RECORD
Some geometric aspects of variational problems in fibred manifolds
Mathematical Physics · Submitted 2 Oct 2001
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This work contains an exposition of foundations of the variational calculus in fibered manifolds.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or differential forms (Lagrangian forms) on the first jet prolongation of a given fibered manifold are studied. Critical points (critical cross sections) of the functionals are examined and the Euler equations for them are derived in a completely invariant manner. The first variation formula is derived by means of the so-called Lepagian forms. All variations appearing in the theory are generated by vector fields. Jet prolongations of projectable vector fields are defined. The Euler form, associated with a given Lagrange function (of Lagrangian form) is introduced by means of the Euler equations of the calculus of variations. Necessary and sufficient conditions for the vanishing of the Euler form are stated in terms of differential forms and their exterior differential. The corresponding conditions for a Lagrange function leading to identically vanishing Euler equations are given. Some special Lepagian forms are studied. Classes of symmetries of a variational problem are defined. Invariant, generalized invariant, and symmetry transformations are characterized in terms of the Lie derivatives. The variational problem with prescribed symmetry transformations is formulated, and necessary and sufficient conditions for its solutions are studied. The geometrical aspects of the so-called generally covariant variational theories are studied. Definitions and theorems are well adapted to the situation in physical field theories.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This work contains an exposition of foundations of the variational calculus in fibered manifolds. | contribution 0.93 | 0.74 | 0.07 | asserted 1.00 |
| s1 | The emphasis is laid on the geometric aspects of the theory. | contribution 0.74 | 0.56 | 0.05 | asserted 0.99 |
| s2 | Especially functionals defined by real functions (Lagrange functions) or differential forms (Lagrangian forms) on the first jet prolongation of a given fibered manifold are studied. | contribution 0.93 | 0.59 | 0.04 | asserted 1.00 |
| s3 | Critical points (critical cross sections) of the functionals are examined and the Euler equations for them are derived in a completely invariant manner. | contribution 0.79 | 0.92 | 0.15 | asserted 1.00 |
| s4 | The first variation formula is derived by means of the so-called Lepagian forms. | method 0.87 | 0.90 | 0.05 | asserted 1.00 |
| s5 | All variations appearing in the theory are generated by vector fields. | method 0.75 | 0.39 | 0.03 | asserted 1.00 |
| s6 | Jet prolongations of projectable vector fields are defined. | contribution 0.67 | 0.78 | 0.02 | asserted 0.99 |
| s7 | The Euler form, associated with a given Lagrange function (of Lagrangian form) is introduced by means of the Euler equations of the calculus of variations. | contribution 0.85 | 0.85 | 0.07 | asserted 1.00 |
| s8 | Necessary and sufficient conditions for the vanishing of the Euler form are stated in terms of differential forms and their exterior differential. | result 0.86 | 0.84 | 0.10 | asserted 1.00 |
| s9 | The corresponding conditions for a Lagrange function leading to identically vanishing Euler equations are given. | result 0.81 | 0.83 | 0.06 | asserted 0.99 |
| s10 | Some special Lepagian forms are studied. | contribution 1.00 | 0.84 | 0.03 | asserted 0.98 |
| s11 | Classes of symmetries of a variational problem are defined. | contribution 0.99 | 0.82 | 0.04 | asserted 0.98 |
| s12 | Invariant, generalized invariant, and symmetry transformations are characterized in terms of the Lie derivatives. | contribution 0.87 | 0.72 | 0.05 | asserted 1.00 |
| s13 | The variational problem with prescribed symmetry transformations is formulated, and necessary and sufficient conditions for its solutions are studied. | contribution 0.98 | 0.83 | 0.07 | asserted 0.98 |
| s14 | The geometrical aspects of the so-called generally covariant variational theories are studied. | contribution 1.00 | 0.79 | 0.06 | asserted 0.97 |
| s15 | Definitions and theorems are well adapted to the situation in physical field theories. | interpretation 0.93 | 0.55 | 0.07 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or differential forms (Lagrangian forms) on the first jet prolongation of a given fibered manifold are studied. Critical points (critical cross sections) of the functionals are examined and the Euler equations for them are derived in a completely invariant manner. The first variation formula is derived by means of the so-called Lepagian forms. All variations appearing in the theory are generated by vector fields. Jet prolongations of projectable vector fields are defined. The Euler form, associated with a given Lagrange function (of Lagrangian form) is introduced by means of the Euler equations of the calculus of variations. Necessary and sufficient conditions for the vanishing of the Euler form are stated in terms of differential forms and their exterior differential. The corresponding conditions for a Lagrange function leading to identically vanishing Euler equations are given. Some special Lepagian forms are studied. Classes of symmetries of a variational problem are defined. Invariant, generalized invariant, and symmetry transformations are characterized in terms of the Lie derivatives. The variational problem with prescribed symmetry transformations is formulated, and necessary and sufficient conditions for its solutions are studied. The geometrical aspects of the so-called generally covariant variational theories are studied. Definitions and theorems are well adapted to the situation in physical field theories.
Human review
References
How the literature uses it
12 arXiv papers cite this in 28 sentences
Papers citing it · 16 sentences without a confident label
Cites it as background 9 papers
-
Among the well-known examples of Lepage forms we mention namely: the Cartan form of classical mechanics and its generalization in higher-order mechanics (Krupka [ref] ); in first-order field theory, the Poincaré–Cartan form (García [ref] ), the Carathéodory form (Carathéodory [ref] ), and the fundamental Lepage form (also known as the Krupka–Betounes form) (Krupka [ref] , Betounes [ref] ); in second-order field theory, the generalized Poincaré–Cartan form (Krupka [ref] ), the generalized Carathéodory form (Crampin and Saunders [ref] , Urban and Volná [ref] ), the fundamental Lepage form for second-order, homogeneous Lagrangians (Saunders and Crampin [ref] ).
The fundamental Lepage form in two independent variables: a generalization using order-reducibility · Introduction · 0.99
-
From the point of view of mathematics, the main idea is to join variational approach, represented by the calculus of variations on fibred manifolds (see [ref] or [ref] ) with nonholonomic geometric mechanics in jet bundles due to [ref] , [ref] (for alternative approaches see e.g. [ref] and others).
The relativistic mechanics in a nonholonomic setting: A unified approach to particles with non-zero mass and massless particles · Mathematical background · 0.94
-
Next, in Section [ref] we recall the concept of Lepage form ( [ref] and introduce the Lepage representation of the variational sequence.
Variational Sequences, Representation Sequences and Applications in Physics · Introduction · 0.80
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.