of complex systems. Galileo published the first modern law of motion, the Law of Fall, that was ideal and simple, laying the foundation upon which Newton built the first theory of dynamics. Early in the twentieth century, geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, forcing light rays to bend past the Sun. Possibly more radical was Feynman's dilemma of quantum particles taking all paths at once-setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant, the need to track ever more complex changes and to capture their essence, to find patterns in the chaos as we try to predict and control our world.
This book deals with the mathematical properties of dimensioned quantities, such as length, mass, voltage, and viscosity.
Beginning with a careful examination of how one expresses the numerical results of a measurement and uses these results in subsequent manipulations, the author rigorously constructs the notion of dimensioned numbers and discusses their algebraic structure. The result is a unification of linear algebra and traditional dimensional analysis that can be extended from the scalars to which the traditional analysis is perforce restricted to multidimensional vectors of the sort frequently encountered in engineering, systems theory, economics, and other applications.
This book describes several mathematical models of the primary visual cortex, referring them to a vast ensemble of experimental data and putting forward an original geometrical model for its functional architecture, that is, the highly specific organization of its neural connections. The book spells out the geometrical algorithms implemented by this functional architecture, or put another way, the “neurogeometry” immanent in visual perception. Focusing on the neural origins of our spatial representations, it demonstrates three things: firstly, the way the visual neurons filter the optical signal is closely related to a wavelet analysis; secondly, the contact structure of the 1-jets of the curves in the plane (the retinal plane here) is implemented by the cortical functional architecture; and lastly, the visual algorithms for integrating contours from what may be rather incomplete sensory data can be modelled by the sub-Riemannian geometry associated with this contact structure.
As such, it provides readers with the first systematic interpretation of a number of important neurophysiological observations in a well-defined mathematical framework. The book’s neuromathematical exploration appeals to graduate students and researchers in integrative-functional-cognitive neuroscience with a good mathematical background, as well as those in applied mathematics with an interest in neurophysiology.
De las grandes civilizaciones, hay una que brilla con luz propia por la monumentalidad de sus edificaciones, la riqueza de su simbolismo, el valor de sus contribuciones al conocimiento y el atractivo de su extraordinaria longevidad: el antiguo Egipto. Imbuida de misterio, probablemente ninguna haya despertado tanto interés en la conciencia occidental.
La obra que el lector tiene entre sus manos es una exhaustiva introducción a la egiptología. Además de ofrecer una completa perspectiva sobre el desarrollo histórico del antiguo Egipto, el libro explora la evolución del arte y de las concepciones religiosas. La primera parte aborda la historia del desciframiento de los jeroglíficos egipcios y la geografía del país del Nilo. La segunda se sumerge en los casi tres mil años de existencia de una de las culturas más destacadas de todos los tiempos, desde el predinástico hasta la época ptolemaica. La historia política se conjuga con un análisis de las estructuras sociales y de la mentalidad, con abundantes referencias a textos egipcios. La tercera parte estudia la sociedad y la cultura de los antiguos egipcios a través de sus creencias religiosas y sus conocimientos científicos.