On the commutator of $C^{\infty }$-symmetries and the reduction of Euler–Lagrange equations

A novel procedure to reduce by four the order of Euler–Lagrange equations associated to nth order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two $C^{\infty }$-symmetries is established. The method is based on a pair of variational $C^{\infty }$-symmetries whose commutators satisfy a certain solvability condition. It allows one to recover a $(2n-2)$-parameter family of solutions for the original 2nth order Euler–Lagrange equation by solving two successive first order ordinary differential equations from the solution of the reduced Euler–Lagrange equation. The procedure is illustrated by two different examples. (paper)