Partial Noether operators and first integrals via partial Lagrangians

The notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations that need not be derivable from variational principles. We obtain a Noether-like theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. Applications given include those that admit a standard Lagrangian such as the harmonic oscillator, modified Emden and Ermakov-Pinney equations and systems of two second-order equations that do not have standard Lagrangians.