general relativityand gravitation the Palatini variation is nowadays thought of as variation of a Lagrangianwith respect to the connection.For general relativity Palatini variation gives that the connection is theChristoffel connection. The reason that Palatini variation is consider important is that it means that the use of the Christoffel connection in general relativity does not have to be added as a separate assumption, the information is already in the Lagrangian. For theories of gravitation which have more complexlagrangians than the Einstein-Hilbert Lagrangian of general relativity Palatini variation sometimes gives more complex connections and sometimes tensorial equations. The history of the subject is not straightfoward, see the references.
A.Palatini (1919) Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend.Circ.Mat.Palermo43,203 [English translation by R.Hojman and C.Mukku in P.G.Bergmann and V.De Sabbata (eds.)Cosmology and Gravitation, Plenum Press, New York (1980)]
M.Ferraris, M.Francaviglia, C.Reina, Variational Formulation of General Relativity from 1915 to 1925 'Palatini's Method' Discovered by Einstein in 1925,Gen.Rel.Grav.14(1982)243-254.
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