  We propose a novel algorithmic method for constructing invariant variational
schemes of systems of ordinary differential equations that are the
Euler-Lagrange equations of a variational principle. The method is based on the
invariantization of standard, non-invariant discrete Lagrangian functionals
using equivariant moving frames. The invariant variational schemes are given by
the Euler-Lagrange equations of the corresponding invariantized discrete
Lagrangian functionals. We showcase this general method by constructing
invariant variational schemes of ordinary differential equations that preserve
variational and divergence symmetries of the associated continuous Lagrangians.
Noether's theorem automatically implies that the resulting schemes are exactly
conservative. Numerical simulations are carried out and show that these
invariant variational schemes outperform standard numerical discretizations.
