scholarly journals Symplectic structures related with higher order variational problems

2015 ◽  
Vol 12 (09) ◽  
pp. 1550084 ◽  
Author(s):  
Jerzy Kijowski ◽  
Giovanni Moreno
Keyword(s):  
Canonical Formalism ◽  
Variational Problems ◽  
Higher Order ◽  
Symplectic Reduction ◽  
Field Theories ◽  
Order System ◽  
Symplectic Structures ◽  
First Order ◽  
Cartan Form ◽  
Finite Dimensional

In this paper, we derive the symplectic framework for field theories defined by higher order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher order system of partial differential equations to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré–Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher order version of the Poincaré–Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

scholarly journals THE POINCARÉ–CARTAN FORM IN SUPERFIELD THEORY

2006 ◽  
Vol 03 (04) ◽  
pp. 775-822 ◽  
Author(s):  
JUAN MONTERDE ◽  
JAIME MUÑOZ MASQUÉ ◽  
JOSÉ A. VALLEJO
Keyword(s):  
Variational Problem ◽  
Variational Problems ◽  
Higher Order ◽  
Noether Theorem ◽  
First Order ◽  
Cartan Form

An intrinsic description of the Hamilton–Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincaré–Cartan form. Noether theorem and examples from superfield theory and supermechanics are also discussed.

First-order System Least Squares on Curved Boundaries: Higher-order Raviart--Thomas Elements

2014 ◽  
Vol 52 (6) ◽  
pp. 3165-3180 ◽  
Author(s):  
Fleurianne Bertrand ◽  
Steffen Münzenmaier ◽  
Gerhard Starke
Keyword(s):  
Least Squares ◽  
Higher Order ◽  
Order System ◽  
Curved Boundaries ◽  
First Order

On the Decomposition of Spinor Fields which satisfy a Nonlinear Higher Order Equation

1983 ◽  
Vol 38 (12) ◽  
pp. 1293-1295
Author(s):  
D. Großer
Keyword(s):  
Field Theory ◽  
Higher Order ◽  
Order Equation ◽  
Field Theories ◽  
First Order ◽  
Spinor Fields ◽  
Fermion Fields ◽  
Higher Derivatives ◽  
Derivatives Of ◽  
Higher Order Equation

Abstract A field theory which is based entirely on fermion fields is non-renormalizable if the kinetic energy contains only derivatives of first order and therefore higher derivatives have to be included. Such field theories may be useful for describing preons and their interaction. In this note we show that a spinor field which satisfies a higher order field equation with an arbitrary nonlinear selfinteraction can be written as a sum of fields which satisfy first order equations.

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 On the relativistic invariance of quantized field theories

1948 ◽  
Vol 195 (1042) ◽  
pp. 365-376 ◽  
Keyword(s):  
Present Author ◽  
Higher Order ◽  
Field Theories ◽  
Relativistic Invariance ◽  
Field Equations ◽  
First Order ◽  
Generalized Poisson ◽  
General Relativistic ◽  
Quantized Field ◽  
Derivatives Of

Heisenberg & Pauli (1929) have shown how to quantize field theories derived from Lagrangians containing first-order derivatives of the field quantities. They showed their quantization to be Lorentz invariant. Fuchs (1939) subsequently showed that the quantized theory was in fact invariant under general transformations of co-ordinates. The present author in another paper has shown how the theory of Heisenberg & Pauli can be extended to field equations derived from higher order Lagrangians, i. e. Lagrangians containing higher deri­vatives than the first of the field quantities. In the present paper the general relativistic invariance of the higher order quantized theories is established, making use of the generalized Poisson brackets introduced by Weiss.

scholarly journals PERTURBATIVE HAMILTONIAN CONSTRAINTS FOR HIGHER-ORDER THEORIES

