On Cartan form and equivalence of variational problems

1997 ◽  
Author(s):  
Chi-wing Wong
Keyword(s):  
Variational Problems ◽  
Cartan Form

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.

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.

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.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
GÜNTHER HÖRMANN ◽  
SANJA KONJIK ◽  
LJUBICA OPARNICA
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Beam Model ◽  
Order Partial Differential Equation ◽  
Viscoelastic Foundation ◽  
Regularization Techniques ◽  
Initial Boundary Value

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

scholarly journals Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka B. Malinowska ◽  
Delfim F. M. Torres
Keyword(s):  
Fractional Integral ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Fractional Integrals ◽  
Natural Boundary ◽  
Free Boundary Value Problems ◽  
Fractional Variational Problems ◽  
Generalized Fractional Integral ◽  
Fractional Calculus Of Variations ◽  
Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

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