Euler–Poincaré reduction on frame bundles

Let μ:FX→Xμ:FX→X be a principal bundle of frames with the structure group Gln(R)Gln(R) and let λ   be a Gln(R)Gln(R)-invariant Lagrangian on J1FXJ1FX. We give an explicit expressions of reduced equations for the associated sections of the corresponding bundle of linear connections which replace the Euler–Lagrange equations for the variational problem defined by λ.