Routh reduction and Cartan mechanics

In the present work a Cartan mechanics version for Routh reduction is considered, as an intermediate step toward Routh reduction in field theory. Motivation for this generalization comes from an scheme for integrable systems [12], used for understanding the occurrence of Toda field theories in so called Hamiltonian reduction of WZNW field theories [11]. As a way to accomplish with this intermediate aim, this article also contains a formulation of the Lagrangian Adler-Kostant-Symes systems discussed in [12] in terms of Routh reduction.Comment: 46 pages, comments are welcome. Version 2 contains an additional section concerning reduced equations of motion in quasicoordinate

Unified formalism for Palatini gravity

This paper is devoted to the construction of a unified formalism for Palatini and unimodular gravity. The idea is to employ a relationship between unified formalism for a Griffiths variational problem and its classical Lepage-equivalent variational problem. The main geometrical tools involved in these constructions are canonical forms living on the first jet of the frame bundle for the spacetime manifold. These forms play an essential role in providing a global version of the Palatini Lagrangian and expressing the metricity condition in an invariant form. With them, we were able to find the associated equations of motion in invariant terms and, by using previous results from the literature, to prove their involutivity. As a bonus, we showed how this construction can be used to provide a unified formalism for the so-called unimodular gravity by employing a reduction of the structure group of the frame bundle to the special linear group.Fil: Capriotti, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentin

Chern-Simons field theory on the general affine group, 3d3d-gravity and the extension of Cartan connections

The purpose of this article is to study the correspondence between $3d$-gravity and the Chern-Simons field theory from the perspective of geometric mechanics, specifically in the case where the structure group is the general affine group. To accomplish this, the paper discusses a variational problem of the Chern-Simons type on a principal fiber bundle with this group as its structure group. The connection to the usual Chern-Simons theory is established by utilizing a generalization, in the context of Cartan connections, of the notion of extension and reduction of connections.Comment: This paper supersedes previous submission: arXiv:2004.1078

AKS systems and Lepage equivalent problems

The integrable systems known as "AKS systems" admit a natural formulation in terms of a Hamiltonian picture. The Lagrangian side of these systems are far less known; a version in these terms can be found in a work of Feher et al. The purpose of these notes in to provide a novel description of AKS systems in terms of a variational problem different from the usual in mechanics. Additionally, and using techniques borrowed from an article of M. Gotay, it was possible to build the Hamiltonian side of this variational problem, allowing us to establish the equivalence with the usual approach to these integrable systems.Comment: 19 pages, no figure

Routh reduction of palatini gravity in vacuum

An interpretation of Einstein–Hilbert gravity equations as Lagrangian reduction of Palatini gravity is made. The main technique involved in this task consists in representing the equations of motion as a set of differential forms on a suitable bundle. In this setting Einstein–Hilbert gravity can be considered as a kind of Routh reduction of the underlying field theory for Palatini gravity. As a byproduct of this approach, a novel set of conditions for the existence of a vielbein for a given metric is found.Fil: Capriotti, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Dirac constraints in field theory and exterior differential systems

The usual treatment of a (first order) classical field theory such as electromagnetism has a little drawback: It has a primary constraint submanifold that arise from the fact that the dynamics is governed by the antisymmetric part of the jet variables. So it is natural to ask if there exists a formulation of this kind of field theories which avoids this problem, retaining the versatility of the known approach. The following paper deals with a family of variational problems, namely, the so called non standard variational problems, which intends to capture the data necessary to set up such a formulation for field theories; moreover, we will formulate a multisymplectic structure for the family of non standard variational problems, and we will relate this with the (pre)symplectic structure arising on the space of sections of the bundle of fields. In this setting the Dirac theory of constraints will be studied, obtaining among other things a novel characterization of the constraint manifold which arises in this theory, as generators of an exterior differential system associated to the equations of motion and the chosen slicing. Several examples of application of this formalism are discussed: Two of them motivated from the physical point of view, that is, electromagnetism and Poisson sigma models, and two examples of mathematical application. In the case of electromagnetism, it is shown that this formulation avoids the problems arising in the usual approach.Comment: 51 pages, aims; added examples and references, typos added, improved some content, to appear in "Journal of Geometric Mechanics"

Griffiths variational multisymplectic formulation for Lovelock gravity

This work is mainly devoted to constructing a multisymplectic description of Lovelock’s gravity, which is an extension of General Relativity. We establish the Griffiths variational problem for the Lovelock Lagrangian, obtaining the geometric form of the corresponding field equations. We give the unified Lagrangian-Hamiltonian formulation of this model and we study the correspondence between the unified formulations for the Einstein-Hilbert and the Einstein-Palatini models of gravity.Fil: Capriotti, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Gaset, Jordi. Universitat Autònoma de Barcelona; EspañaFil: Román Roy, Narciso. Universidad Politécnica de Catalunya; EspañaFil: Salomone, Leandro Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Centro de Matemática de la Plata; Argentin

Ground State Phase Diagram of Frustrated S=1 XXZ chains : Chiral Ordered Phases

The ground-state phase diagram of frustrated S=1 XXZ spin chains with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied using the infinite-system density-matrix renormalization-group method. We find six different phases, namely, the Haldane, gapped chiral, gapless chiral, double Haldane, N\'{e}el, and double N\'{e}el (uudd) phases. The gapped and gapless chiral phases are characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, \kappa_l = S_l^xS_{l+1}^y-S_l^yS_{l+1}^x, whereas the spin correlation decays either exponentially or algebraically. These chiral ordered phases appear in a broad region in the phase diagram for \Delta < 0.95, where \Delta is an exchange-anisotropy parameter. The critical properties of phase transitions are also studied.Comment: 13 pages, 9 figures, to appear in J. Phys. Soc. Jp