Geometrical structures of higher-order dynamical systems and field theories
Pedro D. Prieto-Martínez
备注：
Ph.D. Thesis at Universitat Politècnica de Catalunya - BarcelonaTech. 256 pages. Acknowledgments have been removed with respect to the original version. Qualification: Excellent Cum Laude
In this Thesis we develop the geometric formulations for higher-order autonomous and non-autonomous dynamical systems, and second-order field theories. In all cases, the physical information of the system is given in terms of a Lagrangian function/density, or a Hamiltonian that admits Lagrangian counterpart. These geometric frameworks are used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory.
In the present article, Chern-Simons gauge theory and its relationship with gravity are revisited from a geometrical viewpoint. In this setting, our goals are twofold: In one hand, to show how to represent the family of variational problems for Chern-Simons theory in the set of $K$-structures on the space-time $M$, by means of a unique variational problem of Griffiths type. On the other hand, to make an interpretation of the correspondence of these theories in terms of the correspondence between the frame bundle and its affine counterpart.
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to quantum mechanics. We will concentrate our attention into quantum theories and we will show how to use in a systematic way the transition from algebraic to geometrical structures to explore their geometry, mainly its Jordan-Lie structure.
We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular, applications to the geometrical description of optimal control theory for partial differential equations.
The $β$ function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the $β$-functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.
In this paper, we analyze the stability of the real-valued Maxwell-Bloch equations with a control that depends on state variables quadratically. We also investigate the topological properties of the energy-Casimir map, as well as the existence of periodic orbits and explicitly construct the heteroclinic orbits.
A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a submanifold of the manifold of solutions of the equations describing a field theory. Then, the covariance of the geometrical objects involved, will allow to define equations of motion on a reduced space. The computation of the canonical geometrical structure is performed neatly by using the geometrical framework provided by the multisymplectic description of co…
General disagreement between the Geometrical Description of Dynamical In-stability -using non affine parameterizations- and traditional Tangent Dynamics
We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method we obtain the first-integral of SORE and the results are applied to the study of its Lagrangian and Hamiltonian description. We unveil the relation between the Darboux polynomials and master symmetries associated to second-order Riccati. Using these results we show the existence of a Lagrangian description for the related system, and the Painlev…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative geometry like maps, partitions, Hurwitz numbers, intersection numbers, Gromov-Witten invariants... The problem is thus to understand what they count, or in other words, given a spectral curve, construct an enumerative geometry problem. This is what we do in a semi-heuristic approach in this article. Starting from a spectral curve, i.e. an integrable system, we us…
Recently, M. de León el al. ([8]) have developed a geometrical description of Hamilton-Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field theories. The Hamilton-Jacobi theory for this kind of fields is shown.
Inspired by problems arising in the geometrical treatment of Yang-Mills theories and Palatini's gravity, the covariant formulation of Hamiltonian dynamical systems as a Hamiltonian field theory of dimension $1+0$ on a manifold with boundary is discussed. After a precise statement of Hamilton's variational principle in this context, the geometrical properties of the space of solutions of the Euler-Lagrange equations of the theory are analyzed. A sufficient condition is obtained that guarantees that the set of solutions of the Euler-Lagrange equations at the boundary of the manifold, fill a Lagr…
This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new…
The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects and relations between these objects. These relations are expressed with systems of (mainly second order) differential equations. The solution of these equations is frequently a formidable task, either because the dynamical equations cannot be integrated by standard methods or because the defined dynamical system is non-integrable. Therefore, it is important t…
We state a unified geometrical version of the variational principles for second-order classical field theories. The standard Lagrangian and Hamiltonian variational principles and the corresponding field equations are recovered from this unified framework.
The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show t…
Many important theories in modern physics can be stated using differential geometry. Symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems, both regular and singular. Some of these extensions are the subject of this thesis. Recently there has been a growing interest in studying dissipative mechanical systems from a geometric perspective using contact geometry. In this thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems…
We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We focus on the probabilities of rare events (large deviations). First, we discuss a PT (parity-time) symmetry that appears in ensembles of trajectories where a current is constrained to have a large (non-typical) value. We analyse the heat flow in such ensembles, and compare it with non-equilibrium steady …
Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.
The aim of this paper is to create a large geometrical background on the dual 1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the electromagnetic and gravitational physical fields. Our geometric-physical construction is achieved starting only from a given quadratic Hamiltonian function of polymomenta H, which naturally produces a canonical nonlinear connection N, a canonical Cartan N-linear connection CΓ(N) and their corresponding local distinguished (d-) torsions and curvatures. In such a context, we construct some geometrical electromagnetic-like and gravitational-like field…
This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton--Jacobi theory. The relation with the "classical" Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, and how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classica…
