scholarly journals A geometrical analysis of the field equations in field theory

2002 ◽  
Vol 29 (12) ◽  
pp. 687-699 ◽  
Author(s):  
A. Echeverría-Enríquez ◽  
M. C. Muñoz-Lecanda ◽  
N. Román-Roy
Keyword(s):  
Field Theory ◽  
Classical Field ◽  
Field Theories ◽  
Geometrical Analysis ◽  
Field Equations ◽  
First Order ◽  
Nonuniqueness Of Solutions ◽  
Lagrangian And Hamiltonian Formalisms ◽  
Classical Field Theories ◽  
Geometric Formulation

We give a geometric formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and nonuniqueness of solutions of these equations, as well as their integrability.

scholarly journals PRE-MULTISYMPLECTIC CONSTRAINT ALGORITHM FOR FIELD THEORIES

2005 ◽  
Vol 02 (05) ◽  
pp. 839-871 ◽  
Author(s):  
MANUEL DE LEÓN ◽  
JESÚS MARÍN-SOLANO ◽  
JUAN CARLOS MARRERO ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY
Keyword(s):  
Fiber Bundle ◽  
General Framework ◽  
Vector Fields ◽  
Classical Field ◽  
Field Theories ◽  
Second Step ◽  
Field Equations ◽  
Jet Bundle ◽  
First Order ◽  
Classical Field Theories

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

scholarly journals SYMMETRIES AND CONSERVATION LAWS IN THE GÜNTHER k-SYMPLECTIC FORMALISM OF FIELD THEORY

2007 ◽  
Vol 19 (10) ◽  
pp. 1117-1147 ◽  
Author(s):  
NARCISO ROMÁN-ROY ◽  
MODESTO SALGADO ◽  
SILVIA VILARIÑO
Keyword(s):  
Field Theory ◽  
Conservation Laws ◽  
Classical Field ◽  
Field Theories ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
First Order ◽  
Gauge Symmetries ◽  
Classical Field Theories

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

scholarly journals An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 85
Author(s):  
Narciso Román-Roy
Keyword(s):  
Dynamical Systems ◽  
Classical Field ◽  
Field Theories ◽  
Geometric Description ◽  
Hamiltonian Approach ◽  
Physical Systems ◽  
First Order ◽  
Generic Framework ◽  
Lagrangian And Hamiltonian Formalisms ◽  
Classical Field Theories

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, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

scholarly journals THE POISSON BRACKET FOR POISSON FORMS IN MULTISYMPLECTIC FIELD THEORY

2003 ◽  
Vol 15 (07) ◽  
pp. 705-743 ◽  
Author(s):  
MICHAEL FORGER ◽  
CORNELIUS PAUFLER ◽  
HARTMANN RÖMER
Keyword(s):  
Poisson Bracket ◽  
Differential Forms ◽  
Lie Superalgebra ◽  
Classical Field ◽  
Field Theories ◽  
General Definition ◽  
First Order ◽  
Classical Field Theories ◽  
Definition Of ◽  
Geometric Formulation

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

scholarly journals Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

scholarly journals Lagrangian constraint analysis of first-order classical field theories with an application to gravity

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
Verónica Errasti Díez ◽  
Markus Maier ◽  
Julio A. Méndez-Zavaleta ◽  
Mojtaba Taslimi Tehrani
Keyword(s):  
Classical Field ◽  
Field Theories ◽  
First Order ◽  
Constraint Analysis ◽  
Classical Field Theories

scholarly journals A new application of k-symplectic Lie systems

2015 ◽  
Vol 12 (07) ◽  
pp. 1550071 ◽  
Author(s):  
Javier de Lucas ◽  
Mariusz Tobolski ◽  
Silvia Vilariño
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Classical Field ◽  
Diffusion Equations ◽  
Field Theories ◽  
Symplectic Structures ◽  
First Order ◽  
Classical Field Theories ◽  
Lie Systems ◽  
Geometric Study

The k-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the k-symplectic structures to investigate a certain type of systems of first-order ordinary differential equations, the k-symplectic Lie systems. In particular, we analyze the properties, e.g., the superposition rules, of a new example of k-symplectic Lie system which occurs in the analysis of diffusion equations.

scholarly journals Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Marija Dimitrijević Ćirić ◽  
Grigorios Giotopoulos ◽  
Voja Radovanović ◽  
Richard J. Szabo
Keyword(s):  
Deformation Quantization ◽  
Classical Field ◽  
Field Theories ◽  
Field Equations ◽  
New Class ◽  
Gauge Symmetries ◽  
Classical Field Theories ◽  
Noncommutative Gravity ◽  
Classical Gravity ◽  
Matter Fields

AbstractWe define a new homotopy algebraic structure, that we call a braided $$L_\infty $$ L ∞ -algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel’d twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

scholarly journals Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds

2012 ◽  
Vol 4 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Cédric M. Campos ◽  
◽  
Elisa Guzmán ◽  
Juan Carlos Marrero ◽  
Keyword(s):  
Lagrangian Submanifolds ◽  
Classical Field ◽  
Field Theories ◽  
First Order ◽  
Classical Field Theories

scholarly journals Variational principles for multisymplectic second-order classical field theories

2015 ◽  
Vol 12 (08) ◽  
pp. 1560019 ◽  
Author(s):  
Pedro Daniel Prieto-Martínez ◽  
Narciso Román-Roy
Keyword(s):  
Variational Principles ◽  
Classical Field ◽  
Second Order ◽  
Field Theories ◽  
Unified Framework ◽  
Field Equations ◽  
Classical Field Theories ◽  
Geometrical Version

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.

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