Examples: Natural Lagrange Structures

  • Lepage Research Institute, University of Presov
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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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