Partial Noether operators and first integrals <i>via</i> partial Lagrangians

Abstract

Abstract 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. Copyright © 2007 John Wiley &amp; Sons, Ltd.