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Examples: Natural Lagrange Structures
Abstract
Examples presented in this chapter include typical variational functionals that appear as variational principles in the theory of geometric and physical fields. We begin by the discussion of the well-known Hilbert variational functional for the metric fields, first considered in Hilbert in 1915, whose Euler–Lagrange equations are the Einstein vacuum equations. We give a manifold interpretation of this functional and show that its second-order Lagrangian, the formal scalar curvature, possesses a global first-order Lepage equivalent. The Lagrangian used by Hilbert is an example of a differential invariant of a metric field (and its first and second derivatives). Further examples with similar properties, belonging to the class of natural Lagrange structures, are also considered.
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Higgs fields on gauge-natural prolongations of principal bundles are defined
by invariant variational problems and related canonical conservation laws along
the kernel of a gauge-natural Jacobi morphism.
A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler-Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.
The global variational functional, defined by the Hilbert-Yang-Mills Lagrangian over a smooth manifold, is investigated within the framework of prolongation theory of principal fiber bundles, and global variational theory on fibered manifolds. The principal Lepage equivalent of this Lagrangian is constructed, and the corresponding infinitesimal first variation formula is obtained. It is shown, in particular, that the Noether currents, associated with isomorphisms of the underlying geometric structures, split naturally into several terms, one of which is the exterior derivative of the Komar-Yang-Mills superpotential. Consequences of invariance of the Hilbert-Yang-Mills Lagrangian under isomorphisms of underlying geometric structures such as Noether’s conservation laws for global currents are then established. As an example, a general formula for the Komar-Yang-Mills superpotential of the Reissner-Nordström solution of the Einstein equations is found.
We study geometric aspects concerned with symmetries and conserved quantities in gauge-natural invariant variational problems and investigate implications of the existence of a reductive split structure associated with canonical Lagrangian conserved quantities on gauge-natural bundles. In particular, we characterize the existence of covariant conserved quantities in terms of principal Cartan connections on gauge-natural prolongations.
The Hilbert variational principle
- D Kl
- M Krupka
- Lenc