Lagrange multipliers and gravitational theory

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Abstract
The Lagrange multiplier version of the Palatini variational principle is extended to nonlinear Lagrangians, where it is shown in the case of the quadratic Lagrangians, as expected, that this version of the Palatini approach is equivalent to the Hilbert variational method. The (nonvanishing) Lagrange multipliers for the quadratic Lagrangians are then explicitly obtained in covariant form. It is then pointed out how the Lagrange multiplier approach in the language of the (3+1) ‐formalism developed by Arnowitt, Deser, and Misner permits the recasting of the equations of motion for quadratic and general higher‐order Lagrangians into the ADM canonical formalism. In general without the Lagrange multiplier approach, the higher order ADM problem could not be solved. This is done explicitly for the simplest quadratic Lagrangian (g1/2R2) as an example.

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