Related papers: Symplectic Reduction of Classical Mechanics on Sha…
By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recovers the original system but also yields different real…
This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for…
The Standard Model of the elementary particles is controlled by more than 20 parameters, of which it is not known today how they can be linked to deeper principles. Any attempt to clean up this theory, in general results in producing more…
The goal of this contribution is to introduce the Hamiltonian formalism of theoretical mechanics for analysing motion in generic linear and non-linear dynamical systems, including particle accelerators. This framework allows the derivation…
We examine a very simple conceptual model of stochastic behavior based on a random walk process in velocity space. For objects engaged in classical non-relativistic velocities, this leads under asymmetric conditions to acceleration…
Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows…
We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The…
If we admit that quantum mechanics (QM) is universal theory, then QM should contain also some description of classical mechanical systems. The presented text contains description of two different ways how the mathematical description of…
The existence of the theory of `twisted cotangent bundles' (symplectic groupoids) allows to study classical mechanical systems which are generalized in the sense that their configurations form a Poisson manifold. It is natural to study from…
In this paper we will explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space. This research has direct application to optimal control and control of conservative mechanical…
Symplectic quantization is a functional approach to quantum field theory that allows sampling of quantum fluctuations directly in Minkowski space time by means of a Hamiltonian dynamics in an intrinsic time $\tau$ which samples a…
I investigate useful shape quantities for the classical and quantum mechanics of the relational quadrilateral in 2-d. This is relational in the sense that only relative times, relative ratios of separations and relative angles are…
The purpose of this paper is two-fold. First, to make clear (and de-mystify) the basic concepts of classical thermodynamics, and thus to enable the integration of thermodynamics within systems modeling and control. Second, to demonstrate…
In this PhD thesis, we develop a new approach to classical gravity starting from Mach's principles and the idea that the local shape of spatial configurations is fundamental. This new theory, "shape dynamics", is equivalent to general…
We present a new metaphysical framework for physics that is conceptually clear, ontologically parsimonious, and empirically adequate. This framework relies on the notion of self-subsisting structure, that is, a set of fundamental physical…
Shape dynamics is an alternative background-independent approach to classical dynamics that implements Leibnizian philosophy and Mach's Principles. It is a formulation of the dynamics of the universe in terms of the intrinsic and relational…
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of…
We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an…
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of…
We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational…