Lepage manifolds. (English) Zbl 1393.58004
Falcone, Giovanni (ed.), Lie groups, differential equations, and geometry. Advances and surveys. Cham: Springer; Palermo: Università degli Studi di Palermo (ISBN 978-3-319-62180-7/hbk; 978-3-319-62181-4/ebook). UNIPA Springer Series, 321-361 (2017).
Let \(\pi:Y\to X\) be a fibred manifold (surjective submersion) with fibre dimension \(m\) and base manifold dimension \(n\). All manifolds are assumed to be smooth. Let \(\pi_r:J^r Y\to X\) be the \(r\)-jet prolongation of this fibred manifold. Lepage manifold is a fibred manifold endowed with a certain form – Lepage \((n+1)\)-form. Every \(q\)-form on \(Y\), \(q\geq n\), is a Lepage form, as well as every closed \(q\)-form on \(J^rY\), \(q\geq n\), and many others.
The author motivates to introduce the concept of a Lepage manifold which is based on a proper generalization of Lepage \((n+1)\)-forms to non-variational equations. This generalization appears within the theory of variational sequences. The author aims to show how one can put together in a unified framework, linked together and extended many particular results in the calculus of variations and in the geometric theory of differential equations by using Lepage manifolds. To achieve this goal the author introduces Lepage forms (of any degree) in the context of the variational sequence and reminds the solution of the Inverse Problem of the Calculus of Variations. One section is devoted to Lepage manifolds to show the relationship with the variational equations and with non-variational equations. Regular Lepage manifolds and methods to transfer non-variational Lepage manifolds to the variational ones are considered. Furthermore the author presents some results concerning Lepage manifolds of order zero which are a natural framework for general Hamiltonian systems. Then she presents original results on Legendre transformations and Poisson structure for the generalized (and not necessarily variational) Hamiltonian systems. A new concept of a directional Poisson bracket in covariant Hamiltonian field theory is given. The last two sections are devoted to some applications of Lepage manifolds in classical field theories and mechanics. More precisely, the author presents the Lepage manifolds setting applicable to electromagnetism (Maxwell equations), gravity (Einstein equations), gauge theories, and others. Moreover she considers a fibred manifold over a base of dimension 1. In this case equations for sections of \(\pi\) are ordinary differential equations, and Lepage \((n+1)\)-forms are 2-forms. Finally, the author shows that (and how) Lepage manifolds naturally arise in Riemann and Finsler geometry.
For the entire collection see [Zbl 1381.53010].
The author motivates to introduce the concept of a Lepage manifold which is based on a proper generalization of Lepage \((n+1)\)-forms to non-variational equations. This generalization appears within the theory of variational sequences. The author aims to show how one can put together in a unified framework, linked together and extended many particular results in the calculus of variations and in the geometric theory of differential equations by using Lepage manifolds. To achieve this goal the author introduces Lepage forms (of any degree) in the context of the variational sequence and reminds the solution of the Inverse Problem of the Calculus of Variations. One section is devoted to Lepage manifolds to show the relationship with the variational equations and with non-variational equations. Regular Lepage manifolds and methods to transfer non-variational Lepage manifolds to the variational ones are considered. Furthermore the author presents some results concerning Lepage manifolds of order zero which are a natural framework for general Hamiltonian systems. Then she presents original results on Legendre transformations and Poisson structure for the generalized (and not necessarily variational) Hamiltonian systems. A new concept of a directional Poisson bracket in covariant Hamiltonian field theory is given. The last two sections are devoted to some applications of Lepage manifolds in classical field theories and mechanics. More precisely, the author presents the Lepage manifolds setting applicable to electromagnetism (Maxwell equations), gravity (Einstein equations), gauge theories, and others. Moreover she considers a fibred manifold over a base of dimension 1. In this case equations for sections of \(\pi\) are ordinary differential equations, and Lepage \((n+1)\)-forms are 2-forms. Finally, the author shows that (and how) Lepage manifolds naturally arise in Riemann and Finsler geometry.
For the entire collection see [Zbl 1381.53010].
Reviewer: Neda Bokan (Beograd)
MSC:
| 58A20 | Jets in global analysis |
| 55R10 | Fiber bundles in algebraic topology |
| 55N30 | Sheaf cohomology in algebraic topology |
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 58E30 | Variational principles in infinite-dimensional spaces |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
| 35R01 | PDEs on manifolds |
| 49Q99 | Manifolds and measure-geometric topics |
Keywords:
jet bundles; variational sequence; variational equations; Hamiltonian system; Legendre transformation; Poisson structure; directional Poisson bracket; Lepage manifoldReferences:
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