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Lie Groups, Differential Equations, and Geometry pp 321–361Cite as

Lepage Manifolds

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Abstract

Lepage manifolds represent a geometric structure arising from differential equations, and related with them as close as possible. Geometric properties of Lepage manifolds reflect geometrical, topological and dynamical properties of differential equations, and vice versa. After symplectic geometry, Lepage manifolds represent a next step in understanding interrelations between differential equations, geometry and physics. The aim of this chapter is to introduce the new concept, and to stimulate deeper studies in this direction as well as applications in differential geometry, differential equations, dynamical systems, exterior differential systems, calculus of variations, and mathematical physics.

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Notes

  1. 1.

    The concept introduced by Ehresmann in the 1950s [22, 23]; a standard book on this subject is [86].

  2. 2.

    For basics of sheaf theory needed below we refer to standard books, e.g. [6] or [93].

  3. 3.

    As explained later, for the calculus of variations the most important examples of source forms of degree n + 1 are Euler–Lagrange forms, and of source forms of degree n + 2 the Helmholtz forms.

  4. 4.

    For the sake of simplicity of notations, we shall often omit the pull-back and identify a form with its lift or projection.

  5. 5.

    Helmholtz conditions (k = 1) in the form (3) were first presented in [57]. The coordinate form of the Helmholtz conditions has been first obtained by Helmholtz [37] for second order ODEs, and by Anderson and Duchamp [3] and Krupka [47, 48] for higher order PDEs.

    For k = 2 (and mechanics) the explicit coordinate form of (3) can be found in [64] and [74].

  6. 6.

    A differential form ω of degree q > 2 on a manifold M is called multisymplectic if it is closed and nondegenerate (meaning that the Cauchy distribution of ω is zero, i.e. the condition i ζ ω = 0 implies ζ = 0) [8].

  7. 7.

    In general we cannot take simply dθ λ since θ λ need not be global.

  8. 8.

    As we shall see, in this case the Hamilton equations become of De Donder type.

  9. 9.

    This choice of α in the Lepage class {α} is the most simple. As we know, more generally we can add to α (any) at least 2-contact form, and/or the derivative of a 1-contact form, and the dynamical form ɛ, hence the Euler–Lagrange equations, remain the same. However, the Hamilton equations then may no longer be of De Donder type.

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Acknowledgements

Research supported by grant No. 14-02476S Variations, Geometry and Physics of the Czech Science Foundation, and by the IRSES project LIE-DIFF-GEOM (EU FP7, nr 317721).

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  1. Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Olga Rossi

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    Giovanni Falcone

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Rossi, O. (2017). Lepage Manifolds. In: Falcone, G. (eds) Lie Groups, Differential Equations, and Geometry. UNIPA Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-319-62181-4_13

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