2011 ◽  
Vol 26 (26) ◽  
pp. 4661-4686
Author(s):  
S. A. MARTÍNEZ ◽  
R. MONTEMAYOR ◽  
L. F. URRUTIA
Keyword(s):  
Time Derivative ◽  
Canonical Formalism ◽  
Order Theory ◽  
Higher Order ◽  
Low Energy ◽  
Quantum Level ◽  
First Order ◽  
Present Formalism ◽  
Higher Order Theory ◽  
Higher Order Lagrangian

We present an alternative method for constructing a consistent perturbative low energy canonical formalism for higher-order time-derivative theories, which consists in applying the standard Dirac method to the first-order version of the higher-order Lagrangian, augmented by additional perturbative Hamiltonian constraints. The method is purely algebraic, provides the dynamical formulation directly in phase space and can be used in singular theories without the need of initially fixing the gauge. We apply it to two paradigmatic examples: the Pais–Uhlenbeck oscillator and the Bernard–Duncan scalar field with self-interaction. We also compare the results, both at the classical and quantum level, with the ones corresponding to a direct perturbative construction applied to the exact higher-order theory, after incorporating the projection to the space of physical modes. This comparison highlights the soundness of the present formalism.

AFFINE DUALITY AND LAGRANGIAN AND HAMILTONIAN SYSTEMS

2011 ◽  
Vol 08 (03) ◽  
pp. 669-697 ◽  
Author(s):  
OLGA KRUPKOVÁ ◽  
DAVID J. SAUNDERS
Keyword(s):  
Hamiltonian Systems ◽  
Arbitrary Order ◽  
Variational Problems ◽  
Higher Order ◽  
Jet Bundle ◽  
Jet Bundles ◽  
First Order ◽  
Variational Form ◽  
Independent Variable

We use affine duals of jet bundles to describe how Legendre maps may be used to provide Hamiltonian representations of variational problems in a single independent variable. Such a problem may be given as a Lagrangian (of first-order or of higher-order), or alternatively as a locally variational form on a jet bundle of arbitrary order with no preferred Lagrangian.

THE CARTAN FORM AND ITS GENERALIZATIONS IN THE CALCULUS OF VARIATIONS

2010 ◽  
Vol 07 (04) ◽  
pp. 631-654 ◽  
Author(s):  
DEMETER KRUPKA ◽  
OLGA KRUPKOVÁ ◽  
DAVID SAUNDERS
Keyword(s):  
Field Theory ◽  
Calculus Of Variations ◽  
Differential Forms ◽  
Classical Mechanics ◽  
Variational Problems ◽  
Higher Order ◽  
Jet Bundles ◽  
Basic Properties ◽  
Cartan Form ◽  
Tangent Bundles

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Challenging the Multidimensional Conception of Perceived Person-Environment Fit

2020 ◽  
pp. 1-9
Author(s):  
Julian M. Etzel ◽  
Gabriel Nagy
Keyword(s):  
Higher Education ◽  
University Students ◽  
Educational Outcomes ◽  
Factor Model ◽  
Higher Order ◽  
Substantial Proportion ◽  
Unique Variance ◽  
First Order ◽  
Person Environment Fit ◽  
Order Factor

Abstract. In the current study, we examined the viability of a multidimensional conception of perceived person-environment (P-E) fit in higher education. We introduce an optimized 12-item measure that distinguishes between four content dimensions of perceived P-E fit: interest-contents (I-C) fit, needs-supplies (N-S) fit, demands-abilities (D-A) fit, and values-culture (V-C) fit. The central aim of our study was to examine whether the relationships between different P-E fit dimensions and educational outcomes can be accounted for by a higher-order factor that captures the shared features of the four fit dimensions. Relying on a large sample of university students in Germany, we found that students distinguish between the proposed fit dimensions. The respective first-order factors shared a substantial proportion of variance and conformed to a higher-order factor model. Using a newly developed factor extension procedure, we found that the relationships between the first-order factors and most outcomes were not fully accounted for by the higher-order factor. Rather, with the exception of V-C fit, all specific P-E fit factors that represent the first-order factors’ unique variance showed reliable and theoretically plausible relationships with different outcomes. These findings support the viability of a multidimensional conceptualization of P-E fit and the validity of our adapted instrument.

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