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 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 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 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 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 Geometry of Lagrangian First-order Classical Field Theories

1996 ◽  
Vol 44 (3) ◽  
pp. 235-280 ◽  
Author(s):  
Arturo Echeverría-Enríquez ◽  
Miguel C. Muñoz-Lecanda ◽  
Narciso Román-Roy
Keyword(s):  
Classical Field ◽  
Field Theories ◽  
First Order ◽  
Classical Field Theories

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.

Quasi-Lie schemes for PDEs

2019 ◽  
Vol 16 (07) ◽  
pp. 1950096 ◽  
Author(s):  
J. F. Cariñena ◽  
J. Grabowski ◽  
J. de Lucas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformations ◽  
Partial Differential ◽  
First Order ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
Lie Systems ◽  
Abel Differential Equations

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Kolmogorov and classic diffusion equations

1988 ◽  
Vol 53 (6) ◽  
pp. 1181-1197
Author(s):  
Vladimír Kudrna
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Parabolic Type ◽  
Diffusion Equations ◽  
Stochastic Motion ◽  
Energy Component ◽  
Classic Form

The paper presents alternative forms of partial differential equations of the parabolic type used in chemical engineering for description of heat and mass transfer. It points at the substantial difference between the classic form of the equations, following from the differential balances of mass and enthalpy, and the form following from the concept of stochastic motion of particles of mass or energy component. Examples are presented of the processes that may be described by the latter method. The paper also reviews the cases when the two approaches become identical.

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