Variational Sequences, Representation Sequences and Applications in Physics

Palese, Marcella; Rossi, Olga; Winterroth, Ekkehart Hans Konrad; Musilová, Jana

doi:10.3842/sigma.2016.045

Cited by 12 publications

(41 citation statements)

References 68 publications

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“…The property formulated in Theorem 1 is the most practical one of the three equivalent conditions for a general inverse variational problem proved in [11]. Note that if a globally defined dynamical form E in mechanics is (at least locally) variational then its closed Lepage equivalent is unique and global.…”

Section: The Inverse Problem Of the Calculus Of Variationsmentioning

confidence: 95%

“…A general definition of Lepage form and its properties are introduced and discussed in [11] in terms of the finite order variational sequence with the use of the interior Euler operator (see also [1] and [5]). In this section we recall Lepage forms only to the extent needed in this paper.…”

Section: Lepage Forms and Equivalentsmentioning

confidence: 99%

“…Recall that the Euler-Lagrange form of a r-th order Lagrangian λ is identically zero iff λ 0 = h dη where η is an arbitrary (n − 1)-form on J r−1 Y (see [7] or [11]). Such Lagrangians are called trivial or null Lagrangians.…”

Section: Basic Structuresmentioning

confidence: 99%

“…A universal and elegant approach to this problem is the use of properties of the finite order variational sequence see e.g. [6], [5], in details see [11]. Let us summarize basic definitions and results.…”

Section: The Inverse Problem Of the Calculus Of Variationsmentioning

confidence: 99%

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The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Musilová

Hronek

2016

Communications in Mathematics

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Abstract. As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the "true trajectories" of the physical systems represent stationary points of the corresponding functionals.It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are EulerLagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

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Section: The Inverse Problem Of the Calculus Of Variationsmentioning

confidence: 95%

Section: Lepage Forms and Equivalentsmentioning

confidence: 99%

Section: Basic Structuresmentioning

confidence: 99%

Section: The Inverse Problem Of the Calculus Of Variationsmentioning

confidence: 99%

See 2 more Smart Citations

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Musilová

Hronek

2016

Communications in Mathematics

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“…The fibered splitting induced by the contact structure on finite order jets yields a differential forms sheaf splitting in contact components of different degree, so that a sort of 'horizontalization' h can be suitable defined as the projection on the summand of lesser contact degree; see e.g. [19] and the review in [21]. Now, by an abuse of notation, let us denote by ker h + d ker h the induced sheaf generated by the presheaf ker h + d ker h in the standard way (d is an epimorphism of presheaves, but not of sheaves).…”

Section: Variational Problems On Gauge-natural Prolongations Modulo Cmentioning

confidence: 99%

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

Palese¹,

Winterroth²

2019

Int. J. Geom. Methods Mod. Phys.

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We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang-Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example we consider the 'gluon Lagrangian', i.e. a Yang-Mills Lagrangian on the (1, 1)-order gauge-natural bundle of SU (3)-principal connections, and canonically define a 'gluon' classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

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Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory

Palese

Winterroth

2017

Journal of Mathematical Physics

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We relate the existence of Noether global conserved currents associated with locally variational field equations to existence of global solutions for a local variational problem generating global equations. Both can be characterized as the vanishing of certain cohomology classes.In the case of a 3-dimensional Chern-Simons gauge theory, the variationally featured cohomological obstruction to the existence of global solutions is sharp and equivalent to the usual obstruction in terms of the Chern characteristic class for the flatness of a principal connection.We suggest a parallelism between the geometric interpretation of characteristic classes as obstruction to the existence of flat principal connections and the interpretation of certain de Rham cohomology classes to be the obstruction to the existence of global extremals for a local variational principle.

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Variational Sequences, Representation Sequences and Applications in Physics

Abstract: Abstract. This paper is a review containing new original results on the finite order variational sequence and its different representations with emphasis on applications in the theory of variational symmetries and conservation laws in physics.

Cited by 12 publications

References 68 publications

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory

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