Chapter

Examples: Natural Lagrange Structures

Authors:
  • Lepage Research Institute, University of Presov
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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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